Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Mathematical Engineering

Hung Nguyen-Schäfer Jan-Philip Schmidt Tensor Analysis and Elementary Diff erential Geometry for Physicists and Engineers Second Edition Mathematical Engineering

Series editors Claus Hillermeier, Neubiberg, Germany Jo¨rg Schro¨der, Essen, Germany Bernhard Weigand, Stuttgart, Germany More information about this series at http://www.springer.com/series/8445 Hung Nguyen-Scha¨fer • Jan-Philip Schmidt

Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers

Second Edition Hung Nguyen-Scha¨fer Jan-Philip Schmidt EM-motive GmbH Interdisciplinary Center for Scientiﬁc Computing (IWR) A Joint Company of Daimler University of Heidelberg and Bosch Heidelberg, Germany Ludwigsburg, Germany

ISSN 2192-4732 ISSN 2192-4740 (electronic) Mathematical Engineering ISBN 978-3-662-48495-1 ISBN 978-3-662-48497-5 (eBook) DOI 10.1007/978-3-662-48497-5

Library of Congress Control Number: 2016940970

© Springer-Verlag Berlin Heidelberg 2014, 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg In memory of Gregorio Ricci-Curbastro (1853–1925) and Tullio Levi-Civita (1873–1941), who invented Tensor Calculus, for which Elwin Bruno Christoffel (1829–1900) had prepared the ground; Carl Friedrich Gauss (1777–1855) and Bernhard Riemann (1826–1866), who invented Differential Geometry ThiS is a FM Blank Page Preface to the Second Edition

In the second edition, all chapters are carefully revised in which typos are corrected and many additional sections are included. Furthermore, two new chapters that deal with Cartan differential forms and applications of tensors and Dirac notation to quantum mechanics are added in this book. Contrary to tensors, Cartan differential forms based on exterior algebra in Chap. 4 using the wedge product are an approach to multivariable calculus that is independent of coordinates. Therefore, they are very useful methods for differential geometry, topology, and theoretical physics in multidimensional manifolds. In Chap. 6, the quantum entanglement of a composite system that consists of two entangled subsystems has alternatively been interpreted by means of symmetries. Both mathematical approaches of Dirac matrix and wave formulations are used to analyze and calculate the expectation values, probability density operators, and wave functions for nonrelativistic and relativistic particles in a composite system using time-dependent Schrodinger€ equation (TDSE), Klein–Gordon equation, and Dirac equation as well. I would like to thank Mrs. Eva Hestermann-Beyerle and Mrs. Birgit Kollmar- Thoni at Springer Heidelberg for their invaluable suggestions and excellent cooperation to publish this second edition successfully. Finally, my special thanks go to my wife for her understanding, patience, and endless support as I wrote this book in my leisure and vacation time.

Ludwigsburg, Germany Hung Nguyen-Scha¨fer

vii ThiS is a FM Blank Page Preface to the First Edition

This book represents a joint effort by a research engineer and a mathematician. The initial idea for it arose from our many years of experience in the automotive industry, from advanced research development, and of course from our common research interest in applied mathematics, physics, and engineering. The main reason for this cooperation is the fact that mathematicians generally approach problems using mathematical rigor, but which need not always be practically applicable; at the same time, engineers usually deal with problems involving applied mathematics, which must ultimately work in the “real world” of industry. Having recognized that what mathematicians consider rigor can be more like rigor mortis for engineers and physicists, this joint effort proposes a compromise between the mathematical rigors and less rigorous applied mathematics, incorpo- rating different points of view. Our main aim is to bridge the mathematical gap between where physics and engineering mathematics end and where tensor analysis begins, which we do with the help of a powerful and user-friendly tool often employed in computational methods for physical and engineering problems in any general curvilinear coordinate system. However, tensor analysis has certain strict rules and conventions that must unconditionally be adhered to. This book is intended to support research scientists and practicing engineers in various ﬁelds who use tensor analysis and differential geometry in the context of applied physics and electrical and mechanical engineering. Moreover, it can also be used as a textbook for graduate students in applied physics and engineering. Tensor analysis and differential geometry were pioneered by great mathematicians in the late nineteenth century, chieﬂy Curbastro, Levi-Civita, Christoffel, Ricci, Gauss, Riemann, Weyl, and Minkowski, and later promoted by well-known theoretical physicists in the early twentieth century, mainly Einstein, Dirac, Hei- senberg, and Fermi, working on relativity and quantum mechanics. Since then, tensor analysis and differential geometry have taken on an increasingly important role in the mathematical language used in the modern physics of quantum mechan-

ix x Preface to the First Edition ics and general relativity and in many applied sciences ﬁelds. They have also been applied to computational mechanical and electrical engineering in classical mechanics, aero- and vibroacoustics, computational ﬂuid dynamics (CFD), continuum mechanics, electrodynamics, and cybernetics. Approaching the topics of tensors and differential geometry in a mathematically rigorous way would require an immense amount of effort, which would not be practical for working engineers and applied physicists. As such, we decided to present these topics in a comprehensive and approachable way that will show readers how to work with tensors and differential geometry and to apply them to modeling the physical and engineering world. This book also includes numerous examples with solutions and concrete calculations in order to guide readers through these complex topics step by step. For the sake of simplicity and keeping the target audience in mind, we deliberately neglect certain aspects of mathematical rigor in this book, discussing them informally instead. Therefore, those readers who are more mathematically interested should consult the recommended literature. We would like to thank Mrs. Hestermann-Beyerle and Mrs. Kollmar-Thoni at Springer Heidelberg for their helpful suggestions and valued cooperation during the preparation of this book.

Ludwigsburg, Germany Hung Nguyen-Scha¨fer Heidelberg, Germany Jan-Philip Schmidt Contents

1 General Basis and Bra-Ket Notation ...... 1 1.1 Introduction to General Basis and Tensor Types ...... 1 1.2 General Basis in Curvilinear Coordinates ...... 2 1.2.1 Orthogonal Cylindrical Coordinates ...... 5 1.2.2 Orthogonal Spherical Coordinates ...... 8 1.3 Eigenvalue Problem of a Linear Coupled Oscillator ...... 11 1.4 Notation of Bra and Ket ...... 15 1.5 Properties of Kets ...... 15 1.6 Analysis of Bra and Ket ...... 16 1.6.1 Bra and Ket Bases ...... 16 1.6.2 Gram-Schmidt Scheme of Basis Orthonormalization . . . . 18 1.6.3 Cauchy-Schwarz and Triangle Inequalities ...... 19 1.6.4 Computing Ket and Bra Components ...... 19 1.6.5 Inner Product of Bra and Ket ...... 20 1.6.6 Outer Product of Bra and Ket ...... 22 1.6.7 Ket and Bra Projection Components on the Bases . . . . . 23 1.6.8 Linear Transformation of Kets ...... 23 1.6.9 Coordinate Transformations ...... 25 1.6.10 Hermitian Operator ...... 29 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems ...... 31 References ...... 34 2 Tensor Analysis ...... 35 2.1 Introduction to Tensors ...... 35 2.2 Deﬁnition of Tensors ...... 36 2.2.1 An Example of a Second-Order Covariant Tensor ..... 38 2.3 Tensor Algebra ...... 38 2.3.1 General Bases in General Curvilinear Coordinates ..... 38 2.3.2 Metric Coefﬁcients in General Curvilinear Coordinates ...... 47 2.3.3 Tensors of Second Order and Higher Orders ...... 51

xi xii Contents

2.3.4 Tensor and Cross Products of Two Vectors in General Bases ...... 55 2.3.5 Rules of Tensor Calculations ...... 57 2.4 Coordinate Transformations ...... 65 2.4.1 Transformation in the Orthonormal Coordinates ...... 65 2.4.2 Transformation of Curvilinear Coordinates in EN ...... 68 2.4.3 Examples of Coordinate Transformations ...... 70 2.4.4 Transformation of Curvilinear Coordinates in RN ..... 72 2.5 Tensor Calculus in General Curvilinear Coordinates ...... 74 2.5.1 Physical Component of Tensors ...... 74 2.5.2 Derivatives of Covariant Bases ...... 76 2.5.3 Christoffel Symbols of First and Second Kind ...... 78 2.5.4 Prove That the Christoffel Symbols Are Symmetric . . . . 79 2.5.5 Examples of Computing the Christoffel Symbols ...... 80 2.5.6 Coordinate Transformations of the Christoffel Symbols ...... 82 2.5.7 Derivatives of Contravariant Bases ...... 84 2.5.8 Derivatives of Covariant Metric Coefﬁcients ...... 85 2.5.9 Covariant Derivatives of Tensors ...... 86 2.5.10 Riemann-Christoffel Tensor ...... 90 2.5.11 Ricci’sLemma...... 94 2.5.12 Derivative of the Jacobian ...... 95 2.5.13 Ricci Tensor ...... 97 2.5.14 Einstein Tensor ...... 99 References ...... 101 3 Elementary Differential Geometry ...... 103 3.1 Introduction ...... 103 3.2 Arc Length and Surface in Curvilinear Coordinates ...... 103 3.3 Unit Tangent and Normal Vector to Surface ...... 107 3.4 The First Fundamental Form ...... 108 3.5 The Second Fundamental Form ...... 110 3.6 Gaussian and Mean Curvatures ...... 113 3.7 Riemann Curvature ...... 117 3.8 Gauss-Bonnet Theorem ...... 120 3.9 Gauss Derivative Equations ...... 122 3.10 Weingarten’s Equations ...... 123 3.11 Gauss-Codazzi Equations ...... 124 3.12 Lie Derivatives ...... 126 3.12.1 Vector Fields in Riemannian Manifold ...... 127 3.12.2 Lie Bracket ...... 128 3.12.3 Lie Dragging ...... 129 3.12.4 Lie Derivatives ...... 130 3.12.5 Torsion and Curvature in a Distorted and Curved Manifold ...... 138 Contents xiii

3.12.6 Killing Vector Fields ...... 138 3.13 Invariant Time Derivatives on Moving Surfaces ...... 140 3.13.1 Invariant Time Derivative of an Invariant Field ...... 141 3.13.2 Invariant Time Derivative of Tensors ...... 144 3.14 Tangent, Cotangent Bundles and Manifolds ...... 146 3.15 Levi-Civita Connection on Manifolds ...... 148 References ...... 153 4 Differential Forms ...... 155 4.1 Introduction ...... 155 4.2 Deﬁnitions of Spaces on the Manifold ...... 157 4.3 Differential k-Forms ...... 157 4.4 The Notation ω Á X ...... 161 4.5 Exterior Derivatives ...... 163 4.6 Interior Product ...... 166 4.7 Pullback Operator of Differential Forms ...... 168 4.8 Pushforward Operator of Differential Forms ...... 170 4.9 The Hodge Star Operator ...... 171 4.9.1 Star Operator in Vector Calculus and Differential Forms ...... 173 4.9.2 Star Operator and Inner Product ...... 175 4.9.3 Star Operator in the Minkowski Spacetime ...... 177 References ...... 180 5 Applications of Tensors and Differential Geometry ...... 181 5.1 Nabla Operator in Curvilinear Coordinates ...... 181 5.2 Gradient, Divergence, and Curl ...... 182 5.2.1 Gradient of an Invariant ...... 182 5.2.2 Gradient of a Vector ...... 183 5.2.3 Divergence of a Vector ...... 184 5.2.4 Divergence of a Second-Order Tensor ...... 186 5.2.5 Curl of a Covariant Vector ...... 188 5.3 Laplacian Operator ...... 190 5.3.1 Laplacian of an Invariant ...... 190 5.3.2 Laplacian of a Contravariant Vector ...... 191 5.4 Applying Nabla Operators in Spherical Coordinates ...... 192 5.4.1 Gradient of an Invariant ...... 193 5.4.2 Divergence of a Vector ...... 195 5.4.3 Curl of a Vector ...... 196 5.5 The Divergence Theorem ...... 197 5.5.1 Gauss and Stokes Theorems ...... 197 5.5.2 Green’s Identities ...... 199 5.5.3 First Green’s Identity ...... 199 5.5.4 Second Green’s Identity ...... 200 5.5.5 Differentials of Area and Volume ...... 201 xiv Contents

5.5.6 Calculating the Differential of Area ...... 201 5.5.7 Calculating the Differential of Volume ...... 202 5.6 Governing Equations of Computational Fluid Dynamics ...... 204 5.6.1 Continuity Equation ...... 204 5.6.2 Navier-Stokes Equations ...... 206 5.6.3 Energy (Rothalpy) Equation ...... 210 5.7 Basic Equations of Continuum Mechanics ...... 213 5.7.1 Cauchy’sLawofMotion...... 213 5.7.2 Principal Stresses of Cauchy’s Stress Tensor ...... 218 5.7.3 Cauchy’s Strain Tensor ...... 220 5.7.4 Constitutive Equations of Elasticity Laws ...... 223 5.8 Maxwell’s Equations of Electrodynamics ...... 224 5.8.1 Maxwell’s Equations in Curvilinear Coordinate Systems ...... 225 5.8.2 Maxwell’s Equations in the Four-Dimensional Spacetime ...... 227 5.8.3 The Maxwell’s Stress Tensor ...... 232 5.8.4 The Poynting’s Theorem ...... 237 5.9 Einstein Field Equations ...... 239 5.10 Schwarzschild’s Solution of the Einstein Field Equations ...... 241 5.11 Schwarzschild Black Hole ...... 243 References ...... 246 6 Tensors and Bra-Ket Notation in Quantum Mechanics ...... 249 6.1 Introduction ...... 249 6.2 Quantum Entanglement and Nonlocality ...... 250 6.3 Alternative Interpretation of Quantum Entanglement ...... 252 6.4 The Hilbert Space ...... 254 6.5 State Vectors and Basis Kets ...... 255 6.6 The Pauli Matrices ...... 264 6.7 Combined State Vectors ...... 265 6.8 Expectation Value of an Observable ...... 270 6.9 Probability Density Operator ...... 273 6.9.1 Density Operator of a Pure Subsystem ...... 273 6.9.2 Density Operator of an Entangled Composite System ...... 275 6.10 Heisenberg’s Uncertainty Principle ...... 279 6.11 The Wave-Particle Duality ...... 286 6.11.1 De Broglie Wavelength Formula ...... 287 6.11.2 The Compton Effect ...... 289 6.11.3 Double-Slit Experiments with Electrons ...... 293 6.12 The Schrodinger€ Equation ...... 297 6.12.1 Time Evolution in Quantum Mechanics ...... 298 6.12.2 The Schrodinger€ and Heisenberg Pictures ...... 300 6.12.3 Time-Dependent Schrodinger€ Equation (TDSE) ...... 302 6.12.4 Discussions of the Schrodinger€ Wave Functions ...... 306 Contents xv

6.13 The Klein-Gordon Equation ...... 306 6.14 The Dirac Equation ...... 308 References ...... 311

Appendix A: Relations Between Covariant and Contravariant Bases ...... 313

Appendix B: Physical Components of Tensors ...... 319

Appendix C: Nabla Operators ...... 323

Appendix D: Essential Tensors ...... 329

Appendix E: Euclidean and Riemannian Manifolds ...... 335

Appendix F: Probability Function for the Quantum Interference ..... 359

Appendix G: Lorentz and Minkowski Transformations in Spacetime ...... 361

Appendix H: The Law of Large Numbers in Statistical Mechanics .... 365

Mathematical Symbols in This Book ...... 369

Further Reading ...... 371

Index ...... 373 ThiS is a FM Blank Page About the Authors

Hung Nguyen-Scha¨fer is a senior technical manager in development of electric machines for hybrid and electric vehicles at EM-motive GmbH, a joint company of Daimler and Bosch in Germany. He received B.Sc. and M.Sc. in mechanical engineering with nonlinear vibrations in ﬂuid mechanics from the University of Karlsruhe (KIT), Germany, in 1985 and a Ph.D. degree in nonlinear thermo- and ﬂuid dynamics from the same university in 1989. He joined Bosch Company and worked as a technical manager on many development projects. Between 2007 and 2013, he was in charge of rotordynamics, bearings, and design platforms of automotive turbochargers at Bosch Mahle Turbo Systems in Stuttgart. He is also the author of three other professional engineering books Aero and Vibroacoustics of Automotive Turbochargers, Springer (2013), Rotordynamics of Automotive Turbochargers in Springer Tracts in Mechanical Engineering, Second Edition, Springer (2015), and Computational Design of Rolling Bearings, Springer (2016).

Jan-Philip Schmidt is a mathematician. He studied mathematics, physics, and economics at the University of Heidelberg, Germany. He received a Ph.D. degree in mathematics from the University of Heidelberg in 2012. His doctoral thesis was funded by a research fellowship from the Heidelberg Academy of Sciences, in collaboration with the Interdisciplinary Center for Scientiﬁc Computing (IWR) at the University of Heidelberg. His academic working experience comprises several research visits in France and Israel, as well as project works at the Max-Planck- Institute for Mathematics in the Sciences (MPIMIS) in Leipzig and at the Max- Planck-Institute for Molecular Genetics (MPIMG) in Berlin. He also worked as a research associate in the AVACS program at Saarland University, Cluster of Excellence (MMCI).

xvii Chapter 1 General Basis and Bra-Ket Notation

We begin this chapter by reviewing some mathematical backgrounds dealing with coordinate transformations and general basis vectors in general curvilinear coordinates. Some of these aspects will be informally discussed for the sake of simplicity. Therefore, those readers interested in more in-depth coverage should consult the literature recommended under Further Reading. To simplify notation, we will denote a basis vector simply as basis in the following section. We assume that the reader has already had fundamental backgrounds about vector analysis in ﬁnite N-dimensional spaces with the general bases of curvilinear coordinates. However, this topic is brieﬂy recapitulated in Appendix E.

1.1 Introduction to General Basis and Tensor Types

A physical state generally depending on N different variables is defined as a point P(u1,...,uN) that has N independent coordinates of ui. At changing the variables, such as time, locations, and physical characteristics, the physical state P moves from one position to other positions. All relating positions generate a set of points in an N-dimension space. This is the point space with N dimensions (N-point space). Additionally, the state change between two points could be described by a vector r connecting them that obviously consists of N vector components. All state changes are displayed by the vector field that belongs to the vector space with N dimensions (N-vector space). Generally, a differentiable hypersurface in an N-dimensional space with general curvilinear coordinates {ui} for i ¼ 1, 2,...,N is defined as a differentiable (N À 1)-dimensional subspace with a codimension of one. Subspaces with any codimension are called manifolds of an N-dimensional space (cf. Appendix E). Physically, vectors are invariant under coordinate transformations and therefore do not change in any coordinate system. However, their components change and

© Springer-Verlag Berlin Heidelberg 2017 1 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5_1 2 1 General Basis and Bra-Ket Notation depend on the coordinate system. That means the vector components vary as the coordinate system changes. Generally, tensors are a very useful tool applied to the coordinate transformations between two general curvilinear coordinate systems in finite N-dimensional real spaces. The exemplary second-order tensor can be defined as a multilinear functional T that maps an arbitrary vector in a vector space into the image vector in another vector space. Like vectors, tensors are also invariant under coordinate transformations; however, the tensor components change and depend on the relating transformed coordinate system. Therefore, the tensor components change as the coordinate system varies. Scalars, vectors, and matrices are special types of tensors: – scalar (invariant) is a zero-order tensor, – vector is a first-order tensor, – matrix is arranged by a second-order tensor, – bra and ket are first- and second-order tensors, – Levi-Civita permutation symbols in a three-dimensional space are third-order pseudo-tensors (Table 1.1). We consider two important spaces in tensor analysis: first, Euclidean N-spaces with orthogonal and curvilinear coordinate systems; second, general curvilinear Riemannian manifolds of dimension N (cf. Appendix E).

1.2 General Basis in Curvilinear Coordinates

We consider three covariant basis vectors g1, g2, and g3 to the general curvilinear coordinates (u1,u2,u3) at the point P in Euclidean space E3. The non-orthonormal basis gi can be calculated from the orthonormal bases (e1,e2,e3) in Cartesian coordinates xj ¼ xj(ui) using Einstein summation convention (cf. Sect. 2.1).

Table 1.1 Different types of tensors 0-order 1-order 2-order 3-order Higher-order Tensors of T Ti Tij Tijk Tij...pk Scalar a 2 R X Vector v 2 RN X Matrix M 2 RNÂRN X Bra hB| and ket |Ai2RN, XX RNÂRN Levi-Civita symbols 2 RNÂ X (X) RNÂRN Higher-order tensors 2 RN X Â...ÂRN 1.2 General Basis in Curvilinear Coordinates 3

∂r X3 ∂r ∂xj ∂r ∂xj g ¼ : Á i ∂ui ∂xj ∂ui ∂xj ∂ui j¼1 ð1:1Þ ∂xj ¼ e for j ¼ 1, 2, 3 j ∂ui

The metric coefﬁcients can be calculated by the scalar products of the covariant and contravariant bases in general curvilinear coordinates with non-orthonormal bases (i.e. non-orthogonal and non-unitary). There are the covariant, contravariant, and ij j mixed metric coefﬁcients gij, g , and gi , respectively.

δ j gij ¼ gji ¼ gi Á gj ¼ gj Á gi 6¼ i ij ji i j j i δ j : g ¼ g ¼ g Á g ¼ g Á g 6¼ i ð1 2Þ j j i δ j gi ¼ gi Á g ¼ gj Á g ¼ i

j where the Kronecker delta δi is deﬁned as & 0fori 6¼ j δ j i 1fori ¼ j:

Similarly, the bases of the orthonormal coordinates can be written in the non-orthonormal bases of the curvilinear coordinates ui ¼ ui(xj).

∂r X3 ∂r ∂ui ∂r ∂ui e ¼ Á Á j ∂xj ∂ui ∂xj ∂ui ∂xj i¼1 ð1:3Þ ∂ui ¼ g for i ¼ 1, 2, 3 i ∂xj

The covariant and contravariant bases of the orthonormal coordinates (orthogonal and unitary bases) have the following properties:

δ j; ei Á ej ¼ ej Á ei ¼ i i j j i δ j; : e Á e ¼ e Á e ¼ i ð1 4Þ i i δ j: e Á ej ¼ ej Á e ¼ i

The contravariant basis gk of the curvilinear coordinate uk is perpendicular to the covariant bases gi and gj at the given point P, as shown in Fig. 1.1. The contravariant basis gk can be deﬁned as

∂r ∂r αgk g Â g Â ð1:5Þ i j ∂ui ∂uj where α is a scalar factor; 4 1 General Basis and Bra-Ket Notation

g3 u3

g3 x3 g1 P g e 2 3 g2 2 1 u 0 u g1 e e1 2 x2 x1

Fig. 1.1 Covariant and contravariant bases of curvilinear coordinates

gk is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (1.5) by the covariant basis gk, the scalar factor α results in α k αδ k α gi Â gj Á gk ¼ g Á gk ¼ k ¼ hi ð1:6Þ ; ; gi gj gk

The expression in the square brackets is called the scalar triple product. Therefore, the contravariant bases of the curvilinear coordinates result from Eqs. (1.5) and (1.6).

g Â g g Â g g Â g gi ¼ hij k ; g j ¼ hik i ; gk ¼ hii j ð1:7Þ ; ; ; ; ; ; gi gj gk gi gj gk gi gj gk

Obviously, the relation of the covariant and contravariant bases results from Eq. (1.7).

gi Â gj Á gi gk Á g ¼ hi¼ δ k ð1:8Þ i ; ; i gi gj gk

k where δi is the Kronecker delta. The scalar triple product is an invariant under cyclic permutation; therefore, it has the following properties: 1.2 General Basis in Curvilinear Coordinates 5 : gi Â gj Á gk ¼ ðÞÁgk Â gi gj ¼ gj Â gk Á gi ð1 9Þ

Furthermore, the scalar triple product of the covariant bases of the curvilinear coordinates can be calculated [1]. 1 1 1 ∂x ∂x ∂x ∂u1 ∂u2 ∂u3 ∂xi ∂xj ∂xk ∂x2 ∂x2 ∂x2 ½¼g ; g ; g ε ¼ J ð1:10Þ 1 2 3 ijk ∂ 1 ∂ 2 ∂ 3 u u u ∂u1 ∂u2 ∂u3 ∂ 3 ∂ 3 ∂ 3 x x x ∂u1 ∂u2 ∂u3 where J is the Jacobian, determinant of the covariant basis tensor; εijk is the Levi- Civita symbols in Eq. (A.5), cf. Appendix A. Squaring the scalar triple product in Eq. (1.10), one obtains 1 1 1 2 ∂x ∂x ∂x ∂u1 ∂u2 ∂u3 g11 g12 g13 ; ; 2 ∂x2 ∂x2 ∂x2 ½g g g ¼ 1 2 3 ¼ g g g 1 2 3 ∂u ∂u ∂u 21 22 23 : ∂x3 ∂x3 ∂x3 g g g ð1 11Þ ∂ 1 ∂ 2 ∂ 3 31 32 33 u u u ffiffiffi 2 p ¼ gij g ¼ J ) J ¼Æ g

where gij ¼ gi Á gj are the covariant metric coefﬁcients. Thus, the scalar triple product of the covariant bases results in a right-handed rule coordinate system in which the Jacobian is always positive.

J ¼ ½¼g ; g ; g ðÞÁg Â g g p1ffiffiffi 2 3 1 2 3 ð1:12Þ ¼ g > 0

The covariant and contravariant bases of the orthogonal cylindrical and spherical coordinates will be studied in the following subsections.

1.2.1 Orthogonal Cylindrical Coordinates

Cylindrical coordinates (r,θ,z) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 1.2 shows a point P in the cylindrical coordinates (r,θ,z) which is embedded in the orthonormal Cartesian coordinates (x1,x2,x3). However, the cylindrical coordinates change as the point P varies. 6 1 General Basis and Bra-Ket Notation

x3 (r,θ,z) → (u1,u2,u3): u1 ≡ r ; u2 ≡ θ ; u3 ≡ z

z P g2

R e3 g1 x2 0 e2 e1 θ r

Fig. 1.2 Covariant bases of orthogonal cylindrical coordinates

The vector OP can be written in Cartesian coordinates (x1,x2,x3):

θ θ R ¼ ðÞr cos e1 þ ðÞr sin e2 þ z e3 : 1 2 3 ð1 13Þ x e1 þ x e2 þ x e3 where e1, e2, and e3 are the orthonormal bases of Cartesian coordinates; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u1, u2, and u3 are used for r, θ, and z, respectively. Therefore, the coordinates of P(u1,u2,u3) can be expressed in Cartesian coordinates: 8 9 ÀÁ< x1 ¼ r cos θ u1 cos u2 = 1; 2; 3 2 θ 1 2 : Pu u u ¼ : x ¼ r sin u sin u ; ð1 14Þ x3 ¼ z u3

The covariant bases of the curvilinear coordinates can be computed from

∂R ∂R ∂xj ∂xj g ¼ ¼ : ¼ e for j ¼ 1, 2, 3 ð1:15Þ i ∂ui ∂xj ∂ui j ∂ui

The covariant basis matrix G can be calculated from Eq. (1.15). 1.2 General Basis in Curvilinear Coordinates 7

G ¼ 0½g1 g2 g3 1 ∂x1 ∂x1 ∂x1 0 1 B ∂ 1 ∂ 2 ∂ 3 C B u u u C cos θ Àr sin θ 0 B ∂x2 ∂x2 ∂x2 C ð1:16Þ ¼ B C ¼ @ sin θ r cos θ 0 A B ∂ 1 ∂ 2 ∂ 3 C @ u u u A 001 ∂x3 ∂x3 ∂x3 ∂u1 ∂u2 ∂u3

The determinant of the covariant basis matrix G is called the Jacobian J. 1 1 1 ∂x ∂x ∂x ∂u1 ∂u2 ∂u3 cos θ r sin θ 0 ∂ 2 ∂ 2 ∂ 2 À x x x θ θ : jjG J ¼ ¼ sin r cos 0 ¼ r ð1 17Þ ∂u1 ∂u2 ∂u3 ∂ 3 ∂ 3 ∂ 3 001 x x x ∂u1 ∂u2 ∂u3

The inversion of the matrix G yields the contravariant basis matrix GÀ1. The relation between the covariant and contravariant bases results from Eq. (1.8).

i δ i : g Á gj ¼ j ðÞKronecker delta ð1 18aÞ

At det (G) 6¼ 0 given from Eq. (1.17), Eq. (1.18a) is equivalent to

GÀ1G ¼ I ð1:18bÞ

Thus, the contravariant basis matrix GÀ1 can be calculated from the inversion of the covariant basis matrix G, as given in Eq. (1.16). 0 1 ∂u1 ∂u1 ∂u1 2 3 0 1 B ∂ 1 ∂ 2 ∂ 3 C g1 B x x x C r cos θ r sin θ 0 B ∂u2 ∂u2 ∂u2 C 1 GÀ1 ¼ 4 g2 5 ¼ B C ¼ @ À sin θ cos θ 0 A ð1:19aÞ B ∂ 1 ∂ 2 ∂ 3 C r g3 @ x x x A 00r ∂u3 ∂u3 ∂u3 ∂x1 ∂x2 ∂x3

The contravariant bases of the curvilinear coordinates can be written as

∂ui gi ¼ e for j ¼ 1, 2, 3 ð1:19bÞ ∂xj j

The calculation of the determinant and inversion matrix of G will be discussed in the following section. According to Eq. (1.16), the covariant bases can be rewritten as 8 1 General Basis and Bra-Ket Notation 8 θ θ < g1 ¼ ðÞcos e1 þ ðÞsin e2 þ 0 Á e3 ) jjg1 ¼ 1 θ θ : : g2 ¼ÀðÞr sin e1 þ ðÞr cos e2 þ 0 Á e3 ) jjg2 ¼ r ð1 20Þ g3 ¼ 0 Á e1 þ 0 Á e2 þ 1 Á e3 ) jjg3 ¼ 1

The contravariant bases result from Eq. (1.19b). 8 > g1 ¼ ðÞcos θ e þ ðÞsin θ e þ 0 Á e ) g1 ¼ 1 <> 1 2 3 sin θ cos θ 1 2 2 : > g ¼À e1 þ e2 þ 0 Á e3 ) g ¼ ð1 21Þ :> r r r 3 3 g ¼ 0 Á e1 þ 0 Á e2 þ 1 Á e3 ) g ¼ 1

Not only the covariant bases but also the contravariant bases of the cylindrical coordinates are orthogonal due to

j j δ j; gi Á g ¼ g Á gi ¼ i ; gi Á gj ¼ 0 for i 6¼ j gi Á g j ¼ 0 for i 6¼ j:

1.2.2 Orthogonal Spherical Coordinates

Spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 1.3 shows a point P in the spherical coordinates (ρ,φ,θ) which is embedded in the orthonormal Cartesian coordinates (x1,x2,x3). However, the spherical coordinates change as the point P varies. The vector OP can be rewritten in Cartesian coordinates (x1,x2,x3):

ρ φ θ ρ φ θ ρ φ R ¼ ðÞsin cos e1 þ ðÞsin sin e2 þ cos e3 : 1 2 3 ð1 22Þ x e1 þ x e2 þ x e3 where e1, e2, and e3 are the orthonormal bases of Cartesian coordinates; φ is the equatorial angle; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u1, u2, and u3 are used for ρ, φ, and θ, respectively. Therefore, the coordinates of P(u1,u2,u3) can be expressed in Cartesian coordinates: 1.2 General Basis in Curvilinear Coordinates 9

x3 (ρ,ϕ,θ) → (u1,u2,u3): ρ sinϕ u1 ≡ ρ ; u2 ≡ ϕ ; u3 ≡ θ

g P 3

e3 ϕ ρ e g 0 2 2 x2 e1 θ ρ cosϕ

Fig. 1.3 Covariant bases of orthogonal spherical coordinates

8 9 ÀÁ< x1 ¼ ρ sin φ cos θ u1 sin u2 cos u3 = 1; 2; 3 2 ρ φ θ 1 2 3 : Pu u u ¼ : x ¼ sin sin u sin u cos u ; ð1 23Þ x3 ¼ ρ cos φ u1 cos u2

The covariant bases of the curvilinear coordinates can be computed by means of

∂R ∂R ∂xj g ¼ ¼ Á i ∂ i ∂ j ∂ i u x u ð1:24Þ ∂xj ¼ e for j ¼ 1, 2, 3 j ∂ui

Thus, the covariant basis matrix G can be calculated from Eq. (1.24). 0 1 ∂x1 ∂x1 ∂x1 B C B ∂u1 ∂u2 ∂u3 C B ∂ 2 ∂ 2 ∂ 2 C B x x x C G ¼ ½¼g1 g2 g3 B C @ ∂u1 ∂u2 ∂u3 A ∂x3 ∂x3 ∂x3 ð1:25Þ 0 ∂u1 ∂u2 ∂u3 1 sin φ cos θρcos φ cos θ Àρ sin φ sin θ ¼ @ sin φ sin θρcos φ sin θρsin φ cos θ A cos φ Àρ sin φ 0 10 1 General Basis and Bra-Ket Notation sin φ cos θρcos φ cos θ Àρ sin φ sin θ jjG J ¼ sin φ sin θρcos φ sin θρsin φ cos θ ð1:26Þ cos φ Àρ sin φ 0 ¼ ρ2 sin φ

The determinant of the covariant basis matrix G is called the Jacobian J. Similarly, the contravariant basis matrix GÀ1 is the inversion of the covariant basis matrix. 0 1 ∂u1 ∂u1 ∂u1 2 3 B C 1 B ∂x1 ∂x2 ∂x3 C g 2 2 2 À1 4 5 B ∂u ∂u ∂u C G ¼ g2 ¼ B C B ∂ 1 ∂ 2 ∂ 3 C g3 @ x x x A ∂u3 ∂u3 ∂u3 ð1:27aÞ 0 ∂x1 ∂x2 ∂x3 1 ρ sin φ cos θρsin φ sin θρcos φ B C 1B cosφ cos θ cosφ sin θ À sin φ C ¼ ρ@ sin θ cos θ A À 0 sin φ sin φ

The contravariant bases of the curvilinear coordinates can be written as

∂ui gi ¼ e for j ¼ 1, 2, 3 ð1:27bÞ ∂xj j

The matrix product GGÀ1 must be an identity matrix according to Eq. (1.18b). 0 1 ρ sin φ cos θρsin φ sin θρcos φ B φ θ φ θ φ C À1 1B coscos cossin À sin C G G ¼ ρ@ sin θ cos θ A À 0 0 sin φ sin φ 1 0 1 ð1:28Þ sin φ cos θρcos φ cos θ Àρ sin φ sin θ 100 Á@ sin φ sin θρcos φ sin θρsin φ cos θ A ¼ @ 010A I cos φ Àρ sin φ 0 001

According to Eq. (1.25), the covariant bases can be written as

φ θ φ θ φ g1 ¼ ðÞsin cos e1 þ ðÞsin sin e2 þ cos e3 ) jjg1 ¼ 1 ρ φ θ ρ φ θ ρ φ ρ : g2 ¼ ðÞcos cos e1 þ ðÞcos sin e2 À ðÞsin e3 ) jjg2 ¼ ð1 29Þ ρ φ θ ρ φ θ ρ φ g3 ¼ÀðÞsin sin e1 þ ðÞsin cos e2 þ 0 Á e3 ) jjg3 ¼ sin

The contravariant bases result from Eq. (1.27b). 1.3 Eigenvalue Problem of a Linear Coupled Oscillator 11 1 φ θ φ θ φ 1 g ¼ ðÞsin cos e1 þ ðÞsinsin e2 þ cos e3 ) g ¼ 1 2 1 1 1 2 1 g ¼ cos φ cos θ e1 þ cos φ sin θ e2 À sin φ e3 ) g ¼ ρ ρ ρ ρ 1 sin θ 1 cos θ 1 g3 ¼À e þ e þ 0 Á e ) g3 ¼ ρ sin φ 1 ρ sin φ 2 3 ρ sin φ ð1:30Þ

Not only the covariant bases but also the contravariant bases of the spherical coordinates are orthogonal due to

j j δ j; gi Á g ¼ g Á gi ¼ i ; gi Á gj ¼ 0for i 6¼ j gi Á g j ¼ 0for i 6¼ j:

1.3 Eigenvalue Problem of a Linear Coupled Oscillator

In the following subsection, we will give an example of the application of vector and matrix analysis to the eigenvalue problems in mechanical vibration. Figure 1.4 shows the free vibrations without damping of a three-mass system with the masses m1, m2, and m3 connected by the springs with the constant stiffness k1, k2, and k3.In case of the small vibration amplitudes and constant spring stiffnesses, the vibrations can be considered linear. Otherwise, the vibrations are nonlinear for that the bifurcation theory must be used to compute the responses [2]. Using Newton’s second law, the homogenous vibration equations (free vibration equations) of the three-mass system can be written as [2–6]:

m1€x1 þ k1x1 þ k2ðÞ¼x1 À x2 0 m2€x2 þ k2ðÞþx2 À x1 k3ðÞ¼x2 À x3 0 ð1:31Þ m3€x3 þ k3ðÞ¼x3 À x2 0

Thus,

m1€x1 þ ðÞk1 þ k2 x1 À k2x2 ¼ 0 m2€x2 À k2x1 þ ðÞk2 þ k3 x2 À k3x3 ¼ 0 m3€x3 À k3x2 þ k3x3 ¼ 0

Using the abbreviations of k12 k1 + k2 and k23 k2 + k3, one obtains 12 1 General Basis and Bra-Ket Notation

Fig. 1.4 Free vibrations of = + a three-mass system k12 k1 k2 = + k23 k2 k3 x x x k 1 2 3 1 k2 k3 m x m x m1x1 2 2 3 3

m1 m2 m3

− − k1x1 k2 (x1 x2 ) k3 (x2 x3 ) − − k2 (x2 x1) k3 (x3 x2 )

m1€x1 þ k12x1 À k2x2 ¼ 0 € m2x À k2x1 þ k23x2 À k3x3 ¼ 0 € m3x À k3x2 þ k3x3 ¼ 0

The vibration equations can be rewritten in the matrix formulation: 2 3 0 1 2 3 0 1 0 1 m1 00 €x1 k12 Àk2 0 x1 0 4 5 @ A 4 5 @ A @ A 0 m2 0 Á €x2 þ Àk2 k23 Àk3 Á x2 ¼ 0 ð1:32Þ 00m3 €x3 0 Àk3 k3 x3 0

Thus, ÀÁ x€ þ MÀ1K x ¼ 0 ð1:33Þ , x€ þ Ax ¼ 0 where 2 3 k12 k2 6 À 0 7 6 m1 m1 7 6 k k k 7 À1 6 2 23 3 7 A M K ¼ 6 À À 7 4 m2 m2 m2 5 k k 0 À 3 3 m3 m3

The free vibration response of Eq. (1.33) can be assumed as

λ x ¼ Xe tÀÁ _ λ λt λ ) x ¼ ÀÁXe ¼ x ð1:34Þ ) x€ ¼ λ2 Xeλt ¼ λ2x where λ is the complex eigenvalue that is deﬁned by 1.3 Eigenvalue Problem of a Linear Coupled Oscillator 13

λ ¼ α Æ jω 2 C ð1:35Þ in which ω is the eigenfrequency; α is the growth/decay rate [2]. Substituting Eq. (1.34) into Eq. (1.33) one obtains the eigenvalue problem ÀÁ λ A þ λ2I X e t ¼ 0 ð1:36Þ where X is the eigenvector relating to its eigenvalue λ; I is the identity matrix. For any non-trivial solution of x, the determinant of (A + λ2I) must vanish. ÀÁ det A þ λ2I ¼ 0 ð1:37Þ

Equation (1.37) is called the characteristic equation of the eigenvalue. Obviously, this characteristic equation is a polynomial of λ2N where N is the degrees of freedom (DOF) of the vibration system. In this case, the DOF equals 3 for a three-mass system in the translational vibration. Solving the characteristic equation (1.37), one obtains 6 eigenvalues (¼2*DOF) for the vibration equations of the three-mass system. In the case without damping where the real parts are equal to zero, there are six different eigenvalues, three of them for forward whirls and three for backward whirls.

λ1,2 ¼Æjω1; λ3,4 ¼Æjω2; λ5,6 ¼Æjω3 ð1:38Þ

Substituting the eigenvalue λi into Eq. (1.36), one obtains the corresponding eigenvector Xi. ÀÁ λ2 ... : : A þ i I Xi ¼ 0 for i ¼ 1, 2, ,6 ð1 39Þ

The eigenvectors in Eq. (1.39) relating to the eigenvalues show the vibration modes of the system. It is well known that N ordinary differential equations (ODEs) of second order can be transformed into 2N ODEs of ﬁrst order using the simple trick of adding N identical ODEs of ﬁrst order to the original ODEs. x_ ¼ x_ x_ ¼ x_ , Mx€ þ Kx ¼ 0 x€ ¼ÀMÀ1Kx ! ð1:40Þ x_ 0 I x , ¼ Á x€ ÀMÀ1K 0 x_

Substituting a new (2N Â 1) vector of 14 1 General Basis and Bra-Ket Notation x x_ z ¼ ) z_ ¼ ð1:41Þ x_ x€ into Eq. (1.40), the vibration equations of ﬁrst order can be rewritten down ! x_ 0 I x ¼ Á x€ ÀMÀ1K 0 x_ ð1:42Þ , z_ ¼ Bz

The free vibration response of Eq. (1.42) can be assumed as ÀÁ λ λ z ¼ Ze t ) z_ ¼ λ Ze t ¼ λz ð1:43Þ where λ is the complex eigenvalue given in

λ ¼ α Æ jω 2 C ð1:44Þ

Within ω is the eigenfrequency; α is the growth/decay rate. Substituting Eq. (1.43) into Eq. (1.42) one obtains the eigenvalue problem

λ ðÞB À λI Ze t ¼ 0 ð1:45Þ where Z is the eigenvector relating to its eigenvalue λ; I is the identity matrix. For any non-trivial solution of z, the determinant of (B À λI) must vanish:

detðÞ¼B À λI 0 ð1:46Þ

Equation (1.46) is called the characteristic equation of the eigenvalue that is identical to Eq. (1.37). Obviously, this characteristic equation is a polynomial of λ2N where N is the degrees of freedom (DOF) of the vibration system. In this case, the DOF equals 3 because of the three-mass system. Solving the characteristic equation (1.46), one obtains 6 eigenvalues (¼ 2*DOF) for the vibration equations of the three-mass system. In the case without damping where the real parts are equal to zero, there are six different eigenvalues, three of them for forward whirls and three for backward whirls.

λ1,2 ¼Æjω1; λ3,4 ¼Æjω2; λ5,6 ¼Æjω3 ð1:47Þ

Substituting the eigenvalue λi into Eq. (1.45), it gives the corresponding eigenvector Zi.

ðÞB À λiI Zi ¼ 0 for i ¼ 1, 2, ...,6: ð1:48Þ

The eigenvectors in Eq. (1.48) relating to the eigenvalues describe the vibration modes of the system. 1.5 Properties of Kets 15

Fig. 1.5 Notation of bra and ket Bra BracKet Ket

1.4 Notation of Bra and Ket

The notation of bra and ket was defined by Dirac for applications in quantum mechanics and statistical thermodynamics [7]. Bra and ket are tuples of independent coordinates in a finite N-dimensional space in Riemannian manifold (cf. Appendix E). The name of bra and ket comes from the angle bracket <>,as shown in Fig. 1.5. Dividing the bracket into two parts, one obtains the left one called bra and the right one named ket. In general, bra and ket can be considered as vectors, matrices, and high-order tensors. In contrast to vectors (first-order tensors), bra and ket have generally neither direction nor vector length in the point space. They are only a tuple of N coordinates (dimensions), such as of time, position, momentum, velocity, etc. Bra and ket are independent of any coordinate system, but their components depend on the relating basis of the coordinate system; i.e., they are changed at the new basis by coordinate transformations. Therefore, the bra and ket notation is a powerful tool mostly used in quantum mechanics and statistical thermodynamics in order to describe a physical state as a point of N dimensions in a finite N-dimensional complex space. Some examples of2 bra and3 ket can be written in different types: 1 þ i 6 À1 7 Ket vector jiK ¼ 6 7 ! Bra vector hjK ¼ ½ðÞÀ1 À i 12ðÞþ i 1 ; 4 2 À i 5 1 ! ! 1 þ i À2 1 À i 1 Ket matrix jiM ¼ ! Bra matrix hjM ¼ . 12À i À22þ i

1.5 Properties of Kets

We denote the ﬁnite N-dimensional complex vector space by CN.Aket|Ki can be deﬁned as an N-tuple of the coordinates u1,...,uN: K(u1,...,uN) 2 CN. Given three arbitrary kets |Ai, |Bi, and |Ci2CN and two scalars α, β 2 C, the following properties of kets result in [8]: – Commutative property of ket addition: Aiþ Bi¼ Biþ Ai 16 1 General Basis and Bra-Ket Notation

– Distributive property of ket multiplication by a scalar addition: ðÞα þ β Ai¼α Aiþβ Ai

– Distributive property of multiplication of ket addition by a scalar: À α jiA þ B Þ¼α Aiþα Bi

– Associative property of ket addition: ÀÁ ÀÁ Aiþ jiB þ Ci ¼ jiA þ Bi þ Ci

– Associative property of ket multiplication by scalars: ÀÁ ÀÁ αβAi ¼ βαAi ¼ αβ Ai

– Property of ket addition to the null ket |0 i: Aiþ 0i¼ Ai

– Property of ket multiplication by the null scalar: 0Á Ai¼ 0i

– Property of ket addition to an inverse ket | À Ai: Aiþ ÀAi¼ AiÀ Ai¼ 0i:

1.6 Analysis of Bra and Ket

1.6.1 Bra and Ket Bases

1 N N Ket vector |Ai of the coordinates (u ,...,u ): A(a1,...,aN) 2 V is the sum of its components in the orthonormal ket bases and can be written as 0 1 a1 B C XN B a2 C ; α β : jiA @ : A ¼ aijii ai ðÞ2i þ j i C ð1 49Þ i ¼ 1 aN where 1.6 Analysis of Bra and Ket 17

ai is the ket component in the basis |ii; ai is a complex number, ai 2 C; |ii is the orthonormal basis of the coordinates (u1,...,uN). The ket bases of {|1i, |2i,...,|Ni} in the coordinates (u1,...,uN) are column vectors, as given in 0 1 0 1 0 1 0 1 1 0 0 0 B 0 C B 1 C B 0 C B 0 C ji1 B C; ji2 B C; ...; jii B C; ...; jiN B C ð1:50Þ @ : A @ : A @ 1 A @ : A 0 0 0 1

Bra hA| is deﬁned as the transpose conjugate (also adjoint) |Ai* of the ket |Ai. Therefore, the bra is a row vector and its elements are the conjugates of the ket elements. To formulate bra of |Ai, at ﬁrst ket |Ai must be transposed; then, its complex elements are conjugated into bra elements. 0 1 α β 1 þ j 1 B α β C B 2 þ j 2 C jiA @ ... A ) α β ð1:51Þ ÂÃN þ j N À Á T α β α β ... α β A ¼ ðÞ1 þ j 1 ðÞ2 þ j 2 N þ j N ) * α β α β ... α β jiA ½ðÞ1 À j 1 ðÞ2 À j 2 ðÞN À j N hjA

The ket vector |Ai* is called the transpose conjugate (adjoint) of the ket vector |Ai and equals bra hA|. Thus, bra hA| can be written in the bra bases

XN ÀÁ * : *; * α β : hjA jiA ¼ hjj aj aj j À j j 2 C ð1 52Þ j ¼ 1 where the component a* is the complex conjugate of its component a. Analogously, the bra bases result from Eq. (1.50).

hj1 ½10: 0 ; hj2 ½01: 0 ; ð1:53Þ hjj ½0010; hjN ½0001:

Due to orthonormality, the product of bra and ket is a scalar and obviously equals the Kronecker delta. & 0; i 6¼ j hji Á jij hiijj ¼ δ j ¼ ð1:54Þ i 1; i ¼ j 18 1 General Basis and Bra-Ket Notation

1.6.2 Gram-Schmidt Scheme of Basis Orthonormalization

3 The basis {|gii} is non-orthonormal in the curvilinear coordinates in the space R . Using the Gram-Schmidt scheme [8, 9], the orthonormal bases (|e1i,|e2i,|e3i) are created from the non-orthogonal bases (|g1 i,|g2 i,|g3 i ). The orthonormalization procedure of the basis is discussed in Appendix E.2.6. The ﬁrst orthonormal ket basis is generated by

jig1 jie1 ji1 ¼ jjg1

The second orthonormal ket basis results from

jig2 À hie1jg2 Á jie1 jie2 ji2 ¼ jjjig2 À hie1jg2 Á jie1

The third orthonormal ket basis is similarly calculated in

jig3 À hie1jg3 Á jie1 À hie2jg3 Á jie2 jie3 ji3 ¼ jjjig3 À hie1jg3 Á jie1 À hie2jg3 Á jie2

Generally, the orthonormal ket basis |ej i can be rewritten in the N-dimensional space. E DE jXÀ 1 g À e jg Á jie j i j i i ¼1 ... : ej jij ¼ E DE for j ¼ 1, 2, , N ð1 55Þ jXÀ 1 gj À eijgj Á jiei i ¼1

Using the Gram-Schmidt procedure, the ket orthonormal bases of {|1i, |2i,...,|Ni} in the coordinates (u1,...,uN) are generated from any non-orthonormal bases, as given in Eq. (1.50). The bra orthonormal bases of {h1|, h2|,...,hN|} in the coordinates (u1,...,uN) are the adjoint of the ket orthonormal bases. 1.6 Analysis of Bra and Ket 19

1.6.3 Cauchy-Schwarz and Triangle Inequalities

The Cauchy-Schwarz and triangle inequalities immediately apply to the Bra-Ket notation: 1) Cauchy-Schwarz Inequality The well-known Cauchy-Schwarz inequality provides the relation between the inner product of bra and ket, and their norms.

hiAjB jjjiA Á jjjiB ; jiA , jiB 2 VN ð1:56Þ

2) Triangle Inequality The triangle inequality formulates the inequality between the sum of two kets and the ket norms.

jjjiA þ jiB jjjiA þ jjjiB ; jiA , jiB 2 VN ð1:57Þ

1.6.4 Computing Ket and Bra Components

The component of a ket results from multiplying the ket by a bra basis according to Eqs. (1.49) and (1.54):

XN À Á hjÁj jiA hi¼jjA hjÁj aijii i ¼ 1 XN XN ð1:58Þ δ i ¼ hijji Á ai ¼ j Á ai i ¼ 1 ÀÁi ¼ 1 α β ¼ aj j þ j j

Equation (1.58) indicates that the ket component in the orthogonal bases equals the scalar product between the ket and its relating basis. Similarly, the bra component can be computed by multiplying the bra by a ket basis. 20 1 General Basis and Bra-Ket Notation

XN ÀÁ * hjA Á jij hiAjj ¼ hji ai Ájij i ¼ 1 XN XN ð1:59Þ * δ j * ¼ hiijj Á ai ¼ i Á ai i ¼ 1 ÀÁi ¼ 1 * α β ¼ aj j À j j

It is straightforward that the bra component is equal to the complex conjugate of the relating ket component aj, as given in Eq. (1.52).

1.6.5 Inner Product of Bra and Ket

The inner product of bra hA| and ket |Bi is deﬁned as ! ! XN XN * hiAjB ¼ hji ai Á bjjij i ¼ 1 j ¼ 1 XN XN XN XN * * δ j : ¼ ai bjhiijj ¼ ai bj i ð1 60aÞ i ¼ 1j ¼1 i ¼ 1j ¼1 XN * ¼ ai bi i ¼1

It is obvious that the inner product of bra and ket is a complex number according to Eqs. (1.59) and (1.60a). In case of |Ai¼|Bi, the inner product in Eq. (1.60a) becomes ! ! XN XN * hiAjA ¼ hji ai Á ajjij i ¼ 1 j ¼ 1 XN XN XN XN * * δ j : ¼ ai ajhiijj ¼ ai aj i ð1 60bÞ i ¼ 1j ¼1 i ¼ 1j ¼ 1 XN XN À Á * α2 β2 2 ¼ ai ai ¼ i þ i ¼ jjjiA i ¼ 1 i ¼ 1

Thus, the norm (length) of the ket |Ai is given in pffiffiffiffiffiffiffiffiffiffiffiffi jjjiA ¼ hiAjA ð1:60cÞ

The inner product in Eq. (1.60a) can be rewritten using Eq. (1.52). 1.6 Analysis of Bra and Ket 21 2 3 b1 XN ÂÃ6 7 * * * * * 6 b2 7 hiAjB ¼ ai bi ¼ a1 a2 ai aN Á 4 5 bi ð1:61Þ i ¼ 1 bN ¼ jiA * Á jiB ¼ hjA Á jiB hiAjB

Similarly, the inner product of bra hB| and ket |Ai can be calculated as follows: ! ! XN XN * hiBjA ¼ hjj bj Á aijii j ¼ 1 i ¼ 1 XN XN XN XN * * δ i : ¼ bj aihijji ¼ bj ai j ð1 62Þ i ¼ 1j ¼1 i ¼ 1j ¼1 XN * ¼ bi ai i ¼1

Conjugating Eq. (1.62), one obtains the transpose conjugate of hB|Ai: ! XN * * * hiBjA ¼ bi ai i ¼1 XN ÀÁ XN ð1:63Þ * * * * ¼ ai bi ¼ ai bi i ¼1 i ¼1 ¼ hiAjB

Thus, the inner product is skew-symmetric (anti-symmetric) contrary to the inner product of two regular vectors. Some properties of the inner product (scalar product) are valid: Skew-symmetry: hA|Bi¼hB|Ai*; Positive deﬁniteness: hA|Ai¼||Ai |2 0; Distributive property: h A|(αB+βC) i¼αhA|Bi + βhA|Ci for α, β 2 C. Furthermore, the linear adjoint operator has the following properties: 1. (αβγ)* ¼ [α(βγ)]* ¼ (βγ)*α* ¼ γ*β*α* for α, β, γ 2 C Note that the product order is changed in the adjoint operation of the scalar product. 2. (α|Ai)* ¼ |αAi* ¼ |Ai* α* ¼hA| α* for α 2 C 3. (hA|α)* ¼ α*hA|* ¼ α*|Aifor α 2 C 4. hA|α ** ¼ hA|α for α 2 C 5. hB|Ai* ¼ |Ai*.hB|* ¼ hA|.|BihA|Bi: skew-symmetric (anti-symmetric) 6. hA|α*|Bi* ¼ |Bi*.α.hA|* ¼ hB|.α.|AihB|α|Ai for α 2 C 7. (|AihB|)* ¼ hB|*.|Ai* ¼ |BihA|: the outer product of ket and bra. 22 1 General Basis and Bra-Ket Notation

1.6.6 Outer Product of Bra and Ket

The outer product of ket |Ai and bra hB| is deﬁned as ! ! XN XN * jiA hjB ¼ aijii Á hjj bj i ¼ 1 j ¼ 1 : XN XN ð1 64Þ * ¼ aibj jii hjj i ¼ 1j ¼1 where the product term |ii hj| is called the outer product of the bases |ii and hj|. Contrary to the inner product resulting a scalar of (1 Â 1) matrix 2 V, the outer product is an operator of (N Â N) matrix 2 VNÂN because the ket is an (N Â 1) column vector 2 VN and the bra is a (1 Â N) row vector 2 VN. Now, ket |Ai can be expressed in ket bases:

XN jiA ¼ jii ai ð1:65Þ i ¼ 1

According to Eq. (1.58), the ket component is

ai ¼ hiijA

Substituting ai into Eq. (1.65), one obtains the ket

XN XN jiA ¼ jii hiijA IijiA ð1:66Þ i ¼ 1 i ¼ 1 where Ii is the projection operator (outer product) according to [8], as deﬁned by 2 3 2 3 0 0000 6 0 7 6 00007 I jii hji ¼ 6 7 Á ½¼0010 6 7 ð1:67Þ i 4 1 5 4 00105 0 0000

The element Iij of the projection operator (matrix) is 1 at the i row and j column, as shown in Eq. (1.67); otherwise, other elements are equal to zero. Obviously, the sum of all projection operators is the identity matrix.

XN XN I Ii ¼ jii hji ð1:68Þ i ¼1 i ¼1

According to Eq. (1.66), the identity property of the ket is proved by 1.6 Analysis of Bra and Ket 23

XN jiA ¼ IijiA ¼ IAji: ð1:69Þ i ¼ 1

1.6.7 Ket and Bra Projection Components on the Bases

The projection component of ket |Ai on the basis | i i can be calculated as

jiA i ¼ IijiA ¼ jii hji Á jiA ¼ jii hiijA ð1:70Þ ¼ jii ai

Thus, ket |Ai can be expressed, as shown in Eq. (1.49):

XN XN : jiA ¼ jiA i ¼ jii ai ð1 71Þ i ¼ 1 i ¼ 1

Similarly, the projection component of bra hA| on the basis h i | is computed as

hjA i ¼ hjA Ii ¼ hjA Á jii hji ¼ hiAji hji ð1:72Þ * ¼ hji ai

Bra hA| can be expressed in its projection components, as given in Eq. (1.52).

XN XN * hjA ¼ hjA i ¼ hji ai i ¼ 1 i ¼ 1 ! ð1:73Þ XN * XN * * ¼ jiA ¼ jii ai ¼ ai hji i ¼ 1 i ¼ 1

1.6.8 Linear Transformation of Kets

We consider the complex vector spaces V and V’. Each of them belongs to the ﬁnite N-dimensional complex space CN. The linear transformation T maps the ket |Ai in V into the image ket |A’i in V’, as shown in Fig. 1.6. The image ket |A’i can be written in the bases |ii [8, 9]as 24 1 General Basis and Bra-Ket Notation

Fig. 1.6 Linear T transformation T of a ket |Ai

A A′ = T A

V ⊂ CN V′ ⊂ CN

0 T : jiA ! A ¼ TjiA XN XN : 0 ð1 74Þ A ¼ T jii ai ¼ Tjii ai i ¼ 1 i ¼ 1

In this case, the ket basis |ii is also mapped into the image ket basis |i’i. The transformation operator T for the basis can be written as E E 0 0 T : jii ! i ) i ¼ Tjii ð1:75Þ

The image ket basis |i’i is formulated as a linear combination of the old ket bases |ii.

E XN 0 i ¼ Tjii ¼ Tjijij ; i ¼ 1, 2, ..., N ð1:76Þ j ¼ 1 where the operator element Tji is in the j row and i column of the transformation matrix T of the transformation operator T. Multiplying both sides of Eq. (1.76) by the bra basis hk|, one obtains

DE XN 0 kji ¼ hjk Tjii ¼ hjk Á Tjijij j ¼ 1 : XN XN ð1 77Þ δ j ¼ Tjihikjj ¼ Tji k ¼ Tki j ¼ 1 j ¼ 1

Thus, the operator element results from Eq. (1.77): DE 0 Tki ¼ kji ¼ hjk Tjii ð1:78Þ

Substituting Eq. (1.76) into Eq. (1.74), one obtains the image ket. 1.6 Analysis of Bra and Ket 25

XN 0 A ¼ TjiA ¼ T jii ai i ¼ 1 ! XN XN XN ¼ Tjii ai ¼ Tjijij ai ð1:79Þ i ¼ 1 i ¼!1 j ¼ 1 XN XN XN 0 ¼ Tjiai jij ajjij j ¼ 1 i ¼ 1 j ¼ 1

The component of the image ket in the basis |ji is given from Eqs. (1.78) and (1.79).

XN XN a0 ¼ T a ¼ hjj Tjii a , j ji i i ð1:80Þ i ¼ 1 i ¼ 1 0 A ¼ TNÂNjiA

The image ket in Eq. (1.80) can be rewritten in the transformation matrix TNÂN. 2 3 2 3 2 3 0 : a1 T11 T12 T1i T1N a1 6 0 7 6 7 6 7 6 a 7 6 T21 T22 : T2i T2N 7 6 a2 7 6 2 7 6 7 6 7 6 : 7 ¼ 6 :::::7 Á 6 : 7 ð1:81Þ 4 0 5 4 :: 5 4 5 aj Tj1 Tji TjN ai 0 T T : T T a aN N1 N2 Ni NN N where the matrix element Tji is computed by the ket transformation T, as given in Eq. (1.78).

Tji ¼ hjj Tjii

1.6.9 Coordinate Transformations

À1 The ket basis |eii is transformed into the new ket basis |gii by the transformation S , as shown in Fig. 1.7. The transformed ket basis can be written as [9].

À1 : À1 S jiei ! jigi ) jigi ¼ S jiei , : : ð1 82Þ S jigi ! jiei ) jiei ¼ Sjigi

Analogous to the ket transformation, the old ket basis can be written in a linear combination of the new bases: 26 1 General Basis and Bra-Ket Notation

Fig. 1.7 Coordinate S transformation of bases

g e A e A = S A

−1 ∈ N S g ∈ R N ei R i = −1 gi S ei

E XN ... : jiei ¼ gj Sji for i ¼ 1, 2, , N ð1 83Þ j ¼ 1 in which Sji is the matrix element of the transformation matrix S. Multiplying both sides of Eq. (1.83) by the bra basis hgk|, one obtains the matrix element Ski. E DE XN XN higkjei ¼ hjgk Á gj Sji ¼ gkjgj Sji j ¼ 1 j ¼ 1 : XN ð1 84Þ δ j ¼ kSji ¼ Ski j ¼ 1

Thus, using Eq. (1.82) it gives

: Ski ¼ hi¼gkjei hjgk Sjigi ð1 85Þ

An arbitrary ket |Ai can be expressed linearly in terms of the old ket basis:

XN e : jiA ¼ jiei ai ð1 86Þ i ¼ 1

e where ai is the ket component in the old basis |eii. Substituting Eq. (1.83) into Eq. (1.86), one obtains ! XN XN XN e e jiA ¼ jiei ai ¼ jigk Ski ai i ¼ 1 i ¼!1 k ¼ 1 1:87 XN XN XN ð Þ e g ¼ Skiai jigk ak jigk k ¼ 1 i ¼ 1 k ¼ 1

Therefore, the ket component in the transformed basis |gki can be calculated by Eq. (1.87). 1.6 Analysis of Bra and Ket 27

XN XN g e e g e : ak ¼ Skiai ¼ hjgk Sjigi ai , jiA ¼ SjiA ð1 88Þ i ¼ 1 i ¼ 1

The transformed ket in Eq. (1.88) can be rewritten in the transformation matrix SNÂN. 2 3 2 3 2 3 g e a1 S11 S12 : S1i S1N a1 6 7 6 7 6 7 6 a2 7 6 S21 S22 : S2i S2N 7 6 a2 7 6 7 6 7 6 7 6 : 7 ¼ 6 :::::7 Á 6 : 7 ð1:89Þ 4 5 4 5 4 5 aj Sj1 Sj2 : Sji SjN ai aN SN1 SN2 : SNi SNN aN where the matrix element is given in Eq. (1.89):

: Ski ¼ higkjei ¼ hjgk Sjigi ð1 90Þ

jiA e ¼ SÀ1jiA g ð1:91Þ

The second transformation yields the second transformed ket:

A g

∈ N g −1 g g 1 gi R A′ = (STS ) A ≡ U A S −1 U

e e S g e e −1 g T ′ ′ = ′ A = S A A = T A A S A 2 3

∈ N e ∈ R N g ∈ R N ei R i i

Fig. 1.8 The combined transformation of kets 28 1 General Basis and Bra-Ket Notation E e 0 e A ¼ TjiA ð1:92Þ

The third transformation leads to the third transformed ket: E E g e 0 0 A ¼ S A ð1:93Þ

Finally, the combined transformed ket of three transformations results from Eqs. (1.91), (1.92), and (1.93). E E g e ÀÁ 0 0 e g g A ¼ S A ¼ STjiA ¼ STSÀ1 jiA UjiA ð1:94Þ where the combined transformation U is deﬁned as (STSÀ1). The component of the product of many operators is computed [8, 9] according to Eq. (1.78). ÀÁ À1 Uij ¼ hji Ujij ¼ hji STS jij XN XN ¼ hji Sjik Á hjk Tjil Á hjl SÀ1jij : k ¼1l ¼ 1 ð1 95Þ XN XN À1 ¼ SikTklSlj k ¼1l ¼ 1

The ket transformed component of |A’i g can be obtained from Eqs. (1.94) and (1.95). 0 g g A ¼ UjiA , ! XN XN XN XN : 0 g À1 g ð1 96Þ aig ¼ Uijaj ¼ SikTklSlj aj j ¼ 1 j ¼ 1 k ¼1l ¼ 1

The transformed ket in Eq. (1.96) can be rewritten in the transformation matrix UNÂN. 2 3 2 3 2 3 0 g : g a1 U11 U12 U1i U1N a1 6 0 7 6 7 6 7 6 a 7 6 U21 U22 : U2i U2N 7 6 a2 7 6 2 7 6 7 6 7 6 : 7 ¼ 6 :::::7 Á 6 : 7 ð1:97Þ 4 0 5 4 :: 5 4 5 aj Uj1 Uji UjN ai 0 U U : U U a aN N1 N2 Ni NN N 1.6 Analysis of Bra and Ket 29

1.6.10 Hermitian Operator

Hermitian operator plays a key role in eigenvalue problems using in quantum { mechanics. Let T be a matrix; the adjoint T (spoken T dagger) is deﬁned as the { transpose conjugate (adjoint) of the matrix T. In quantum mechanics, the operator T of the matrix T is called a Hermitian operator (or self-adjoint operator) if it equals { { the operator T of the matrix T (i.e., T ¼ T or T ¼ T), cf. Chap. 6. Let T be an operator (matrix or second-order tensor). Its transpose is written as

T T ¼ Tij ) T ¼ Tji

The complex conjugate of T is deﬁned as

* * T ¼ Tij

The transpose conjugate of T is called the adjoint T{ that is calculated as ÀÁ ÀÁ * * { TT ¼ T ¼ T* T{ ¼ T ji ji ij * * * { , T ¼ Tji ¼ T ji ÀÁ ij * , TT ¼ T{

If T is hermitian, then T is self-adjoint; i.e.,

{ { : Tij ¼ Tij , T ¼ T

An arbitrary ket |Bi can be transformed by the linear operator T into a ket:

T : jiB 2 RN ! TjiB 2 RN ð1:98Þ

The inner product of bra hA| and transformed ket T|Bi can be written using the { hermitian matrix T ¼ T*: ÀÁ A TB A* TB A*T B hij ¼ ÀÁÁ ¼ ÀÁÁ * * { * { : ¼ T A Á B ¼ T A Á B ¼ T A Á jiB ð1 99Þ ¼ T{AjB

{ This result shows that the inner product between the transformed bra hT A| and ket | Bi is the same inner product of the bra hA| and transformed ket T|Bi. As a rule of thumb, the inner product does not change when moving the operator T from the second ket into the ﬁrst bra and changing T into T{. There are some properties of the inner product with a complex number α and its conjugate α*: 30 1 General Basis and Bra-Ket Notation hiAjα B ¼ α hiAjB ¼ α*AjB for α 2 C ð1:100Þ hiα AjB ¼ α*hiAjB ¼ Ajα*B for α 2 C

The eigenvalue problem derives from the characteristic equation:

TjiA ¼ λjiA ; jiA 6¼ 0,forλ 2 C ð1:101Þ

The inner product between the bra hA| and its transformed ket T|Ai results as

hiAjTA ¼ hiAjλA ¼ λhiAjA ð1:102Þ

Some characteristics of the eigenvalue problem are discussed in the following section [8, 9]: 1) Eigenvalue of the Hermitian operator is a real number. According to Eq. (1.99), the inner product in Eq. (1.102) with the Hermitian { operator T (¼ T) becomes hi¼AjTA T{AjA hiTAjA ð1:103Þ ¼ hiλAjA ¼ λ*hiAjA

Comparing Eq. (1.102) to Eq. (1.103), one obtains

λ* ¼ λ ð1:104Þ

This result proves that the eigenvalue λ must be a real number. 2) Eigenkets of the Hermitian operator are orthogonal. Given two eigenkets with their different eigenvalues λ and μ, the eigenvalue problems can be formulated:

TjiA ¼ λjiA ; jiA 6¼ 0, ð1:105Þ TjiB ¼ μjiB ; jiB 6¼ 0;

The inner product between the kets |Ai and T|Bi can be given according to Eq. (1.100).

hiAjTB ¼ hiAjμ B ¼ μhiAjB ð1:106Þ

Similarly, the inner product between the kets T|Ai and |Bi can be rewritten according to Eq. (1.100).

hi¼TAjB hi¼λAjB λ*hiAjB ð1:107Þ

{ Comparing Eq. (1.106)toEq.(1.107), one obtains the Hermitian operator T (¼ T) according to Eq. (1.99): 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems 31 hiAjTB ¼ T{AjB ¼ hiTAjB , ð1:108Þ μhiAjB ¼ λ*hiAjB

Therefore, ÀÁ μ À λ* hiAjB ¼ 0 ð1:109Þ

Because the eigenvalues μ and λ* are different, the inner product hA|Bi must be zero according to Eq. (1.109). This result indicates that the eigenkets |Ai and |Bi are orthogonal. 3) Hermitian matrix is diagonalizable in the normalized basis. For the eigenvalue problem in Eq. (1.101), there exists an eigenket (eigenvector) relating to its eigenvalue. Instead of the formulation given in Eq. (1.105), the Hermitian matrix can be easily written in the orthonormal basis |eii for the eigenvalue λi: 2 3 2 3 2 3 λ1 000 0 0 6 7 6 7 6 7 6 0 λ 007 6 0 7 6 0 7 λ 6 2 7 6 7 λ 6 7 ... Tejii ¼ ijiei , 4 5 Á 4 5 ¼ i4 5 for i ¼ 1, 2, , N 00λi 0 1 1

000λN 0 0 ð1:110Þ

Therefore, the Hermitian matrix T is obviously diagonalizable at changing the eigenvector basis |Xii into the orthonormal basis |eii. In this case, the eigenvalues locate on the main diagonal and other matrix elements are zero in the Hermitian matrix, as shown in Eq. (1.110)[9].

1.7 Applying Bra and Ket Analysis to Eigenvalue Problems

Many problems in physics and engineering can be formulated similar to X_ ¼ TXji ð1:111Þ

The solution of Eq. (1.111) can be assumed as

λ jiX ¼ jiE e t ð1:112Þ where |Ei is the eigenvector, λ is the complex eigenvalue, λ ¼ α +jω 2 C in which ω is the eigenfrequency and α is the growth/decay rate. Calculating the ﬁrst derivative of the solution, one obtains 32 1 General Basis and Bra-Ket Notation λ X_ ¼ λjiE e t ¼ λjiX ð1:113Þ

Substituting Eq. (1.113) into Eq. (1.111), the eigenvalue problem is given as

TXji¼ λjiX ð1:114Þ

Equation (1.114) can be rewritten in the matrix form with the identity matrix I.

ðÞT À λI jiX ¼ ji0 ð1:115Þ

For nontrivial solutions of Eq. (1.115), the eigenvalue-related determinant must be zero.

detðÞT À λI jjT À λI ¼ 0 ð1:116Þ

Equation (1.116) is called the characteristic equation whose solutions are the eigenvalues. The characteristic equation is the polynomial of λn; n equals two times of the degrees of freedom (DOF) of the system.

n n nÀ1 PðÞλ anλ þ anÀ1λ þ :::: þ a1λ þ a0 ¼ 0 ð1:117Þ

There exists an eigenvector (eigenket) for each eigenvalue. The eigenvector results from Eq. (1.115).

λ ðÞT À λ I jiX ¼ ðÞT À λ I jiE e it ¼ ji0 i i i i ð1:118Þ 8λi 2 C ) ðÞT À λiI jiEi ¼ ji0

An example for the eigenvalue problem will be given in the following subsection. Let T be the system matrix; it can be written as 2 3 10 0 T ¼ 4 00À1 5 01 1

The characteristic equation of the eigenvalues λ yields 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems 33 ðÞ1 À λ 00 jjT À λI ¼ 0 Àλ À1 011 ðÞÀ λ λ λ À À1 λ λλ ¼ ðÞ1 À λ ¼ ðÞ1 À ½ðÞþÀ 1 1 ÀÁ11ðÞÀ ¼ ðÞ1 À λ λ2 À λ þ 1 pffiffiffi pffiffiffi 1 þ j 3 1 À j 3 ¼ ðÞ1 À λ λ À Á λ À ¼ 0 2 2

Thus, there are three eigenvalues as follows:

λ ¼ 1; 1 pffiffiffi 1 3 ; λ ¼ þ j 2 2 2 : pffiffiffi 1 3 λ ¼ À j 3 2 2

Using Eq. (1.118), one obtains the eigenvectors of the eigenvalue problem:

• For λ ¼ λ1 ¼ 1: 2 3 2 3 2 3 00 0 x1 0 4 5 4 5 4 5 0 À1 À1 : x2 ¼ 0 01 0 x3 0

Therefore, 8 0 1 < 0x1 ¼ 0 ! x1 1 1 @ A : Àx2 À x3 ¼ 0 ! x3 ¼Àx2 ¼ 0 ) jiE1 ¼ 0 x2 ¼ 0 ! x2 ¼ 0 0

pffiffi λ λ 1 3 • For ¼ 2 ¼ 2 þ j 2 : 2 pffiffiffi 3 1 3 6 À j 007 2 3 2 3 6 2 2 ffiffiffi 7 6 p 7 x1 0 6 1 3 7 4 5 4 5 6 0 À þ j À1 7 Á x2 ¼ 0 6 2 2 7 4 pffiffiffi 5 x3 0 1 3 01À j 2 2 34 1 General Basis and Bra-Ket Notation

Thus, 8 pffiffiffi > 1 3 > À j x ¼ 0 ! x ¼ 0 > 2 2 1 1 0 1 <> pffiffiffi 0 1 3 @ A > À þ j x2 À x3 ¼ 0 ! x2 2j ) jiE2 ¼ pffiffiffi2j > 22 ffiffiffi > p ffiffiffi 3 À j > 1 3 p : x þ À j x ¼ 0 ! x ¼ 3 À j 2 2 2 3 3

pffiffi λ λ 1 3 • For ¼ 3 ¼ 2 À j 2 : 2 pffiffiffi 3 1 3 6 þ j 007 2 3 2 3 6 2 2 ffiffiffi 7 6 p 7 x1 0 6 1 3 7 4 5 4 5 6 0 À À j À1 7 Á x2 ¼ 0 6 2 2 7 4 pffiffiffi 5 x3 0 1 3 01þ j 2 2

Therefore, 8 pffiffiffi > 1 3 > þ j x ¼ 0 ! x ¼ 0 > 2 2 1 1 0 1 <> pffiffiffi 0 1 3 @ A > À À j x2 À x3 ¼ 0 ! x2 À2j ) jiE3 ¼ pÀffiffiffi2j > 22 ffiffiffi > p ffiffiffi 3 þ j > 1 3 p : x þ þ j x ¼ 0 ! x ¼ 3 þ j: 2 2 2 3 3

References

1. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 2. Nguyen-Scha¨fer, H.: Rotordynamics of Automotive Turbochargers. Springer, Berlin (2012) 3. Kraemer, E.: Rotordynamics of Rotors and Foundations. Springer, Heidelberg (1993) 4. Muszy´nska, A.: Rotordynamics. CRC, Taylor and Francis, London (2005) 5. Vance, J.: Rotordynamics of Turbomachinery. Wiley, New York, NY (1988) 6. Yamamoto, T., Ishida, Y.: Linear and Nonlinear Rotordynamics. Wiley, New York, NY (2001) 7. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford Science Publications, Oxford (1958) 8. Shankar, R.: Principles of Quantum Mechanics, 2nd edn. Springer, Berlin (1980) 9. Grifﬁths, D.J.: Introduction to Quantum Mechanics, 2nd edn. Pearson Prentice Hall, Upper Saddle River, NJ (2005) 10. Longair, M.: Quantum Concepts in Physics. Cambridge University Press, Cambridge (2013) Chapter 2 Tensor Analysis

2.1 Introduction to Tensors

Tensors are a powerful mathematical tool that is used in many areas in engineering and physics including general relativity theory, quantum mechanics, statistical thermodynamics, classical mechanics, electrodynamics, solid mechanics, and ﬂuid dynamics. Laws of physics and physical invariants must be independent of any arbitrarily chosen coordinate system. Tensors describing these characteristics are invariant under coordinate transformations; however, their tensor components heavily depend on the coordinate bases. Therefore, the tensor components change as the coordinate system varies in the considered spaces. Before going into details, we provide less experienced readers with some examples. Different tensors are listed in Table 2.1, which can be expressed in different chosen bases for any curvilinear coordinate. Using Einstein summation convention, the notation can be shortened. Note that Einstein summation convention is only valid for the same indices in the lower and upper positions. The relating contravariant or covariant tensor components can be expressed in the covariant or contravariant bases (cf. Appendix E). The tensor order is determined by the number of the coordinate basis. Thus, the component of a ﬁrst-order tensor has only one dummy index i relating to a single basis. In case of a second-order tensor, its component contains two dummy indices i and j that relate to the double bases. Similarly, the component of an N-order tensor has N dummy indices relating to N bases. The dummy indices (inner indices) are the repeated indices running from the values from 1 to N in Einstein summation convention. The free index (outer index) can be independently chosen for any value from 1 to N; i.e., for any tensor component in the particular coordinate, as shown in the below example. Note that the dimensions of the dummy and free indices must be the same value of the space dimensions.

© Springer-Verlag Berlin Heidelberg 2017 35 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5_2 36 2 Tensor Analysis

Table 2.1 Tensors in general curvilinear coordinates Type Component Basis Tensor First-order Ti, T g , gi ðÞ1 i i i i T ¼ T gi, Tig tensors 2 RN ij i i ðÞ2 ij i j i j j i Second-order T , Tij, T j, gi, g , T ¼ T g g ,Tijg g ,Tjg g , T ig g j j i j i j tensors T i gj, g 2 RN Â RN ijk i ðÞ3 ijk i j k ik j j i k Third-order ten- T , Tijk, gi, g , T ¼ T g g g , Tijkg g g , T :jg g g , T ik:g g g ik j j i j k i k j sors T . j, T ik. gj, g , N N N k 2 R Â R Â R gk, g ... N-order tensors Tijk n, g , gi, ðÞN ijk...n ... ik...n ... j i T ¼ T gigjgk gn, T :jgigk gng N N j 2 R Â ...Â R Tijk ...n, gj, g , ik ...n k T . j, gk, g j T ik ...n. , ... n gn, g

XN XN ij ij ; : T ¼ T gigj T gigj i, j dummy indices j¼1 i¼1 XN j ij ij ; : : T ¼ T gi T gi i dummy, j free index i¼1

2.2 Deﬁnition of Tensors

The definition of tensors is based on multilinear algebra by a multilinear map. We consider the linear vector space L and its dual vector space L*. Each of the vector spaces belongs to the finite N-dimensional space RN, the image space W, to the real space R. A mixed tensor of type (m, n) is a multilinear functional T which maps an (m + n) tuple of vectors of the vector spaces L and L* into the real space W [1] (cf. Fig. 2.1): ÀÁ T m T : ðÞL ÂÁÁÁÂL Â L* ÂÁÁÁÂL* ! W n |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} n copies m copies N N N N R|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}ÂÁÁÁÂR Â R|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}ÂÁÁÁÂR ! R n copies m copies

T : ðÞ!u1; ...; un; v1; ...; vm TuðÞ21; ...; un; v1; ...; vm R ð2:1Þ

im Mapping the covariant basis {gjn}2 L and contravariant basis {g }2 L* by the multilinear functional T of the tensor type (m, n), one obtains its images in the real 2.2 Deﬁnition of Tensors 37

T(u ,...,u ; v ,..., v ) (u1,...,un ; v1,..., vm ) 1 n 1 m

T : (L×...× L)×(L* ×...× L* ) W ⊂ R ⊂ RN×...×RN × RN×...×RN n copies m copies

Fig. 2.1 Multilinear mapping functional T space W R. The real images are called the components of the (m + n)-order mixed tensor T with respect to the relating bases: ; ...; ; i1; ...; im i1...im : Tgj1 gjn g g Tj1...jn 2 W R ð2 2Þ

m Thus, the (m + n)-order mixed tensor T of type (m, n) 2 Tn (L) can be expressed in the covariant and contravariant bases. In total, the (m + n)-order tensor T has Nm+n components.

i1...im ... j1 ... jn m : T ¼ Tj1...jn gi1 gimg g 2 Tn ðÞL ð2 3Þ

m where Tn (L) is called the tensor space that consists of all tensors T of type (m, n): ÀÁ T mðÞ¼L ðÞL ÂÁÁÁÂL Â L* ÂÁÁÁÂL* n |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} m copies n copies

In case of covariant and contravariant tensors T, only the basis of the dual vector space L* or real vector space L are respectively considered in Eq. (2.3).

• n-order covariant tensors of type (0, n) in the tensor space Tn(L): ÀÁ j1 jn * * T ¼ T ... g ... g 2 T ðÞ¼L L ÂÁÁÁÂL ð2:4Þ j1 jn n |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} n copies

• m-order contravariant tensors of type (m, 0) in the tensor space Tm(L):

i1...im ... m : T ¼ T gi1 gim 2 T ðÞ¼L |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}ðÞL ÂÁÁÁÂL ð2 5Þ m copies 38 2 Tensor Analysis

2.2.1 An Example of a Second-Order Covariant Tensor

An arbitrary vector v can be expressed in the covariant basis gk in the N-dimensional vector space V as

k ... : v ¼ v gk for k ¼ 1, 2, , N ð2 6Þ

Applying the bilinear mapping T to the vector v and using the Kronecker delta, one obtains its mapping image Tv. Straightforwardly, this is a tensor of one lower order compared to the mapping tensor T. ÀÁ i j k Tv Tijg g Á v gk k i j ¼ Tijv ðÞg Á gk g k i j ¼ Tijv δ g for i ¼ k ð2:7aÞ ÀÁk k j ¼ Tkjv g for j, k ¼ 1, 2, ..., N * j ... Tj g for j ¼ 1, 2, , N whereas the second-order covariant tensor T can be expressed as

i j T ¼ Tijg g for i, j ¼ 1, 2, ..., N; T 2 RN Â RN

Note that in case of a three-dimensional vector space R3 (N ¼ 3), there are nine covariant components Tij. The number of the tensor components can be calculated by Nn (32 ¼ 9), in which n is the number of indices i and j (n ¼ 2). Obviously, that the mapping image Tv is also a tensor of one lower order * compared to the tensor T. The covariant tensor component Tj can be calculated by

* k : : Tj Tkjv ¼ gj Á Tv ð2 7bÞ

2.3 Tensor Algebra

2.3.1 General Bases in General Curvilinear Coordinates

The vector r can be written in Cartesian coordinates of Euclidean space E3,as displayed in Fig. 2.2. 2.3 Tensor Algebra 39

u3 g3

g3 x3 g1

P g2 e 3 g2 u2 1 0 u g1 e e1 2 x2 x1

Fig. 2.2 Bases of general curvilinear coordinates in the space E3

i r ¼ x ei ð2:8Þ

The differential dr results from Eq. (2.8)in

∂r dr ¼ e dxi ¼ dxi ð2:9Þ i ∂xi

i Using the chain rule of differentiation, the orthonormal bases ei of the coordinates x are deﬁned by

∂r ∂r ∂uj e ¼ i ∂ i ∂ j ∂ i x u x ð2:10Þ ∂uj g for j ¼ 1, 2, ..., N j ∂xi

Analogously, the bases of the curvilinear coordinates ui can be calculated in the curvilinear coordinate system of EN

∂r ∂r ∂xk g ¼ j ∂ j ∂ k ∂ j u x u ð2:11Þ ∂xk ¼ e for k ¼ 1, 2, ..., N k ∂uj

The curvilinear coordinates ui are functions of the coordinates xi, the covariant bases in Eq. (2.11) can be calculated using the chain rule of differentiation. 40 2 Tensor Analysis

∂r ∂r ∂xi g ¼ ¼ j ∂uj ∂xi ∂uj ∂xi ð2:12Þ ¼ e i ∂uj i ... eixj for i ¼ 1, 2, , N

Thus, the curvilinear basis gj can be written in a linear combination of the ortho- i normal basis ei according to Eq. (2.12). The derivative xj is called the shift tensor between the orthonormal and curvilinear coordinates. i Generally, the basis gi of the curvilinear coordinate u can be rewritten in a linear j j combination of the basis g’j of other curvilinear coordinate u’ . The derivative u’i is deﬁned as the shift tensor between both curvilinear coordinates.

0 j ∂r ∂u g ¼ i 0 j ∂ i ∂u u 2:13 0 j ð Þ 0 ∂u 0 0 j ¼ g g u for j ¼ 1, 2, ..., N j ∂ui j i

In the curvilinear coordinate system (u1,u2,u3) of Euclidean space E3, its basis is generally non-orthogonal and non-unitary (non-orthonormal basis); i.e., the bases are not mutually perpendicular and their vector lengths are not equal to one [2– 4]. In this case, the curvilinear coordinate system (u1,u2,u3) has three covariant 1 2 3 bases g1, g2, and g3 and three contravariant bases g , g , and g at the origin P, as shown in Fig. 2.2. Generally, the origin P of the curvilinear coordinates could move everywhere in Euclidean space. Therefore, the bases of the curvilinear coordinates only depend on the respective origin P. For this reason, the curvilinear bases are not ﬁxed in the whole curvilinear coordinates like in Cartesian coordinates. The vector r of the point P(u1,u2,u3) can be written in covariant and contravariant bases.

1 2 3 r ¼ u g1 þ u g2 þ u g3 : 1 2 3 ð2 14Þ ¼ u1g þ u2g þ u3g where u1, u2, u3 are the contravariant vector components of the coordinates (u1,u2,u3); 1 2 3 g1, g2, g3 are the covariant bases of the coordinate system (u ,u ,u ); 1 2 3 u1, u2, u3 are the covariant vector components of the coordinates (u ,u ,u ); g1, g2, g3 are the contravariant bases of the coordinate system (u1,u2,u3). The covariant base gi is deﬁned by the tangential vector to the corresponding i curvilinear coordinate u for i ¼ 1, 2, 3. Both bases g1 and g2 generates a tangential surface to the curvilinear surface (u1u2) at the considered origin P. Note that the basis g1 is not perpendicular to the bases g2 and g3. However, the contravariant basis 3 g is perpendicular to the tangential surface (g1g2) at the origin P. Generally, the 2.3 Tensor Algebra 41 contravariant basis (gk) results from the cross product of the other covariant bases (gi Â gj).

α k : g ¼ gi Â gj for i, j, k ¼ 1, 2, 3 ð2 15Þ where α is a scalar factor. Multiplying Eq. (2.15) by the covariant basis gk, the scalar factor α can be calculated as ÀÁ α k αδk α g Á gk ¼ k ¼ ¼ gi Â gj Á gk hi ð2:16Þ α ; ; ) ¼ gi Â gj Á gk gi gj gk

The scalar factor α equals the scalar triple product that is given in [3]: α ; ; ½¼g1 g2 g3 gi Â gj Á gk ¼ ðÞÁgk Â gi gj ¼ gj Â gk Á gi 1 1 1 g g g 2 g g g 2 g g g 2 11 12 13 31 32 33 21 22 23 ¼ g21 g22 g23 ¼ g11 g12 g13 ¼ g31 g32 g33 ð2:17Þ rg31ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig32 g33 g21 g22 g23 g11 g12 g13 ffiffiffi p > ¼ det gij g ¼ J 0 where J is defined as the Jacobian, as given in 1 1 1 ∂x ∂x ∂x ∂u1 ∂u2 ∂u3 ∂xi ∂xj ∂xk ∂x2 ∂x2 ∂x2 J ε ¼ ð2:18Þ ijk ∂ 1 ∂ 2 ∂ 3 u u u ∂u1 ∂u2 ∂u3 ∂ 3 ∂ 3 ∂ 3 x x x ∂u1 ∂u2 ∂u3

Thus, εijk εijk gk ¼ pffiffiffi g Â g ¼ g Â g ð2:19Þ g i j J i j where εijk is the Levi-Civita permutation symbols in Eq. (A.5), cf. Appendix A. According to Eq. (2.19), the contravariant basis gk is perpendicular to both k covariant bases gi and gj. Additionally, the contravariant basis g is chosen such that the vector length of the contravariant basis equals the inversed vector length of its relating covariant basis. Therefore, 42 2 Tensor Analysis

gi Â gj Á gi gk Á g ¼ hi¼ δ k ð2:20Þ i ; ; i gi gj gk

As a result, the relation between the contravariant and covariant bases is given in the general curvilinear coordinate system (u1,...,uN). & k k δ k ... gi Á g ¼ g Á gi ¼ i for i, k ¼ 1, 2, , N : δ k ... ð2 21Þ gi Á gk ¼ gk Á gi 6¼ i for i, k ¼ 1, 2, , N

j The basis gi is called dual to the basis g [5]if

j δ j ... : gi Á g ¼ i for i, j ¼ 1, 2, , N ð2 22Þ

j where δi is the Kronecker delta. i Let {g1, g2,...,gN} be a covariant basis of the curvilinear coordinates {u }. The contravariant basis {g1, g2,...,gN}, the dual basis to the covariant basis, can be written in the matrix formulation as 2 3 g1 6 7 6 g2 7 À1 6 7 À1 G ¼ ½g g : g g ; G ¼ 6 : 7 ) G G ¼ I ð2:23Þ 1 2 i N 4 5 g j gN

j À1 where g is the j row vector of G ; gi is the i column vector of G. The covariant and contravariant bases (dual bases) of the orthogonal cylindrical and spherical coordinates are computed in the following section.

2.3.1.1 Orthogonal Cylindrical Coordinates

The cylindrical coordinates (r,θ,z) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 2.3 shows a point P in the cylindrical coordinates (r,θ,z) embedded in the orthonormal Cartesian coordinates (x1,x2,x3). However, the cylindrical coordinates change as the point P varies. The vector OP can be written in Cartesian coordinates (x1,x2,x3):

θ θ R ¼ ðÞr cos e1 þ ðÞr sin e2 þ z e3 : 1 2 3 ð2 24Þ x e1 þ x e2 þ x e3 where e1, e2, and e3 are the orthonormal bases of Cartesian coordinates; θ is the polar angle. 2.3 Tensor Algebra 43

x3 (r,θ,z) → (u1,u2,u3): u1 ≡ r ; u2 ≡ θ ; u3 ≡ z

z P g2

R e3 g1 x2 0 e2 e1 θ r

Fig. 2.3 Covariant bases of orthogonal cylindrical coordinates

To simplify the formulation with Einstein symbol, the coordinates of u1, u2, and u3 are used for r, θ, and z, respectively. Therefore, the coordinates of P(u1,u2,u3) are given in Cartesian coordinates: 8 9 ÀÁ< x1 ¼ r cos θ u1 cos u2 = 1; 2; 3 2 θ 1 2 : Pu u u ¼ : x ¼ r sin u sin u ; ð2 25Þ x3 ¼ z u3

The covariant bases of the curvilinear coordinates are computed from

∂R ∂R ∂xj ∂xj g ¼ ¼ Á ¼ e for j ¼ 1, 2, 3 ð2:26Þ i ∂ui ∂xj ∂ui j ∂ui

The covariant basis matrix G yields from Eq. (2.26):

G ¼ 0½g1 g2 g3 1 ∂x1 ∂x1 ∂x1 0 1 B ∂ 1 ∂ 2 ∂ 3 C B u u u C cos θ Àr sin θ 0 B ∂x2 ∂x2 ∂x2 C ð2:27Þ ¼ B C ¼ @ sin θ r cos θ 0 A B ∂ 1 ∂ 2 ∂ 3 C @ u u u A 001 ∂x3 ∂x3 ∂x3 ∂u1 ∂u2 ∂u3

The determinant of G is called the Jacobian J. 44 2 Tensor Analysis 1 1 1 ∂x ∂x ∂x ∂u1 ∂u2 ∂u3 cos θ r sin θ 0 ∂ 2 ∂ 2 ∂ 2 À x x x θ θ : jjG J ¼ ¼ sin r cos 0 ¼ r ð2 28Þ ∂u1 ∂u2 ∂u3 ∂ 3 ∂ 3 ∂ 3 001 x x x ∂u1 ∂u2 ∂u3

The relation between the covariant and contravariant bases yields from Eq. (2.22):

i δ i : g Á gj ¼ j ðÞKronecker delta ð2 29Þ

Thus, the contravariant basis matrix GÀ1 results from the inversion of the covariant basis matrix G, as given in Eq. (2.27). 2 3 0 1 1 r cos θ r sin θ 0 g 1 GÀ1 ¼ 4 g2 5 ¼ @ À sin θ cos θ 0 A ð2:30Þ r g3 00r

The calculation of the determinant and inversion matrix of G will be discussed in the following section. According to Eq. (2.27), the covariant bases can be denoted as 8 θ θ < g1 ¼ ðÞcos e1 þ ðÞsin e2 þ 0 Á e3 ) jjg1 ¼ 1 θ θ : : g2 ¼ÀðÞr sin e1 þ ðÞr cos e2 þ 0 Á e3 ) jjg2 ¼ r ð2 31Þ g3 ¼ 0 Á e1 þ 0 Á e2 þ 1 Á e3 ) jjg3 ¼ 1

The contravariant bases result from Eq. (2.30). 8 > g1 ¼ ðÞcos θ e þ ðÞsin θ e þ 0 Á e ) g1 ¼ 1 <> 1 2 3 sin θ cos θ 1 2 2 : > g ¼À e1 þ e2 þ 0 Á e3 ) g ¼ ð2 32Þ :> r r r 3 3 g ¼ 0 Á e1 þ 0 Á e2 þ 1 Á e3 ) g ¼ 1

Not only the covariant bases but also the contravariant bases of the cylindrical coordinates are orthogonal due to

j j δ j gi Á g ¼ g Á gi ¼ i ; gi Á gj ¼ 0for i 6¼ j gi Á g j ¼ 0for i 6¼ j: 2.3 Tensor Algebra 45

x3 (ρ,ϕ,θ ) → (u1,u2,u3): ρ sinϕ u1 ≡ ρ ; u2 ≡ ϕ ; u3 ≡ θ

g P 3

e3 ϕ ρ e g2 0 2 x2 e1 θ ρ cosϕ x1

Fig. 2.4 Covariant bases of orthogonal spherical coordinates

2.3.1.2 Orthogonal Spherical Coordinates

The spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. The spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 2.4 shows a point P in the spherical coordinates (r,θ,z) embedded in the orthonormal Cartesian coordinates (x1,x2,x3). However, the spherical coordinates change as the point P varies. The vector OP can be written in Cartesian coordinates (x1,x2,x3):

ρ φ θ ρ φ θ ρ φ R ¼ ðÞsin cos e1 þ ðÞsin sin e2 þ cos e3 : 1 2 3 ð2 33Þ x e1 þ x e2 þ x e3 where e1, e2, and e3 are the orthonormal bases of Cartesian coordinates; φ is the equatorial angle; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u1, u2, and u3 are used for ρ, φ, and θ, respectively. Therefore, the coordinates of P(u1,u2,u3) are given in Cartesian coordinates: 46 2 Tensor Analysis 8 9 ÀÁ< x1 ¼ ρ sin φ cos θ u1 sin u2 cos u3 = 1; 2; 3 2 ρ φ θ 1 2 3 : Pu u u ¼ : x ¼ sin sin u sin u cos u ; ð2 34Þ x3 ¼ ρ cos φ u1 cos u2

The covariant bases of the curvilinear coordinates are computed from

∂R ∂R ∂xj ∂xj g ¼ ¼ Á ¼ e for j ¼ 1, 2, 3 ð2:35Þ i ∂ui ∂xj ∂ui j ∂ui

Thus, the covariant basis matrix G can be calculated from Eq. (2.35). 0 1 ∂x1 ∂x1 ∂x1 B C B ∂u1 ∂u2 ∂u3 C B ∂ 2 ∂ 2 ∂ 2 C B x x x C G ¼ ½¼g1 g2 g3 B C @ ∂u1 ∂u2 ∂u3 A ∂x3 ∂x3 ∂x3 ð2:36Þ 0 ∂u1 ∂u2 ∂u3 1 sin φ cos θρcos φ cos θ Àρ sin φ sin θ ¼ @ sin φ sin θρcos φ sin θρsin φ cos θ A cos φ Àρ sin φ 0

The determinant of the covariant basis matrix G is called the Jacobian J. sin φ cos θρcos φ cos θ Àρ sin φ sin θ jjG J ¼ sin φ sin θρcos φ sin θρsin φ cos θ ð2:37Þ cos φ Àρ sin φ 0 ¼ ρ2 sin φ

Similarly, the contravariant basis matrix GÀ1 is the inversion of the covariant basis matrix G. 0 1 2 3 ρ φ θρ φ θρ φ 1 sin cos sin sin cos g 1 B φ θ φ θ φ C À1 4 2 5 B coscos cossin À sin C : G ¼ g ¼ ρ @ sin θ cos θ A ð2 38Þ g3 À 0 sin φ sin φ

The matrix product GGÀ1 must be an identity matrix according to Eq. (2.23). 2.3 Tensor Algebra 47 0 1 ρ sin φ cos θρsin φ sin θρcos φ B φ θ φ θ φ C À1 1B coscos cossin À sin C G G ¼ ρ@ sin θ cos θ A À 0 0 sin φ sin φ 1 0 1 ð2:39Þ sin φ cos θρcos φ cos θ Àρ sin φ sin θ 100 Á@ sin φ sin θρcos φ sin θρsin φ cos θ A ¼ @ 010A I cos φ Àρ sin φ 0 001

According to Eq. (2.36), the covariant bases can be written as

φ θ φ θ φ g1 ¼ ðÞsin cos e1 þ ðÞsin sin e2 þ cos e3 ) jjg1 ¼ 1 ρ φ θ ρ φ θ ρ φ ρ : g2 ¼ ðÞcos cos e1 þ ðÞcos sin e2 À ðÞsin e3 ) jjg2 ¼ ð2 40Þ ρ φ θ ρ φ θ ρ φ g3 ¼ÀðÞsin sin e1 þ ðÞsin cos e2 þ 0 Á e3 ) jjg3 ¼ sin

The contravariant bases result from Eq. (2.38). 1 φ θ φ θ φ 1 g ¼ ðÞsin cos e1 þ ðÞsinsin e2 þ cos e3 ) g ¼ 1 2 1 1 1 2 1 g ¼ cos φ cos θ e1 þ cos φ sin θ e2 À sin φ e3 ) g ¼ ρρ ρ ρ 1 sin θ 1 cos θ 1 g3 ¼À e þ e þ 0 Á e ) g3 ¼ ρ sin φ 1 ρ sin φ 2 3 ρ sin φ ð2:41Þ

Not only the covariant bases but also the contravariant bases of the spherical coordinates are orthogonal due to

j j δ j; gi Á g ¼ g Á gi ¼ i ; gi Á gj ¼ 0fori 6¼ j gi Á g j ¼ 0fori 6¼ j:

2.3.2 Metric Coefﬁcients in General Curvilinear Coordinates

1 The covariant basis vectors g1, g2, and g3 to the general curvilinear coordinates (u , 2 3 u ,u ) at the point P can be calculated from the orthonormal bases (e1,e2,e3)in Cartesian coordinates xj ¼ xj(ui), as shown in Fig. 2.2. 48 2 Tensor Analysis

∂r ∂r ∂xk g ¼ i ∂ i ∂ k ∂ i u x u ð2:42Þ ∂xk ¼ e for k ¼ 1, 2, 3 k ∂ui

The covariant metric coefﬁcients gij are deﬁned as

∂r ∂r g g Á g ¼ ¼ g Á g g ij i j ∂ui ∂uj j i ji ∂xk ∂xl ∂xk ∂xl ¼ e e ¼ δ ð2:43Þ ∂ui ∂uj k l ∂ui ∂uj kl ∂xk ∂xk ¼ ∂ui ∂uj

Similarly, the contravariant metric coefﬁcients gij can be denoted as

gij gi Á g j ¼ g j Á gi ¼ gji ð2:44Þ

Furthermore, the contravariant basis can be rewritten as a linear combination of the covariant bases.

i ij : g ¼ A gj ð2 45Þ

According to Eq. (2.44) and using Eqs. (2.21) and (2.45), the contravariant metric coefﬁcients can be expressed as

gik i k Aij k g Á g ¼ gj Á g : ijδ k ik ð2 46Þ ¼ A j ¼ A

Thus,

i ij : g ¼ g gj for j ¼ 1, 2, 3 ð2 47Þ

Analogously, one obtains the covariant basis

l : gk ¼ gklg for l ¼ 1, 2, 3 ð2 48Þ

The mixed metric coefﬁcients can be deﬁned by 2.3 Tensor Algebra 49 i i ij g g Á gk ¼ g gj Á gk k ÀÁ ij l ij l ¼ g gj Á gklg ¼ g gkl gj Á g ð2:49aÞ ij δ l ij ¼ g gkl j ¼ g gkj δ i ¼ k

Thus,

ij ij δ i : g gkj ¼ gkjg ¼ k ð2 49bÞ

Therefore, the contravariant metric tensor is the inverse of the covariant metric tensor.

ij ij δ i À1 À1 : g gkj ¼ gkjg ¼ k , M M ¼ MM ¼ I ð2 50Þ where MÀ1 and M are the contravariant and covariant metric tensors. Thus, 2 3 2 3 11 12 : 1N : g g g g11 g12 g1N 6 21 22 : 2N 7 6 : 7 À1 6 g g g 7 6 g21 g22 g2N 7 M M ¼ 4 ::ij : 5 Á 4 :: : 5 g gij N1 N2 : NN : g g g2 gN1 3gN2 gNN : : ð2 51Þ 10 0 ÀÁ 6 01: 0 7 ¼ gijg ¼ δ i ¼ 6 7 ¼ I kj k 4 0 : 105 00: 1

According to Eqs. (2.42) and (2.49a), the contravariant bases of the curvilinear coordinates can be derived as

∂xj δ k gk Á g ¼ gk Á e ) i i ∂ui j ∂xj ∂xj δ k Á e ¼ gk Á e Á e ¼ gk Á ) ð2:52Þ i j ∂ui j j ∂ui ∂ui ∂uk gk ¼ δ k e ¼ e i ∂xj j ∂xj j

Thus,

∂uj g j ¼ e for i ¼ 1, 2, 3 ð2:53Þ ∂xi i 50 2 Tensor Analysis

Generally, the covariant and contravariant metric coefﬁcients of the general curvilinear coordinates have the following properties: 8 > δ j : < gji ¼ gij ¼ gi Á gj 6¼ i ðÞcov metric coefficient gji ¼ gij ¼ gi Á g j 6¼ δ j ðÞcontrav: metric coefficient ð2:54Þ :> i j j δ j gi ¼ gi Á g ¼ i ðÞmixed metric coefficient

Using Eqs. (2.42) and (2.53), one obtains the Kronecker delta

∂xk ∂uj ∂xk ∂uj δ j ¼ g Á g j ¼ e Á e ¼ δ l i i ∂ui ∂xl k l ∂ui ∂xl k ð2:55aÞ ∂xk ∂uj ∂uj ¼ ¼ ∂ui ∂xk ∂ui

The Kronecker delta is deﬁned by & 0fori 6¼ j δ j ð2:55bÞ i 1fori ¼ j

As an example, the covariant metric tensor M in the cylindrical coordinates results from Eq. (2.31). 2 3 2 3 g11 g12 g13 100 4 5 4 2 5 : M ¼ g21 g22 g23 ¼ 0 r 0 ð2 56aÞ g31 g32 g33 001

The contravariant metric coefﬁcients in the contravariant metric tensor MÀ1 are calculated from inverting the covariant metric tensor M. 2 3 2 3 g11 g12 g13 100 MÀ1 ¼ 4 g21 g22 g23 5 ¼ 4 0 rÀ2 0 5 ð2:56bÞ g31 g32 g33 001

Analogously, the covariant metric tensor M in the spherical coordinates results from Eq. (2.40). 2 3 2 3 g11 g12 g13 10 0 4 5 4 ρ2 5 : M ¼ g21 g22 g23 ¼ 0 0 ð2 57aÞ ρ φ 2 g31 g32 g33 00ðÞsin

The contravariant metric coefﬁcients in the contravariant metric tensor MÀ1 are calculated from inverting the covariant metric tensor M. 2.3 Tensor Algebra 51 2 3 2 3 g11 g12 g13 10 0 MÀ1 ¼ 4 g21 g22 g23 5 ¼ 4 0 ρÀ2 0 5 ð2:57bÞ g31 g32 g33 00ðÞρ sin φ À2

2.3.3 Tensors of Second Order and Higher Orders

Mapping an arbitrary vector x 2 RN by a linear functional T, one obtains its image vector y ¼ Tx [2, 3].

T : RN ! RN T : x ! y ¼ Tx T Á xjg ¼ xjT Á g j j ð2:58Þ j i k j δ k i j i ¼ x Tikg g Á gj ¼ x Tik j g ¼ Tijx g where T is a second-order tensor 2 RNÂ RN. It is obvious that the image vector y ¼ Tx is a tensor of one lower order compared to the tensor T that can be considered as a linear operator. The second-order tensor T can be generated from the tensor product (dyadic product) of two vectors u and v, as denoted in Eq. (2.60). Let u and v be two arbitrary vectors. They can be written in the covariant and contravariant bases as & i i N; u ¼ u gi ¼ uig 2 R : j j N ð2 59Þ v ¼ v gj ¼ vjg 2 R

The tensor product of two vectors u and v results in the second-order tensor T.

T : u, v 2 RN ! T u v 2 RN Â RN i j i j ij : T ¼ u v gi gj u v gigj T gigj ð2 60Þ i j i j i j T ¼ uivjg g uivjg g Tijg g

i j Note that the terms gigj and g g are called the covariant and contravariant basis tensors. Hence, they are not the same notations as the covariant and contravariant ij metric coefﬁcients gij and g , respectively.

gigj 6¼ gi Á gj gij gig j 6¼ gi Á g j gij

Similarly, one obtains the properties of the mixed basis tensors: 52 2 Tensor Analysis

i i i δ i g gj 6¼ g Á gj gj ¼ j j j j δ j gig 6¼ gi Á g gi ¼ i

ij Each of the covariant and contravariant tensor components Tij and T contains nine independent elements (N2 ¼ 9) 2 R3 Â R3 in a three-dimensional space (N ¼ 3).

Tij ¼ uivj; ð2:61Þ Tij ¼ uivj

An example of the tensor product of two contravariant vectors u and v in a three-dimensional space R3 is given. 0 1 0 1 u1 v1 T ¼ u v ¼ @ u2 A @ v2 A 0 u3 v3 1 0 1 u1v1 u1v2 u1v3 T11 T12 T13 @ A @ A ¼ u2v1 u2v2 u2v3 ¼ T21 T22 T23 ; 8ui, vj, Tij 2 R u3v1 u3v2 u3v3 T31 T32 T33

The second-order tensor T consists of nine tensor components (N2 ¼ 32) 2 R3 Â R3 in a three-dimensional space R3 (N ¼ 3) with two indices i and j. The basis gj of the general curvilinear coordinates is mapped by the linear functional T in Eq. (2.58) into the image vector Tj that can be written according to Eq. (2.2)as

: Tj T Á gj ð2 62Þ

i Each vector Tj can be expressed in a linear combination of the contravariant basis g as

i Tj ¼ Tijg ð2:63Þ where Tij is the covariant tensor component of the second-order tensor T. Multiplying Eq. (2.63) by the covariant basis gi and using Eq. (2.62), the covariant tensor component Tij results in ÀÁ T gi Á g ¼ T Á g Á g ¼ g Á T Á g ij i j i i j ð2:64Þ δ i ) Tij i ¼ Tij ¼ gi Á T Á gj

Equation (2.64) can be written in the contravariant bases gi and gj as follows: 2.3 Tensor Algebra 53

i j i j TijðÞ¼g g gi Á T Á gjðÞg g i j δ i δ j : ¼ ðÞgi Á g TgÁ gj ¼ i T j ¼ T ð2 65Þ i j ) T ¼ Tijg g

Similarly, the vector Tj is formulated in a linear combination of the covariant basis gi.

j j ij : T ¼ T Á g ¼ T gi ð2 66Þ in which Tij is the contravariant component of the second-order tensor T. Multiplying Eq. (2.66) by the contravariant basis gi, the contravariant tensor component Tij can be computed as ÀÁ ij i j i i j T gi Á g ¼ ðÞÁT Á g g ¼ g Á T Á g : ijδ i ij i j ð2 67Þ ) T i ¼ T ¼ g Á T Á g

Similarly, Eq. (2.67) can be written in the covariant bases gi and gj Tij g g ¼ gi:T:gj g g i j i j i: : j δ i δ j ð2:68Þ ¼ ðÞg gi Tgj g ¼ i T j ¼ T ij ) T ¼ T gigj

Alternatively, the vector component can be rewritten in a linear combination of the mixed tensor component.

j j j i : T ¼ T Á g ¼ Ti:g ð2 69Þ

j Multiplying Eq. (2.69) by the covariant basis gi, the mixed tensor component Ti. can be calculated as T j gi Á g ¼ ðÞÁT Á g j g ¼ g Á T Á g j i: i i i ð2:70Þ j δ i j j ) Ti: i ¼ Ti: ¼ gi Á T Á g

Note that in Eq. (2.70), the dot after the lower index indicates the position of the basis of the upper index locating after the tensor T. In this case, the tensor T is j located between the lower basis gi and upper basis g [4–6]. i Equation (2.70) can be written in the covariant and contravariant bases gj and g as follows: 54 2 Tensor Analysis T j gig ¼ g Á T Á g j gig i: j i j i j δ i δ j ð2:71Þ ¼ ðÞgi Á g Tgj Á g ¼ i T j ¼ T j i ) T ¼ Ti:g gj

i Analogously, one obtains the mixed tensor component T.j . T i g j Á g ¼ ðÞÁT Á gi g ¼ T Á gi Á g :j j j j i i ð2:72Þ ¼ T Á gj Á g ¼ g Á T Á gj i δ j i i ) T:j j ¼ T:j ¼ g Á T Á gj

Note that in Eq. (2.72), the dot before the lower index indicates the position of the basis of the upper index locating in front of the tensor T. In this case, the tensor T is i located between the upper basis g and lower basis gj [4–6]. j Equation (2.72) can be written in the covariant and contravariant bases gi and g as follows:

i j i j T:jðÞ¼gig g Á T Á gjðÞgig i j δ i δ j : ¼ ðÞg Á gi TgÁ gj ¼ i T j ¼ T ð2 73Þ i j ) T ¼ T:jgig

Brieﬂy, the second-order tensor can be written in different expressions according to the covariant, contravariant, and mixed components. 8 i j; > Tijg g Tij ¼ g Á T Á g <> i j Tijg g ; Tij ¼ gi Á T Á g j TðÞ2 ¼ i j ð2:74Þ > T j gig ; T j ¼ g Á T Á g j :> i: j i: i i j; i i T:jgig T:j ¼ g Á T Á gj

Note that if the second-order tensor T is symmetric, then

; ij ji; j j ; i i : Tij ¼ Tji T ¼ T Ti: ¼ T:i T:j ¼ Tj: ð2 75Þ

Compared to the second-order tensors, the ﬁrst-order tensor T(1) has only one dummy index, as shown in & i; ðÞ1 Tig Ti ¼ T Á gi : T ¼ i ; i i ð2 76Þ T gi T ¼ T Á g

An N-order tensor T(N ) is the tensor product of the N covariant, contravariant, and mixed bases of the coordinates: 2.3 Tensor Algebra 55 8 ij...n ... < T gigj gn ðÞN i j ... n : T ¼ : Tij...ng g g ð2 77Þ ij:: ... l ... n Tl...ngigj g g

The N-order tensors contain the 2N expressions in total. Two of them are in respect of the covariant and contravariant tensor components; and (2N À 2) expressions, in respect of the mixed tensor components [3]. In case of a second-order tensor T(2) for N ¼ 2, there are four expressions: two with the covariant and contravariant tensor components and two with the mixed tensor components, as displayed in Eq. (2.74).

2.3.4 Tensor and Cross Products of Two Vectors in General Bases

2.3.4.1 Tensor Product

Let u and v be two arbitrary vectors in the ﬁnite N-dimensional vector space RN. They can be written in the covariant and contravariant bases as & i i N; u ¼ u gi ¼ uig 2 R : j j N ð2 78Þ v ¼ v gj ¼ vjg 2 R

The tensor product T of two vectors generates a second-order tensor that can be deﬁned by the linear functional T. • In the covariant bases:

: N i j i j N N T u, v 2 R ! T u v ¼ u v gigj ¼ uivjg g 2 R Â R i j ij : T ¼ u v gigj T gigj ð2 79Þ i j i j ¼ uivjg g Tijg g where Tij is the contravariant component of the second-order tensor T; Tij is the covariant component of the second-order tensor T; • In the covariant and contravariant bases:

: N i j j i N N T u, v 2 R ! T u v ¼ u vjgig ¼ uiv g gj 2 R Â R i j i j : T ¼ u vjgig Tj gig ð2 80Þ j i j i ¼ uiv g gj Ti g gj

i j where Tj and Ti are the mixed components of the second-order tensor T. 56 2 Tensor Analysis

The tensor product T of two vectors in an orthonormal basis (e.g. Euclidean coordinate system) is an invariant (scalar). The invariant is independent of the coordinate system and has an intrinsic value in any coordinate transformations. In Newtonian mechanics, the mechanical work W that is created by the force vector F and path vector x does not change in any chosen coordinate system. This mechanical work W ¼ F Á x is called an invariant and has an intrinsic value of energy. Given three arbitrary vectors u, v, and w in RN and a scalar α in R, the tensor product of two vectors has the following properties [3]: • Distributive property

uvðÞ¼þ w uv þ uw ðÞu þ v w ¼ uw þ vw

• Associative property

ðÞα u v ¼ uðÞ¼α v α uv

2.3.4.2 Cross Product

The cross product of two vectors u and v can be deﬁned by a linear functional T. : N i j N : T u, v 2 R ! T ¼ u Â v ¼ u v gi Â gj 2 R ð2 81Þ

Obviously, the cross product T of two vectors is a vector (first-order tensor) of which the direction is perpendicular to the bases of gi and gj. Using the scalar triple product in Eq. (1.10), the cross product of the bases can be written as ffiffiffiffi ε p k ε k : gi Â gj ¼ ijk g g ¼ ijkJ g ð2 82Þ

where εijk is the Levi-Civita permutation symbol; J is the Jacobian. Thus, the cross product in Eq. (2.81) can be expressed as

k T ¼ ÀÁu Â v ffiffiffi Tkg ÀÁ : p i j k i j k ð2 83Þ ¼ εijk g u v g ¼ εijkJuv g

The covariant component Tk of the first-order tensor T results from Eqs. (2.81)to (2.83). 2.3 Tensor Algebra 57 i j i j k k T ¼ u v gi Â gj ¼ u v gi Â gj Á gk g Tkg ffiffiffi ð2:84Þ i j ε p i j ε i j ) Tk ¼ u v gi Â gj Á gk ¼ ijk g u v ¼ ijkJuv

The Levi-Civita permutation symbol (pseudo-tensor) can be deﬁned as 8 < þ1ifðÞi, j, k is an even permutation; ε ; : ijk ¼ : À1ifðÞi, j, k is an odd permutation ð2 85Þ 0ifi ¼ j,ori ¼ k; or j ¼ k

Therefore, & εijk ¼ εjki ¼ εkij ðÞeven permutation ; εijk ¼ ð2:86Þ Àεikj ¼Àεkji ¼Àεjik ðÞodd permutation

n 3 The permutation symbol εijk contains totally 27 elements (N ¼ 3 ) for i, j, k (n ¼ 3) in a 27-dimensional tensor space R3 Â R3 Â R3. Note that the permutation symbol is used in Eq. (2.83) because the direction of the cross-product vector is opposite if the dummy indices are interchanged with each other in Einstein summation convention (cf. Appendix A). pffiffiffi g gk ¼ J gk ¼ g Â g ¼À g Â g i j j i ε ε ijk gi Â gj ijk gi Â gj : ) gk ¼ pffiffiffi ¼ ð2 87Þ g J ffiffiffi ε p k ε k ) gi Â gj ¼ ijk gg ¼ ijkJg where J denotes the Jacobian.

2.3.5 Rules of Tensor Calculations

In order to carry out the tensor calculations, some fundamental rules must be taken into account in tensor calculus [11–13].

2.3.5.1 Calculation of Tensor Components

Let T be a second-order tensor; it can be written in different tensor forms: 58 2 Tensor Analysis

T ¼ Tijg g ¼ T ig g j i j j i : i j j i ð2 88Þ ¼ Tijg g ¼ Ti gjg

Multiplying the ﬁrst row of Eq. (2.88) by the covariant basis gk, one obtains Tij g Á g g ¼ Tijg g ¼ T ig k j i kj i k i ð2:89aÞ i ij ... ) Tk ¼ T gkj for j ¼ 1, 2, , N

Analogously, multiplying the second row of Eq. (2.88) by the contravariant basis gk, one obtains ÀÁ T gk Á g j gi ¼ T gkjgi ¼ T kgi ij ij i 2:89b k kj ... ð Þ ) Ti ¼ Tijg for j ¼ 1, 2, , N

Multiplying Eq. (2.89a)bygkj, the contravariant tensor components result in

ij i kj ... : T ¼ Tkg for k ¼ 1, 2, ,N ð2 90aÞ

Multiplying Eq. (2.89b)bygkj, one obtains the covariant tensor components

k ... : Tij ¼ Ti gkj for k ¼ 1, 2, , N ð2 90bÞ

Substituting Eqs. (2.89a) and (2.90a), one obtains the contraction rules between the contravariant tensor components.

ij ip kj ... : T ¼ T gpkg for k, p ¼ 1, 2, , N ð2 91aÞ

Similarly, the contraction rules between the covariant tensor components result from substituting Eqs. (2.89b) and (2.90b).

pk ... : Tij ¼ Tipg gkj for k, p ¼ 1, 2, , N ð2 91bÞ

Analogously, the contraction rules between the mixed tensor components can be derived as

i i pk ... : Tj ¼ Tpg gkj for k, p ¼ 1, 2, , N ð2 92aÞ

Similarly, one obtains

j p kj ... : Ti ¼ Ti gpkg for k, p ¼ 1, 2, , N ð2 92bÞ 2.3 Tensor Algebra 59

2.3.5.2 Addition Law

Tensors of the same orders and types can be added together. The resulting tensor has the same order and type of the initial tensors. The tensor resulted from the addition of two covariant or contravariant tensors A and B can be calculated as ÀÁ i j k i j k C ¼ A þ B ¼ Aijk þ Bijk g g g ¼ Cijkg g g ¼ B þ A ) C ¼ A þ B ¼ B þ A ; ijk ijk ÀÁijk ijk ijk : ijk ijk ijk ð2 93Þ C ¼ A þ B ¼ A þ B gigjgk ¼ C gigjgk ¼ B þ A ) Cijk ¼ Aijk þ Bijk ¼ Bijk þ Aijk

Similarly, the tensor resulted from the addition of two mixed tensors A and B can be written as pq pq i j k C ¼ A þ B ¼ Aijk þ Bijk gpgqg g g pq i j k : ¼ Cijkgpgqg g g ¼ B þ A ð2 94Þ pq pq pq pq pq ) Cijk ¼ Aijk þ Bijk ¼ Bijk þ Aijk

Straightforwardly, the addition of tensors is commutative, as proved in Eqs. (2.93) and (2.94).

2.3.5.3 Outer Product

On the contrary, the outer product can be carried out at tensors of different orders and types. The tensor components resulted from the outer product of two mixed tensors A and B can be calculated as ÀÁ pq i j rts k l AB ¼ Aij gpgqg g Bkl g g grgtgs pqrst i j k l : ¼ Cijkl gpgqg g g g grgtgs 6¼ BA ð2 95Þ pqrst pq rts rts pq ) Cijkl ¼ Aij Bkl ¼ Bkl Aij

The outer product of two tensors results a tensor with the order that equals the sum of the covariant and contravariant indices. The outer product is not commutative, but their tensor components are commutative, as shown in Eq. (2.95). In this example, the resulting ninth-order tensor is generated from the outer product of the mixed fourth-order tensor A and mixed ﬁfth-order tensor B. Obviously, the outer product of tensors A, B, and C is associative, i.e. A(BC) ¼ (AB)C. 60 2 Tensor Analysis

2.3.5.4 Contraction Law

The contraction operation can be only carried out at the mixed tensor types of different orders. The tensor contraction is operated in many contracting steps where the tensor order is shortened by eliminating the same covariant and contravariant indices of the tensor components. We consider a mixed tensor of high orders. In this example, the mixed ﬁfth-order tensor A of type (2, 3) can be transformed from the coordinates {ui} into the barred coordinates { ui}. The transformed tensor components can be calculated according to the transformation law in Eq. (2.144).

i j r s t ij ∂u ∂u ∂u ∂u ∂u A ¼ Apq ð2:96Þ klm ∂up ∂uq ∂uk ∂ul ∂um rst

Carrying out the ﬁrst contraction of A¯ in Eq. (2.96)atl ¼ i, one obtains the tensor components

i j r s t ij ∂u ∂u ∂u ∂u ∂u A ¼ Apq kim ∂up ∂uq ∂uk ∂ul ∂um rst ∂uj ∂ur ∂ut ∂ui ∂us ¼ Apq ∂uq ∂uk ∂um ∂up ∂ui rst ∂uj ∂ur ∂ut ¼ δ sApq ð2:97aÞ ∂uq ∂uk ∂um p rst ∂uj ∂ur ∂ut ¼ Apq ∂uq ∂uk ∂um rpt j r t j ∂u ∂u ∂u , B ¼ B q km ∂uq ∂uk ∂um rt

As a result, the resulting tensor components B are the third-order tensor type after the ﬁrst contraction of A at s ¼ p:

pq δ p pq q : Arst s ¼ Arpt Brt ð2 97bÞ

Further contracting the tensor components B in Eq. (2.97a)atk ¼ j, one obtains t j r j ∂u ∂u ∂u B ¼ B q jm ∂um ∂uq ∂uj rt ∂ t u δ r q ¼ q Brt ∂um ð2:98aÞ ∂ut ¼ B q ∂um qt ∂ut , C ¼ C m ∂um t 2.3 Tensor Algebra 61

As a result, the resulting tensor components C are the ﬁrst-order tensor type after the second contraction of B in Eq. (2.98a)atr ¼ q:

qδ q q : Brt r ¼ Bqt Ct ð2 98bÞ

2.3.5.5 Inner Product

The inner product of tensors comprises two basic operations of the outer product and at least one contraction of tensors. As an example, the outer product of two third-order tensors A and B results in a sixth-order tensor. ÀÁ AB ¼ Ampg g gq B r g gsgt q m p st r ð2:99Þ mp r q s t ¼ Aq Bstg grgmgpg g

Using the ﬁrst tensor contraction in Eq. (2.99)atr ¼ q, one obtains the resulting fourth-order tensor components of the inner product.

mp r δ r mp q mp : Aq Bst q ¼ Aq Bst Cst ð2 100Þ

Similarly, using the second tensor contraction law in Eq. (2.100)atp ¼ s, the resulting second-order tensor components result in

mpδ p ms m : Cst s ¼ Cst Dt ð2 101Þ

Finally, applying the third tensor contraction law to Eq. (2.101)atm ¼ t, the resulting tensor component is an invariant (zeroth-order tensor).

mδ m t : Dt t ¼ Dt D ð2 102Þ

In another approach, one can calculate the tensor components of the inner product of two contravariant tensors A and B multiplying by the metric tensor.

ijk lm ijk lm : A B ! A B glk ð2 103Þ

Using Eq. (2.89a), one obtains the resulting tensor components

lm m : B glk ¼ Bk ð2 104Þ

Substituting Eq. (2.104) into Eq. (2.103) and using the tensor contraction law, one obtains the resulting tensor components 62 2 Tensor Analysis

ijm ijk m m ijk : C A Bk ¼ Bk A ð2 105Þ

Equation (2.105) denotes that the inner product of the tensor components is commutative.

2.3.5.6 Indices Law

Using the metric tensors, the operation of moving indices enables changing indices of the tensor components from the upper into lower positions and vice versa. Multiplying a tensor component by the metric tensor components, the lower index (covariant index) is moved into the upper index (contravariant index) and vice versa. – Moving covariant indices i, j to the upper position:

k k il kl kl jm klm : Aij ! Aijg ¼ Aj ! Aj g ¼ A ð2 106aÞ

– Moving contravariant indices i, j to the lower position:

ij ij i i : Ak ! Ak gjl ¼ Akl ! Aklgim ¼ Aklm ð2 106bÞ

2.3.5.7 Quotient Law

The quotient law of tensors postulates that if the tensor product of AB and B are tensors, A must be a tensor.

ðÞ\AB ¼ C ∴tensor ðÞ)B ∴tensor ðÞA∴tensor ð2:107aÞ

Proof Using the contraction law, the barred components in the transformation coordinates { ui} of the tensor product AB result in

k il l k il l : AijBk ¼ Cj ) AijBk ¼ Cj ð2 107bÞ

According to the transformation law (2.144), the transformed components in the coordinates { ui} of the tensors B and C can be calculated as 2.3 Tensor Algebra 63

i l p il ∂u ∂u ∂u B ¼ Bmn ; k p ∂um ∂un ∂uk : l q ð2 107cÞ l ∂u ∂u C ¼ C n j q ∂un ∂uj

Substituting Eq. (2.107c) into Eq. (2.107b) and using the contraction law, one obtains i l p l q k ∂u ∂u ∂u ∂u ∂u A Bmn ¼ C n ij p ∂um ∂un ∂uk q ∂un ∂uj ð2:107dÞ ∂ul ∂uq ¼ A p Bmn mq p ∂un ∂uj

Rearranging the terms of Eq. (2.107d), one obtains ∂ i ∂ p ∂ q ∂ l k u u p u u mn Aij À Amq Bp ¼ 0 ∂um ∂uk ∂uj ∂un : l i p q ð2 107eÞ ∂u k ∂u ∂u ∂u )8Bmn; 6¼ 0 : A ¼ A p p ∂un ij ∂um ∂uk mq ∂uj k Applying the inner product by ∂ur ∂u to Eq. (2.107e), one obtains the barred ∂ui ∂us components of A at r ¼ m and s ¼ p. ∂ q ∂ r ∂ k kδ r δ p p u u u Aij m s ¼ Amq ∂uj ∂ui ∂us : k m q ð2 107fÞ k ∂u ∂u ∂u ) A ¼ A p ij mq ∂up ∂ui ∂uj

Equation (2.107f) proves that A is a mixed third-order tensor of type (1, 2), cf. Eq. (2.107c).

2.3.5.8 Symmetric Tensors

Tensor T is called symmetric in the given basis if two covariant or contravariant indices of the tensor component can be interchanged without changing the tensor component value. 64 2 Tensor Analysis

ijij jiji : Tij ¼ Tji symmetric in i and j Tij ¼ Tji : symmetric in i and j ð2:108Þ ijk ikj : Tpq ¼ Tpq symmetric in j and k ijk ijk : Tpq ¼ Tqp symmetric in p and q

In case of a second-order tensor, the tensor T is symmetric if T equals its transpose.

T ¼ TT ð2:109Þ

2.3.5.9 Skew-Symmetric Tensors

The sign of the tensor component is opposite if a pair of the covariant or contravariant indices are interchanged with each other. In this case, the tensor is skew-symmetric (anti-symmetric). Tensor T is deﬁned as a skew-symmetric tensor (anti-symmetric) if

Tij ¼ÀTji : skew-symmetric in i and j Tij ¼ÀTji : skew-symmetric in i and j : ijk ikj : ð2 110Þ Tpq ¼ÀTpq skew- symmetric in j and k ijk ijk : Tpq ¼ÀTqp skew-symmetric in p and q

In case of a second-order tensor, the tensor T is skew-symmetric if T is opposite to its transpose.

T ¼ÀTT ð2:111Þ

An arbitrary tensor T can be generally decomposed into the symmetric and skew- symmetric tensors: ÀÁÀÁ 1 1 T ¼ T þ TT þ T À TT T þ T ð2:112Þ 2 2 sym skew

Proof • The ﬁrst tensor Tsym is symmetric: ÀÁ 1 T T þ TT ¼ T T ðÞqed sym 2 sym 2.4 Coordinate Transformations 65

• The second tensor Tskew is skew-symmetric: ÀÁ 1 T T À TT ¼ÀT T ðÞqed skew 2 skew

2.4 Coordinate Transformations

Tensors are tuples of independent coordinates in a finite multifold N-dimensional tensor space (RN Â ...Â RN). The tensor describes physical states generally depending on different variables (dimensions). Each physical state can be defined as the point P(u1,...,uN) with N coordinates of ui. By changing the variables, such as time, locations, and physical characteristics (e.g. pressure, temperature, density, velocity), the physical state point varies in the multifold N-dimensional space. The tensor does not change itself and is invariant in any coordinate system. However, its components change in the new basis by the coordinate transformation since the basis changes as the coordinate system varies. In this case, applications of tensor analysis have been used to describe the transformation between two general curvilinear coordinate systems in the multifold N-dimensional spaces. Hence, tensors are a very useful tool applied to the coordinate transformations in the multifold N-dimensional tensor spaces. High-order tensors can be generated by a multilinear map between two multifold N-dimensional spaces (cf. Sect. 2.2). Their components change in the relating bases by the coordinate transformations, as displayed in Fig. 2.5. In the following section, the relations between the tensor components in different curvilinear coordinates of the finite N-dimensional spaces will be discussed.

2.4.1 Transformation in the Orthonormal Coordinates

The simple coordinate transformation of rotation between the orthonormal coordinates xi and uj in Euclidean coordinate system is carried out. An arbitrary vector r (ﬁrst-order tensor) can be written in both coordinate systems:

r ¼ x e þ x e x e 1 1 2 2 i i ð2:113Þ ¼ u1g1 þ u2g2 ujgj 66 2 Tensor Analysis

Tensor components Basis{}gi ,g j Tij

Tensor T invariant Tensor components = i j = i j T Tijg g Tij g g in any basis dependent on the basis

Tensor components Basis{}g i , g j changed ≠ Tij Tij

Fig. 2.5 Tensor and tensor components in different bases

P(x j ) x2 r e2 g2

0 x1 e1 x1 g 1 θ =θ 11 u1

Fig. 2.6 Two-dimensional coordinate transformation of rotation

The vector components in the coordinate uj can be calculated in (Fig. 2.6) & u ¼ cos θ x þ cos θ x 1 11 1 12 2 ð2:114Þ u2 ¼ cos θ21x1 þ cos θ22x2

Thus, ! ! ! u cos θ cos θ x 1 ¼ 11 12 Á 1 , u ¼ Tx ð2:115Þ u2 cos θ21 cos θ22 x2 2.4 Coordinate Transformations 67 where T is the transformation matrix. Setting θ11 ¼ θ, one obtains π cos θ ¼ cos θ; cos θ ¼ cos θ þ ¼Àsin θ 11 12 2 π cos θ ¼ cos θ À ¼ sin θ; cos θ ¼ cos θ 21 2 22

Therefore, the transformation matrix T becomes ! ! cos θ cos θ cos θ À sin θ T ¼ 11 12 ¼ ð2:116Þ cos θ21 cos θ22 sin θ cos θ

The transformed coordinates can be computed by the transformation T: ! ! ! u cos θ À sin θ x T : x ! u ¼ Tx : 1 ¼ Á 1 ð2:117Þ u2 sin θ cos θ x2 where θ is the rotation angle of the rotating coordinates uj. Transforming backward Eq. (2.117), one obtains the coordinates xi ! ! ! x cos θ sin θ u TÀ1 : u ! x ¼ TÀ1u : 1 ¼ Á 1 ð2:118Þ x2 À sin θ cos θ u2

The vector component on the basis is obtained multiplying Eq. (2.113) by the relating basis ei or gj. ; xi ¼ r Á ei i ¼ 1, 2 : ; ð2 119Þ uj ¼ r Á gj j ¼ 1, 2

Substituting Eq. (2.119) into Eq. (2.117), one obtains the transformation matrix between two coordinate systems. ! ! ! θ θ g1 cos À sin e1 : ¼ θ θ Á , g ¼ T Á e ð2 120Þ g2 sin cos e2

Similarly, ! ! ! θ θ e1 cos sin g1 À1 : ¼ θ θ Á , e ¼ T Á g ð2 121Þ e2 À sin cos g2 68 2 Tensor Analysis

2.4.2 Transformation of Curvilinear Coordinates in EN

In the following section, second-order tensors T are used in the transformation of general curvilinear coordinates in Euclidean space EN, as shown in Fig. 2.7. i The basis gi of the curvilinear coordinate {u } can be transformed into the new i basis gi of the curvilinear coordinate { u } using the linear transformation S. The new covariant basis can be rewritten as a linear combination of the old basis.

: j : S gi ! gi ¼ Si gj , G ¼ GS ð2 122Þ

j where Si are the mixed transformation components of the second-order tensor S. The old covariant basis results can be calculated as

i ÀÁ ÀÁ i ∂r ∂r ∂u i i ∂u g ¼ ¼ g SÀ1 ) SÀ1 ð2:123Þ j ∂uj ∂ui ∂uj i j j ∂uj

Inversing the basis matrix in Eq. (2.122), the new contravariant basis can be calculated as ÀÁ À1 À1 À1 À1 i À1 i j : G ¼ ðÞGS ¼ S G ) g ¼ S j g ð2 124Þ

Multiplying Eq. (2.124) by the linear transformation S, the old contravariant basis results in

À1 À1 i i j : G ¼ SG ) g ¼ Sj g ð2 125Þ

According to Eqs. (2.11) and (2.122), the new covariant basis can be calculated as

∂r ∂r ∂uk ∂uk g ¼ ¼ g S k ) S k ð2:126aÞ j ∂uj ∂uk ∂uj k j j ∂uj

Combining Eqs. (2.123), (2.124) and (2.126a) and using the chain rule of differentiation, one obtains the relation of the mixed transformation components between two general curvilinear coordinates in Euclidean space EN.

Fig. 2.7 Basis transformation of general N curvilinear coordinates in E (2) N×N T(2) ∈EN×N T ∈E

{} S {} gi gi 2.4 Coordinate Transformations 69

ÀÁ ÀÁ i i À1 i k l À1 i kδ l gj Á g ¼ g Á gj ¼ S l Sj g Á gk ¼ S l Sj k ÀÁ i k i i ∂u ∂u ∂u : ¼ SÀ1 S k ¼ ¼ ¼ δ i ð2 126bÞ k j ∂uk ∂uj ∂uj j , SSÀ1 ¼ SÀ1S ¼ I

Therefore, the transformation tensor S can be written as 2 3 ∂u1 ∂u1 ∂u1 6 : 7 6 ∂u1 ∂u2 ∂uN 7 6 ∂ 2 ∂ 2 ∂ 2 7 6 u u u 7 N N S ¼ 6 : 7 2 R Â R ð2:127aÞ 6 ∂u1 ∂u2 ∂uN 7 4 ::::5 ∂ N ∂ N ∂ N u u : u ∂u1 ∂u2 ∂uN

The transformation tensor S in Eq. (2.127a) is identical to the Jacobian matrix between two coordinate systems {ui} and { ui}. Inverting the transformation tensor S, the back transformation results in 2 3 ∂u1 ∂u1 ∂u1 6 : 7 6 ∂u1 ∂u2 ∂uN 7 6 ∂ 2 ∂ 2 ∂ 2 7 À1 6 u u u 7 N N S ¼ 6 : 7 2 R Â R ð2:127bÞ 6 ∂u1 ∂u2 ∂uN 7 4 ::::5 ∂ N ∂ N ∂ N u u : u ∂u1 ∂u2 ∂uN

The relation between the new and old components of an arbitrary vector v (ﬁrst- order tensor) is similarly given in the coordinate transformation S according to Eqs. (2.122) and (2.124). ÀÁ v ¼ g Á v ¼ S jg Á v g j ¼ S jv i i ÀÁi j j i j ÀÁ : i i À1 i j j À1 i j ð2 128Þ v ¼ g Á v ¼ S j g Á v gj ¼ S j v

Using Eq. (2.74), the relation of the components of the second-order tensor T can be derived. – Covariant metric tensor components: ÀÁ T ¼ g Á T Á g ¼ S kg Á T Á S lg ij i j i k j l ð2:129Þ k l k l ¼ Si Sj ðÞ¼gk Á T Á gl Si Sj Tkl

– Contravariant metric tensor components: 70 2 Tensor Analysis

ÀÁ ÀÁ ij i j T ¼ gi Á T Á gj ¼ SÀ1 gk Á T Á SÀ1 gl ÀÁÀÁÀÁk ÀÁl ÀÁ ð2:130Þ À1 i À1 j k l À1 i À1 j kl ¼ S k S l g Á T Á g ¼ S k S l T

– Mixed metric tensor components: ÀÁ i i k À1 i l T : ¼ g Á T Á g ¼ S g Á T Á S g j j ÀÁÀÁj k ÀÁl ð2:131Þ k À1 i l k À1 i l ¼ Sj S l gk Á T Á g ¼ Sj S l Tk:

Note that in Eq. (2.131), the dot after the lower index indicates the position of the basis of the upper index locating after the tensor T. In this case, the tensor T is l located between the upper basis g and lower basis gk. ÀÁ i i À1 i k l T: ¼ g Á T Á g ¼ S g Á T Á S g j ÀÁjÀÁk ÀÁj l ð2:132Þ À1 i l k À1 i l k ¼ S kSj g Á T Á gl ¼ S kSj T:l

Note that in Eq. (2.132), the dot before the lower index indicates the position of the basis of the upper index locating in front of the tensor T. In this case, the tensor T is k located between the upper basis g and lower basis gl.

2.4.3 Examples of Coordinate Transformations

2.4.3.1 Cylindrical Coordinates

The transformation S from Cartesian {ui} to cylindrical coordinates { ui}: 8 < u1 ¼ r cos θ u1 cos u2 : 2 θ 1 2 S : u ¼ r sin u sin u u3 ¼ z ¼ u3

The covariant transformation matrix S can be calculated as 2 3 ∂u1 ∂u1 ∂u1 0 1 6 ∂ 1 ∂ 2 ∂ 3 7 6 u u u 7 cos θ Àr sin θ 0 6 ∂u2 ∂u2 ∂u2 7 S ¼ 6 7 ¼ @ sin θ r cos θ 0 A 6 ∂ 1 ∂ 2 ∂ 3 7 4 u u u 5 001 ∂u3 ∂u3 ∂u3 ∂u1 ∂u2 ∂u3

The determinant of S is called the Jacobian J. 2.4 Coordinate Transformations 71 ∂ 1 ∂ 1 ∂ 1 u u u ∂u1 ∂u2 ∂u3 θ θ ∂ 2 ∂ 2 ∂ 2 cos Àr sin 0 u u u θ θ jjS ¼ J ¼ sin r cos 0 ¼ r ∂u1 ∂u2 ∂u3 ∂ 3 ∂ 3 ∂ 3 001 u u u ∂u1 ∂u2 ∂u3

The contravariant transformation matrix SÀ1 results from the inversion of the covariant matrix S. 2 3 ∂u1 ∂u1 ∂u1 6 7 0 1 6 ∂u1 ∂u2 ∂u3 7 r cos θ r sin θ 0 6 ∂u2 ∂u2 ∂u2 7 1 SÀ1 ¼ 6 7 ¼ @ À sin θ cos θ 0 A 6 ∂ 1 ∂ 2 ∂ 3 7 r 4 u u u 5 00r ∂u3 ∂u3 ∂u3 ∂u1 ∂u2 ∂u3

2.4.3.2 Spherical Coordinates

The transformation S from Cartesian {ui} to spherical coordinates { ui}: 8 < u1 ¼ ρ sin φ cos θ u1 sin u2 cos u3 : 2 ρ φ θ 1 2 3 S : u ¼ sin sin u sin u cos u u3 ¼ ρ cos φ ¼ u1 cos u2

The covariant transformation matrix S can be calculated as 2 3 ∂u1 ∂u1 ∂u1 0 1 6 ∂ 1 ∂ 2 ∂ 3 7 6 u u u 7 sin φ cos θρcos φ cos θ Àρ sin φ sin θ 6 ∂u2 ∂u2 ∂u2 7 S ¼ 6 7 ¼ @ sin φ sin θρcos φ sin θρsin φ cos θ A 6 ∂ 1 ∂ 2 ∂ 3 7 4 u u u 5 cos φ Àρ sin φ 0 ∂u3 ∂u3 ∂u3 ∂u1 ∂u2 ∂u3

The determinant of S is called the Jacobian J. 72 2 Tensor Analysis

∂ 1 ∂ 1 ∂ 1 u u u ∂u1 ∂u2 ∂u3 ∂ 2 ∂ 2 ∂ 2 u u u jjS ¼ J ∂u1 ∂u2 ∂u3 ∂ 3 ∂ 3 ∂ 3 u u u ∂ 1 ∂ 2 ∂ 3 u u u sin φ cos θρcos φ cos θ Àρ sin φ sin θ ¼ sin φ sin θρcos φ sin θρsin φ cos θ ¼ ρ2 sin φ cos φ Àρ sin φ 0

The contravariant transformation matrix results from the inversion of the matrix S. 2 3 ∂u1 ∂u1 ∂u1 0 1 6 7 ρ φ θρ φ θρ φ 6 ∂u1 ∂u2 ∂u3 7 sin cos sin sin cos 2 2 2 B φ θ φ θ φ C À1 6 ∂u ∂u ∂u 7 1 B coscos cossin À sin C S ¼ 6 7 ¼ @ A 6 ∂ 1 ∂ 2 ∂ 3 7 ρ sin θ cos θ 4 u u u 5 À 0 ∂u3 ∂u3 ∂u3 sin φ sin φ ∂u1 ∂u2 ∂u3

2.4.4 Transformation of Curvilinear Coordinates in RN

In the following section, second-order tensors T will be used in the transformation of general curvilinear coordinates in Riemannian manifold RN, as shown in Fig. 2.8. i i In Riemannian manifold, the bases gi and gi of the curvilinear coordinates u and u do not exist any longer. Instead of the metric coefﬁcients, the transformation coefﬁcients that depend on the relating coordinates have been used in Riemannian manifold [3]. The new barred curvilinear coordinate ui is a function of the old curvilinear coordinates uj, j ¼ 1,2,...,N. Therefore, it can be written in a linear function of uj.

ui ¼ uiðÞu1; ...; uN i i j ... ) u ¼ aj u for j ¼ 1, 2, , N ð2:133Þ i i j ... ) du ¼ aj du for j ¼ 1, 2, , N

i where a¯j is the transformation coefﬁcient in the coordinate transformation S. Using the chain rule of differentiation, one obtains 2.4 Coordinate Transformations 73

Fig. 2.8 Basis transformation of general N curvilinear coordinates in R (2) N×N T(2) ∈ R N×N T ∈ R

{} S {} ai ai

∂ui dui ¼ duj ¼ a iduj ∂ j j u ð2:134Þ ∂ui ) a i ¼ j ∂uj

Analogously, the transformation coefﬁcients of the back transformation result in

∂ui dui ¼ duj ¼ a iduj ∂ j j u ð2:135Þ ∂ui ) a i ¼ j ∂uj

Combining Eqs. (2.134) and (2.135) and using the Kronecker delta, one obtains the relation between the transformation coefﬁcients

∂ui ∂uj ∂ui a ia j ¼ ¼ ¼ δ i j k ∂uj ∂uk ∂ukk ð2:136Þ i j j i , aj ak ¼ ak aj ¼ I

The relation of the second-order tensor components between the new and old curvilinear coordinates can be calculated using Eq. (2.133). TðÞ2 ¼ T ukul ¼ T a ka l uiuj kl kl i j ð2:137Þ i j Tiju u

Thus,

∂uk ∂ul T ¼ a ka lT ¼ T ; ij i j kl ∂ i ∂ j kl u u ð2:138Þ ∂uk ∂ul T ¼ a ka lT ¼ T ij i j kl ∂ui ∂uj kl

In the same way, the covariant, contravariant, and mixed components of the second- order tensor T between both coordinates in the transformation can be derived as: 74 2 Tensor Analysis

• Covariant tensor components:

k l k l : Tij ¼ ai aj Tkl , Tij ¼ ai aj Tkl ð2 139Þ

• Contravariant tensor components:

ij i j kl ij i j kl : T ¼ akal T , T ¼ akal T ð2 140Þ

• Mixed tensor components:

j k j l j k j l : Ti: ¼ ai al Tk: , Ti: ¼ ai al Tk: ð2 141Þ i i l k i i l k : T:j ¼ akaj T:l , T:j ¼ akaj T:l ð2 142Þ

Generally, the transformation coefﬁcients of high-order tensors can be alternatively computed as ∂ k ∂ l ∂ m ∂ s ∂ t klm u u u u u pqr Tij ¼ p q r Tst ∂u ∂u ∂u ∂ui ∂uj ð2:143Þ k l m s t pqr ¼ ap aqar ai aj Tst

Therefore, k l m s t klm ∂u ∂u ∂u ∂u ∂u pqr Tij ¼ Tst ∂up ∂uq ∂ur ∂ui ∂uj ð2:144Þ k l m s t pqr ¼ ap aqar ai aj Tst

2.5 Tensor Calculus in General Curvilinear Coordinates

In the following sections, some necessary symbols, such as the Christoffel symbols, Riemann-Christoffel tensor, and fundamental invariants of the Nabla operator have to be taken into account in the tensor applications to ﬂuid mechanics and other working areas.

2.5.1 Physical Component of Tensors

Various types of the second-order tensors are shown in Eq. (2.74). The physical tensor component is deﬁned as the tensor component on its covariant unitary basis 2.5 Tensor Calculus in General Curvilinear Coordinates 75

* gi . Therefore, the basis of the general curvilinear coordinates must be normalized (cf. Appendix B). Dividing the covariant basis by its vector length, the covariant unitary basis (covariant normalized basis) results in * gi gi * : gi ¼ ¼ pffiffiffiffiffiffiffiffi ) gi ¼ 1 ð2 145Þ jjgi gðÞii

The covariant basis norm |gi| can be considered as a scale factor hi. pffiffiffiffiffiffiffiffi hi ¼ jjgi ¼ gðÞii ðno summation over iÞ

Thus, the covariant basis can be related to its covariant unitary basis by the relation of pffiffiffiffiffiffiffiffi * * : gi ¼ gðÞii gi ¼ higi ð2 146Þ

The contravariant basis can be related to its covariant unitary basis using Eqs. (2.47) and (2.146).

i ij ij * : g ¼ g gj ¼ g hjgj ð2 147Þ

The contravariant second-order tensor can be written in the covariant unitary bases using Eq. (2.146). ÀÁ ij ij * * *ij * * : T ¼ T gigj ¼ T hihj gi gj T gi gj ð2 148Þ

Thus, the physical contravariant tensor components result in

*ij ij T T hihj ð2:149Þ

The covariant second-order tensor can be written in the contravariant unitary bases using Eq. (2.147). ÀÁ i j ik jl * * * * * : T ¼ Tijg g ¼ Tijg g hkhl gkgl Tijgk gl ð2 150Þ

Similarly, the physical covariant tensor component results in

* ik jl : Tij ¼ Tijg g hkhl ð2 151Þ

The mixed tensors can be written in the covariant unitary bases using Eqs. (2.146) and (2.147) 76 2 Tensor Analysis ÀÁ T ¼ T ig g j ¼ T ig gjkg j ÀÁi ÀÁj i k i * jk * i jk * * ¼ T hig g hkg ¼ T g hihk g g : j i k j i k ð2 152Þ * i * * Tj gi gk

Thus, the physical mixed tensor component results from Eq. (2.152): * i i jk : Tj ¼ Tj g hihk ð2 153Þ

2.5.2 Derivatives of Covariant Bases

i Let gi be a covariant basis in the curvilinear coordinates {u }. The derivative of the covariant basis with respect to the time variable t can be computed as ∂g ∂ ∂r g_ ¼ i ¼ r_ ð2:154Þ i ∂t ∂t ∂ui ,i

Due to ui is a differentiable function of t, Eq. (2.154) can be rewritten as

∂g ∂g ∂uj g_ ¼ i ¼ i g u_ j ð2:155Þ i ∂t ∂uj ∂t i, j where gi,j is called the derivative of the covariant basis gi of the curvilinear coordinates {ui}. Using the chain rule of differentiation, the covariant basis of the curvilinear coordinates {ui} can be calculated in Cartesian coordinates {xi}.

∂r ∂r ∂xp g ¼ ¼ ¼ e x p ð2:156Þ i ∂ui ∂xp ∂ui p ,i

Similarly, one obtains the covariant basis of the coordinates {xi}.

∂r ∂r ∂uk e ¼ ¼ ¼ g u k ð2:157Þ p ∂xp ∂uk ∂xp k , p

The derivative of the covariant basis of the coordinates {ui} can be obtained from Eqs. (2.156) and (2.157). 2.5 Tensor Calculus in General Curvilinear Coordinates 77 ÀÁ p p ∂g ∂ epx ∂x g ¼ i ¼ ,i ¼ e , i i,j ∂uj ∂uj p ∂uj ! ∂ p ∂ k ∂2 p x,i u x ð2:158Þ ¼ u k g ¼ g ,p ∂uj k ∂xp ∂ui∂uj k Γ k ... ijgk for k ¼ 1, 2, , N

k The symbol Γ ij in Eq. (2.158) is deﬁned as the second-kind Christoffel symbol, which has 27 (¼33) components for a three-dimensional space (N ¼ 3). Thus, the second-order Christoffel symbols that only depend on both coordinates of {ui} and {xi} can be written as

∂uk ∂2xp Γ k ¼ ij ∂ p ∂ i∂ j x u u ð2:159Þ ∂uk ∂2xp ¼ ¼ Γ k ∂xp ∂uj∂ui ji

The result of Eq. (2.159) proves that the second-kind Christoffel symbols are symmetric with respect to i and j. The second-kind Christoffel symbols are given by multiplying both sides of Eq. (2.158) by the contravariant basis gl. ÀÁ Γ k g gl Γ kδ l gl g ij k Á ¼ ij k ¼ Á i, j : Γ l l ð2 160Þ ) ij ¼ g Á gi,j

Substituting Eq. (2.158) into Eq. (2.155), one obtains the relation between the covariant basis time derivative and the Christoffel symbol.

_ _ j Γ k _ j : g i ¼ gi,ju ¼ iju gk ð2 161Þ

Furthermore, the covariant basis derivative can be calculated in Cartesian coordinate {xi} using Eq. (2.156). ÀÁ ÀÁ p p ∂g ∂ epx ∂ x g ¼ i ¼ , i ¼ e , i ¼ e x p ð2:162Þ i,j ∂uj ∂uj p ∂uj p , ij

According to Eq. (2.159), the second-kind Christoffel symbols can be rewritten as

∂uk ∂2xp ∂uk ∂2xp Γ k ¼ ¼ ij ∂xp ∂ui∂uj ∂xp ∂uj∂ui k p k p : ¼ u, px,ij ¼ u,px,ji ð2 163Þ 78 2 Tensor Analysis

2.5.3 Christoffel Symbols of First and Second Kind

According to Eq. (2.160), the second-kind Christoffel symbol can be deﬁned as &' 2 2 k ∂ r ∂ r Γ k ¼ gk Á ¼ gk Á ij ij ∂ui∂uj ∂uj∂ui ð2:164Þ k k Γ k ¼ g Á gi,j ¼ g Á gj,i ¼ ji

Equation (2.164) reconﬁrms the symmetric property of the Christoffel symbols with respect to the indices i and j. Obviously, the Christoffel symbols are coordinate dependent; therefore, they are not tensors. In order to compute the second-kind Christoffel symbols in the covariant metric k coefﬁcients, the derivative of gij with respect to u has to be taken into account.

gij ¼ gi Á gj ∂ ð2:165Þ gij ) gij,k ¼ gi Á gj ¼ gi,k Á gj þ gi Á gj,k ∂uk ,k

Using Eq. (2.158) at changing the index j into k; then, i into j, one obtains the following relations

Γ p ; Γ p : gi,k ¼ ikgp gj,k ¼ jkgp ð2 166Þ

Substituting Eq. (2.166) into Eq. (2.165), one obtains the derivative of gij with respect to uk.

gij,k ¼ gi,k Á gj þ gj, k Á gi Γ p Γ p ¼ ikgp Á gj þ jkgp Á gi ð2:167Þ Γ p Γ p ¼ ikgpj þ jkgpi

Interchanging k with i in Eq. (2.167), one obtains

Γ p Γ p : gkj, i ¼ kigpj þ ji gpk ð2 168Þ

Analogously, one reaches the relation interchanging k with j in Eq. (2.167).

Γ p Γ p : gik, j ¼ ij gpk þ kjgpi ð2 169Þ

Combining Eqs. (2.167)–(2.169), the Christoffel symbols can be written in the derivatives of the covariant metric coefﬁcients. 1 g Γ p ¼ g þ g À g ð2:170Þ pj ik 2 ij,k kj,i ik,j 2.5 Tensor Calculus in General Curvilinear Coordinates 79

Multiplying Eq. (2.170)bygqj, the Christoffel symbols result according to Eq. (2.50)in

Γ p g gqj ¼ Γ p δ q ik pj ik p 1 ð2:171Þ ) Γ q ¼ gqj g þ g À g ik 2 ij, k kj, i ik, j

Changing j into p, k into j, and q into k in Eq. (2.171), one obtains 1 Γ k ¼ g þ g À g gkp ij 2 ip,j jp,i ij,p ð2:172Þ kp Γijpg

Changing the index p into k, the ﬁrst-kind Christoffel symbol Γijp in Eq. (2.172) that has 27 (¼33) components for a three-dimensional space (N ¼ 3) is deﬁned as Γ ijk ½ij, k 1 g þ g À g ð2:173aÞ 2 ik,j jk,i ij,k Γ p ... ¼ gpk ij for p ¼ 1, 2, , N

Other expressions of the Christoffel symbols can be found in some literature.

Γ ijk ½ij&', k p p ð2:173bÞ ¼ g g Γ for p ¼ 1, 2, ..., N pk ij pk ij

2.5.4 Prove That the Christoffel Symbols Are Symmetric

1. The ﬁrst-kind Christoffel symbol is symmetric with respect to i, j According to Eq. (2.173a), the ﬁrst-kind Christoffel symbol can be written as 1 Γ ¼ g þ g À g ijk 2 ik,j jk,i ij,k

Interchanging i with j in the equation, one obtains 1 1 Γ ¼ g þ g À g ¼ g þ g À g ¼ Γ ðÞq:e:d: jik 2 jk,i ik,j ji,k 2 ik,j jk,i ij,k ijk

2. The second-kind Christoffel symbol is symmetric with respect to i, j Using Eq. (2.172), the second-kind Christoffel symbol can be expressed as 80 2 Tensor Analysis

Γ k kp Γ ij ¼ g ijp

Due to the symmetry of the ﬁrst-kind Christoffel symbol, the second-kind Christoffel symbol results in

Γ k kp Γ kp Γ ij ¼ g ijp ¼ g jip Γ k : : : ¼ ji ðÞq e d

2.5.5 Examples of Computing the Christoffel Symbols

Given a curvilinear coordinate{ui} with u1 ¼ u;u2 ¼ v;u3 ¼ w in another coordinate {xi}, the relation between two coordinate systems can be written as 8 < x1 ¼ uv 2 : x ¼ w x3 ¼ u2 À v

The covariant basis matrix G can be calculated from 0 1 ∂x1 ∂x1 ∂x1 B C 2 3 B ∂u1 ∂u2 ∂u3 C vu0 B ∂x2 ∂x2 ∂x2 C G ¼ ½¼g g g B C ¼ 4 0015 1 2 3 B ∂ 1 ∂ 2 ∂ 3 C @ u u u A 2u À10 ∂x3 ∂x3 ∂x3 ∂u1 ∂u2 ∂u3

Therefore, the covariant bases can be given in 8 < g1 ¼ ðÞv,0,2u : g2 ¼ ðÞu,0, À 1 g3 ¼ ðÞ0, 1, 0

The determinant of G that equals the Jacobian J of vu0 2 jjG ¼ J ¼ 001 ¼ 2u þ v 6¼ 0 2u À10

The contravariant basis matrix GÀ1 is the inverse matrix of the covariant basis matrix G. 2.5 Tensor Calculus in General Curvilinear Coordinates 81 0 1 ∂u1 ∂u1 ∂u1 2 3 B ∂ 1 ∂ 2 ∂ 3 C g1 B x x x C B ∂u2 ∂u2 ∂u2 C GÀ1 ¼ 4 g2 5 ¼ B C B ∂ 1 ∂ 2 ∂ 3 C g3 @ x x x A ∂u3 ∂u3 ∂u3 ∂x1 ∂x2 ∂x3 2 3 2 3 10u JÀ1 0 uJÀ1 14 5 4 5 ¼ 2u 0 Àv ¼ 2uJÀ1 0 ÀvJÀ1 J 02ðÞu2 þ v 0 010

Thus, 1 1 1 GÀ ¼ ¼ ðÞ2u2 þ v J

1 Testing: GÀ Á jjG ¼ JÀ1J ¼ jjI ¼ 1 ðÞq:e:d:

Thus, the contravariant bases result in 8 < g1 ¼ JÀ1ðÞ1, 0, u 2 À1 : g ¼ J ðÞ2u,0, À v g3 ¼ JÀ1ðÞ0, 2u2 þ v,0

Some examples of the second-kind Christoffel symbols of 27 components can be computed from Eq. (2.160).

Γ k k ij ¼ gi,j g ) Γ1 1 À1 À1 11 ¼ g1,1 g ¼ J ðÞ¼0 Á 1 þ 0 Á 0 þ 2 Á u 2uJ Γ1 1 À1 À1 12 ¼ g1,2 g ¼ J ðÞ¼1 Á 1 þ 0 Á 0 þ 0 Á u J Γ1 1 À1 13 ¼ g1,3 g ¼ J ðÞ¼0 Á 1 þ 0 Á 0 þ 0 Á u 0 ... Γ3 3 À1 2 32 ¼ g3,2 g ¼ J ðÞ¼0 Á 0 þ 0 Á ðÞþ2u þ v 0 Á 0 0 Γ3 3 À1 2 33 ¼ g3,3 g ¼ J ðÞ¼0 Á 0 þ 0 Á ðÞþ2u þ v 0 Á 0 0

The ﬁrst-kind Christoffel symbols containing 27 components in R3 can be computed from Eq. (2.173a). Γ Γ p Γ p ijk ¼ gpk ij ¼ gp Á gk ij for p ¼ 1, 2, 3 82 2 Tensor Analysis

2.5.6 Coordinate Transformations of the Christoffel Symbols

The second-kind Christoffel symbols like tensor components strongly depend on the coordinates at the coordinate transformations. The curvilinear coordinates {ui} are transformed into the new barred curvilinear coordinates {ui}. Therefore, the old basis is also changed into the new basis. The second-kind Christoffel symbols can be written in the new basis of the barred coordinates {ui}.

k ∂g Γ ¼ gk Á i ¼ gk Á g ð2:174Þ ij ∂uj i, j

Using the chain rule of differentiation, the basis of the coordinates {ui} can be calculated as

∂r ∂r ∂uk g ¼ ¼ p ∂ p ∂ k ∂ p u u u ð2:175Þ ∂uk ¼ g k ∂up

Multiplying Eq. (2.175) by the new contravariant basis of the coordinates {ui}, one obtains changing the indices m into k, and p into l.

∂ k ∂ k ∂ m m m u δ m u u g Á gp ¼ g Á gk ¼ k ¼ ∂up∂up ∂up ∂um ) gm Á g gp ¼ gm g Á gp ¼ gmδ p ¼ gp ð2:176Þ p p p ∂up ∂uk ) gk ¼ gl ∂ul

The new covariant basis of the coordinates {ui} can be calculated as

∂r ∂r ∂up ∂up g ¼ ¼ ¼ g ð2:177Þ i ∂ui ∂up ∂ui p ∂ui

Thus, the new covariant basis derivative with respect to j of the coordinates {ui} results in p 2 p p ∂ ∂g ∂ ∂u ∂ u ∂u gp g ¼ i ¼ g ¼ g þ ð2:178Þ i, j ∂uj ∂uj ∂ui p ∂uj∂ui p ∂ui ∂uj

Substituting Eqs. (2.176) and (2.178) into Eq. (2.174), one obtains the second-kind Christoffel symbols in the new basis. 2.5 Tensor Calculus in General Curvilinear Coordinates 83

Γ k k ij ¼ g Á gi,j ! ∂uk ∂2up ∂up ∂g ð2:179Þ ¼ gl Á g þ p ∂ul ∂uj∂ui p ∂ui ∂uj

Using Eq. (2.166) and the chain rule of differentiation, the second term in the parentheses of Eq. (2.179) can be computed as

∂g ∂g ∂uq p ¼ p ∂ j ∂ q ∂ j u u u ð2:180Þ ∂uq ∂uq ¼ g ¼ Γ r g ∂uj p,q ∂uj pq r

Inserting Eq. (2.180) into Eq. (2.179), the transformed Christoffel symbols in the new barred coordinates can be calculated as

Γ k k ij ¼ g Á g i, j ! ∂uk ∂2up ∂up ∂uq ÀÁ ¼ gl Á g þ Γ r gl Á g ∂ l ∂ j∂ i p ∂ i ∂ j pq r u u u u u! ∂uk ∂2up ∂up ∂uq ð2:181Þ ¼ δ l þ Γ r δ l ∂ l ∂ j∂ i p ∂ i ∂ j pq r u u u u u ! ∂uk ∂2ul ∂up ∂uq ¼ þ Γ l ∂ul ∂ui∂uj ∂ui ∂uj pq

Therefore, the transformed Christoffel symbols in the new barred coordinates {ui} result in

k p q k 2 l k ∂u ∂u ∂u ∂u ∂ u Γ ¼ Γ l þ ð2:182Þ ij pq ∂ul ∂ui ∂uj ∂ul ∂ui∂uj

Rearranging the terms in Eq. (2.181), the second derivatives of ul with respect to the new barred coordinates result in ! l 2 l p q ∂u k ∂ u ∂u ∂u Γ ¼ þ Γ l ∂uk ij ∂ui∂uj ∂ui ∂uj pq ð2:183Þ 2 l l p q ∂ u ∂u k ∂u ∂u ) ¼ Γ À Γ l ∂ui∂uj ∂uk ij ∂ui ∂uj pq

Using Eq. (2.160), all Christoffel symbols in Cartesian coordinates {xi} vanish j because the basis ei does not change in any coordinate x . 84 2 Tensor Analysis

∂e Γ k ¼ ek Á e ¼ ek Á i ¼ 0 ð2:184Þ ij i,j ∂xj

2.5.7 Derivatives of Contravariant Bases

Like Eq. (2.158), the derivative of the contravariant basis of the curvilinear coordinates {ui} with respect to uj can be deﬁned as

i ∂g i g i ¼ Γ^ gk ð2:185Þ ,j ∂uj jk

Γ^ i k where jk are the second-kind Christoffel symbols in the contravariant bases g . In order to compute those Christoffel symbols, some calculating steps are carried out in the following section. The derivative of the product between the covariant and contravariant bases with respect to uj can be computed using Eqs. (2.156), (2.164) and (2.185). i i i δ i g Á gj ¼ g, k Á gj þ g Á gj,k ¼ j ,k ,k Γ^ i l Γ l i ¼ kl g Á gj þ jkðÞgl Á g i ð2:186Þ Γ^ δ l Γ l δ i ¼ kl j þ jk l Γ^ i Γ i Γ^ i Γ i ¼ kj þ jk ¼ kj þ kj ¼ 0

Thus, the relation between the Christoffel symbols of two coordinates results in

Γ^ i Γ i Γ i : kj ¼À kj ¼À jk ð2 187Þ

Using Eqs. (2.164) and (2.187), one obtains

Γ^ i Γ i Γ i Γ^ i : kj ¼À kj ¼À jk ¼ jk ð2 188Þ

Γ^ i It proves that the Christoffel symbol jk is symmetric with respect to j and k. Finally, the derivatives of the contravariant basis gi with respect to uj result from Eqs. (2.185) and (2.187). 2.5 Tensor Calculus in General Curvilinear Coordinates 85

i g i ¼ Γ^ gk ¼ÀΓ i gk ,j jk jk ð2:189Þ Γ^ i k Γ i k ¼ kjg ¼À kjg

2.5.8 Derivatives of Covariant Metric Coefﬁcients

The derivatives of the covariant metric coefﬁcient can be derived from the ﬁrst-kind Christoffel symbols written as 1 1 Γ ¼ g þ g À g ; Γ ¼ g þ g À g ð2:190Þ ikj 2 ij, k kj, i ik, j jki 2 ji,k ki,j jk,i

The derivative of the covariant metric coefﬁcient results by adding both Christoffel symbols given in Eq. (2.190). 1 1 Γ þΓ ¼ g þg Àg þ g þg Àg ikj jki 2 ij,k kj,i ik,j !2 ji ,k ki,j jk,i ! 1 1 ð2:191Þ ¼ gij,k þ gkj,i À gik,j þ gij,k þ gik,j Àgkj,i 2 ::::::::: ÀÀÀÀÀ 2 ÀÀÀÀÀ ::::::::: ¼ gij,k

k Therefore, the derivatives of the covariant metric coefﬁcient gij with respect to u can be expressed in the ﬁrst-kind Christoffel symbols.

∂g g ij ¼ Γ þ Γ ð2:192Þ ij,k ∂uk ikj jki

Using Eq. (2.173a), Eq. (2.192) can be rewritten in the second-kind Christoffel symbols.

g ¼ Γ þ Γ ij, k ikj jki : Γ l Γ l ð2 193Þ ¼ gjl ik þ gil jk

Similar to Eq. (2.192), one can write

Γ Γ : gjk,i ¼ jik þ kij ð2 194Þ

Subtracting Eq. (2.192) from Eq. (2.194), one obtains the relation 86 2 Tensor Analysis

Γ Γ Γ Γ gij, k À gjk,i ¼ ikj þ jki À jik À kij ÀÀÀÀ ÀÀÀÀ : ¼ Γjki À Γjik ð2 195Þ ¼ Γkji À Γijk

2.5.9 Covariant Derivatives of Tensors

2.5.9.1 Contravariant First-Order Tensors with Components Ti

The contravariant ﬁrst-order tensor (vector) T can be written in the covariant basis.

i : T ¼ T gi ð2 196Þ

Using Eq. (2.158), the derivative of the contravariant tensor T with respect to uj results in ÀÁ i i i T, j ¼ T g ¼ T g þ T g i , j ,j i i,j 2:197 i iΓ k ð Þ ¼ T, jgi þ T ijgk

The derivative of the contravariant tensor component Ti with respect to uj in Eq. (2.197) can be deﬁned as

∂Ti T i ð2:198Þ ,j ∂uj

Interchanging i with k in the second term on the RHS of Eq. (2.197), one obtains

TiΓ kg TkΓ i g ij k ¼ kj i : kΓ i ð2 199Þ ¼ T jkgi

Substituting Eq. (2.199) into Eq. (2.197), one obtains the derivative of T with respect to uj.

i kΓ i T, j ¼ T, jgi þ T jkgi T i Γ i Tk g ð2:200Þ ¼ , j þ jk i i T jgi

Therefore, the covariant derivative with respect to uj of the contravariant ﬁrst-order tensor (vector) can be written as 2.5 Tensor Calculus in General Curvilinear Coordinates 87 i i Γ i k i : T j ¼ T, j þ jkT ¼ T,j Á g ð2 201Þ

The covariant derivative of the contravariant ﬁrst-order tensor component is transformed in the new barred coordinates {ui}.

i n j ∂u ∂u Ti ¼ T k ∂ j ∂ k n u u ð2:202aÞ ∂uj ∂uk j i ) T ¼ T n n k ∂ui ∂u

Proof The ﬁrst-order tensor component can be written as

i j ∂u Ti ¼ T ð2:202bÞ ∂uj

Differentiating Ti with respect to uk and using the chain rule of differentiation, one obtains i i ∂T j ∂u ¼ T ∂uk ∂uj ,!k ! : j n i 2 i n ð2 202cÞ ∂T ∂u ∂u j ∂ u ∂u ¼ þ T ∂un ∂uk ∂uj ∂uj∂un ∂uk

Using Eq. (2.183), we have

2 i i p q ∂ u k ∂u ∂u ∂u ¼ Γ À Γ i ð2:202dÞ ∂un∂uj nj ∂uk pq ∂un ∂uj

Substituting Eq. (2.202d) into Eq. (2.202c) and interchanging the indices k with j and j with m, one obtains ! i j n i i p q n ∂T ∂T ∂u ∂u j k ∂u ∂u ∂u ∂u ¼ þ T Γ À Γ i ∂uk ∂un ∂uk ∂uj nj ∂uk pq ∂un ∂uj ∂uk

Thus, i p q n j i n i n ∂T j ∂u ∂u ∂u ∂T ∂u ∂u j k ∂u ∂u þ T Γ i ¼ þ T Γ ∂uk pq ∂un ∂uj ∂uk ∂un ∂uj ∂uk nj ∂uk ∂uk ð2:202eÞ j i n i n ∂T ∂u ∂u m j ∂u ∂u ¼ þ T Γ ∂un ∂uj ∂uk nm ∂uj ∂uk

The terms on the RHS of Eq. (2.202e) can be written as 88 2 Tensor Analysis

j i n i n hii n ∂T ∂u ∂u m j ∂u ∂u j m j ∂u ∂u þ T Γ ¼ T þ T Γ ∂ n ∂ j ∂ k nm ∂ j ∂ k ,n nm ∂ j ∂ k u u u u u u u ð2:202fÞ i n j ∂u ∂u ¼ T n ∂uj ∂uk

Using Eq. (2.202b) and interchanging the indices p with k and q with m, the terms on the LHS of Eq. (2.202e) are rearranged in ∂ i ∂ p ∂ q ∂ n ∂ j ∂ p ∂ q ∂ n T jΓ i u u u i m u Γ i u u u þ T pq n ¼ T,k þ T pq n ∂uk ∂u ∂uj ∂uk ∂um ∂u ∂uj ∂uk ∂uk ∂um ∂un ∂uj : i mΓ i ð2 202gÞ ¼ T, k þ T km ÂÃ∂un ∂uj ∂uk ∂um i mΓ i i ¼ T,k þ T km T k

Substituting Eqs. (2.202f) and (2.202g) into Eq. (2.202e), one obtains Eq. (2.202a).

i n j ∂u ∂u Ti ¼ T k j ∂ k n ∂u u ∂uj ∂uk j i : : : ) T ¼ T n ðÞq e d n k ∂ui ∂u

2.5.9.2 Covariant First-Order Tensors with Components Ti

The covariant ﬁrst-order tensor (vector) T can be written in the contravariant basis.

i T ¼ Tig ð2:203Þ

Using Eq. (2.189), the partial derivative of the tensor T results in

T T gi T gi T g i ,j ¼ ðÞi ,j ¼ i,j þ i , j : i Γ i k ð2 204Þ ¼ Ti,jg À Ti jkg

j The partial derivative of the covariant tensor component Ti with respect to u in Eq. (2.204) can be deﬁned as

∂T T i ð2:205Þ i, j ∂uj

Interchanging i with k in the second term on the RHS of Eq. (2.204), one obtains 2.5 Tensor Calculus in General Curvilinear Coordinates 89

Γ i k Γ k i Γ k i : Ti jkg Tk jig ¼ Tk ijg ð2 206Þ

Substituting Eq. (2.206) into Eq. (2.204), one obtains the derivative of the ﬁrst- order tensor component (vector) T with respect to uj.

i Γ k i T,j ¼ Ti,jg À Tk ijg T Γ kT gi ð2:207Þ ¼ i,j À ij k i Tijg

j Therefore, the covariant derivative of the tensor component Ti with respect to u can be deﬁned as Γ k : Tij¼ Ti,j À ijTk ¼ T,j Á gi ð2 208Þ

The covariant derivative of the ﬁrst-order tensor component Ti is transformed in the new barred coordinates {ui}, similarly to Eq. (2.202a), cf. [4, 7].

i j ∂u ∂u Tklj ¼ Tij ∂uk ∂ul : k l ð2 209Þ ∂u ∂u ) T ¼ T j ij kl∂ui ∂uj

2.5.9.3 Second-Order Tensors

Second-order tensors can be written in different expressions with covariant and contravariant bases.

i j ij j i i j : T ¼ Tijg g ¼ T gigj ¼ Ti g gj ¼ Tj gig ð2 210Þ

Similarly, the covariant derivatives with respect to uk of the second-order tensor components of T can be calculated as, cf. [3, 4, 7]

Γ m Γ m Tijj k ¼ Tij,k À ik Tmj À jk Tim Tij Tij Γ i Tmj Γ j Tim jk ¼ , k þ km þ km : i i Γ i m Γ m i ð2 211aÞ Tj jk ¼ Tj, k þ kmTj À jk Tm j j Γ m j Γ j m Ti jk ¼ Ti, k À ik Tm þ kmTi

ij i k where Tij,k, T ,k, and T j,k are the partial derivatives with respect to u of the covariant, contravariant, and mixed tensor components. Note that they are different to the covariant derivatives of the tensor components, as deﬁned in Eq. (2.211a). 90 2 Tensor Analysis

In the coordinate transformation from the curvilinear coordinates {ui} to the new barred curvilinear coordinates {uα}, the covariant derivative of the covariant second-order tensor with respect to uγ can be calculated using the chain rule of differentiation, similarly to Eq. (2.202a), cf. [4, 7].

i j k ∂u ∂u ∂u Tαβ ¼ T ð2:211bÞ γ ij k ∂uα ∂uβ ∂uγ α where the partial derivatives u;i are called the shift tensor between two coordinate systems. This relation in Eq. (2.211b) is the chain rule of the covariant derivatives of the second-order tensors in the coordinate transformation. Analogously, the covariant derivatives of the second-order tensors of different types in the new barred curvilinear coordinates are calculated using the shift tensors.

i j k ∂u ∂u ∂u Tαβ ¼ T ; γ ij k ∂uα ∂uβ ∂uγ α β αβ ∂ ∂ ∂ k ij u u u : T ¼ T γ ; ð2 211cÞ γ k ∂ui ∂uj ∂u α α ∂u ∂uj ∂uk i : Tβ ¼ Tj β γ γ k ∂ui ∂u ∂u

2.5.10 Riemann-Christoffel Tensor

The Riemann-Christoffel tensor is closely related to the Gaussian curvature of the surface in differential geometry that will be discussed in Chap. 3. At first, let us look into the second covariant derivative of an arbitrary first-order tensor. The covariant derivative of the tensor with respect to uj has been derived in Eq. (2.208). Γ k : Tij¼ Ti,j À ijTk ð2 212Þ Obviously, the covariant derivative Ti j is a second-order tensor component. k Differentiating Ti j with respect to u , the covariant derivative of the second- order tensor (component) Ti j is the second covariant derivative of an arbitrary first- order tensor (component) Ti. This second covariant derivative has been given from Eq. (2.211a)[3]. 2.5 Tensor Calculus in General Curvilinear Coordinates 91 À Á Tijk ÀTij Ájk Γ m Γ m : ¼ Tij , k À ik TmjÀ jk Timj ð2 213Þ Γ m Γ m ¼ Tij, k À ik TmjÀ jk Timj

Equation (2.212) delivers the relations of Γ m Γ m : Tij, k ¼ Ti, jk À ij,kTm þ ij Tm, k ð2 214aÞ Γ m Γ m Γ n : ik Tmj¼ ik Tm, j À mjTn ð2 214bÞ ÀÁ Γ m Γ m Γ n : jk Timj¼ jk Ti,m À imTn ð2 214cÞ

Inserting Eqs. (2.214a)–(2.214c) into Eq. (2.213), one obtains the second covariant derivative of Ti. Γ m Γ m Tijk¼ Tij, k Àik TmjÀ jk Timj Γ m Γ m ¼ Ti,jk À ij, kTm þ ij Tm, k ÀÁ Γ m Γ n Γ m Γ n ð2:215Þ À ik Tm,j À mjTn À jk Ti,m À imTn Γ m Γ m ¼ Ti, jk À ij, kTm À ij Tm, k Γ m Γ mΓ n Γ m Γ mΓ n À ik Tm,j þ ik mjTn À jk Ti,m þ jk imTn where the second partial derivative of Ti is symmetric with respect to j and k:

∂2T ∂2T T i ¼ i T ð2:216Þ i, jk ∂uj∂uk ∂uk∂uj i, kj

Interchanging the indices j with k in Eq. (2.215), one obtains Γ m Γ m Tikj¼ Ti,kj À ik,jTm À ik Tm, j : Γ m Γ mΓ n Γ m Γ mΓ n ð2 217Þ À ij Tm, k þ ij mkTn À kj Ti, m þ kj imTn

Using the symmetry properties given in Eqs. (2.164) and (2.216), Eq. (2.217) can be rewritten as Γ m Γ m Tikj¼ Ti,jk À ik,jTm À ik Tm, j : Γ m Γ mΓ n Γ m Γ mΓ n ð2 218Þ À ij Tm, k þ ij mkTn À jk Ti, m þ jk imTn

In a ﬂat space, the second covariant derivatives in Eqs. (2.215) and (2.218) are identical. However, they are not equal in a curved space because of its surface curvature. The difference of both second covariant derivatives is proportional to the curvature tensor. Subtracting Eq. (2.215) from Eq. (2.218), the curvature tensor results in 92 2 Tensor Analysis

T À T ¼ Γ n À Γ n þ Γ mΓ n À Γ mΓ n T ijk ikj ik, j ij,k ik mj ij mk n 2:219 n ð Þ RijkTn

The Riemann-Christoffel tensor (Riemann curvature tensor) can be expressed as

n Γ n Γ n Γ mΓ n Γ mΓ n : Rijk ik,j À ij, k þ ik mj À ij mk ð2 220Þ

Straightforwardly, the Riemann-Christoffel tensor is a fourth-order tensor with respect to the indices of i, j, k, and n. They contain 81 (¼34) components in a three-dimensional space. In Eq. (2.220), the partial derivatives of the Christoffel symbols are deﬁned by

∂Γ n ∂Γ n Γ n ¼ ik ; Γ n ¼ ij ð2:221Þ ik, j ∂uj ij, k ∂uk

Furthermore, the covariant Riemann curvature tensor of fourth order is deﬁned by the Riemann-Christoffel tensor and covariant metric coefﬁcients.

n n ln : Rlijk ¼ glnRijk , Rijk ¼ g Rlijk ð2 222Þ

The Riemann curvature tensor has four following properties using the relation given in Eq. (2.222)[4]: • First skew-symmetry with respect to l and i:

Rlijk ¼ÀRiljk ð2:223Þ

• Second skew-symmetry with respect to j and k:

Rlijk ¼ÀRlikj; n n ð2:224Þ Rijk ¼ÀRikj

• Block symmetry with respect to two pairs (l, i) and ( j, k):

Rlijk ¼ Rjkli ð2:225Þ

• Cyclic property in i, j, k:

Rlijk þ Rljki þ Rlkij ¼ 0; n n n ð2:226Þ Rijk þ Rjki þ Rkij ¼ 0

Equation (2.226) is called the Bianchi ﬁrst identity. 2.5 Tensor Calculus in General Curvilinear Coordinates 93

Resulting from these properties, there are six components of Rlijk in the three- dimensional space as follows [3]:

Rlijk ¼ R3131, R3132, R3232, R1212, R3112, R3212 ð2:227Þ

In Cartesian coordinates, all Christoffel symbols equal zero according to Eq. (2.184). Therefore, the Riemann-Christoffel tensor, as given in Eq. (2.220) must be equal to zero.

R n ¼ Γ n À Γ n þ Γ mΓ n À Γ mΓ n ijk ik,j ij, k ik mj ij mk ð2:228Þ ¼ 0

Thus, the Riemann surface curvature tensor can be written as

n : Rlijk ¼ glnRijk ¼ 0 ð2 229Þ

In this case, Euclidean N-space with orthonormal Cartesian coordinates is considered as a ﬂat space because the Riemann curvature tensor there equals zero. In the following section, the Riemann curvature tensor can be calculated from the Christoffel symbols of ﬁrst and second kinds. From Eq. (2.222) the Riemann curvature tensor can be rewritten as

l Rhijk ¼ ghlRijk

Using Eq. (2.220), the Riemann curvature tensor results in

n Rhijk ¼ghnRijk ¼ g Γ n À Γ n þ Γ mΓ n À Γ mΓ n hn ik,j ij, k ik mjij mk ÀÁ n ∂ Γ n ∂ g Γ ghn ij ¼ hn ik À Γ n g À þ Γ ng þ g Γ mΓ n À g Γ mΓ n ∂uj ik hn, j ∂uk ij hn,k hn ik mj hn ij mk ÀÁ n ∂ Γ n ∂ g Γ ghn ij ¼ hn ik À Γ n g À þ Γ ng þ Γ mΓ À Γ mΓ ∂uj ik hn, j ∂uk ij hn,k ik mjh ij mkh ð2:230aÞ

Changing the index m into n in both last terms on the RHS of Eq. (2.230a), one obtains 94 2 Tensor Analysis ÀÁ n ∂ Γ n ∂ g Γ ghn ij R ¼ hn ik À Γ n g À þ Γ ng þ Γ n Γ À Γ nΓ hijk ∂uj ikhn, j ∂uk ij hn,k ik njh ij nkh ÀÁ n ∂ Γ n ∂ g Γ ghn ij ÀÁ ¼ hn ik À À Γ n g À Γ þ Γ n g À Γ ∂uj ∂uk ik hn,j njh ij hn,k nkh ð2:230bÞ

Using Eq. (2.192) for the ﬁrst-kind Christoffel symbols in Eq. (2.230b), the Riemann curvature tensor becomes

∂Γ ∂Γ R ¼ g R n ¼ ikh À ijh hijk hnijk ∂uj ∂uk : Γ n Γ Γ Γ Γ n Γ Γ Γ ð2 230cÞ À ik hjn þ njh À njh þ ij hkn þ nkh À nkh ÀÀÀÀ ÀÀÀÀ Γ Γ Γ n Γ Γ nΓ ¼ ikh,j À ijh,k À ik hjn þ ij hkn

2.5.11 Ricci’s Lemma

k The covariant derivative of the metric covariant coefﬁcient gij with respect to u results from Eq. (2.211a) changing Tij into gij. Then, using Eq. (2.193), one obtains ∂g g j ¼ ij À g Γ m þ g Γ n ij k ∂uk mj ik in jk ð2:231aÞ : : : ¼ gij,k À gij,k ¼ 0 ! ðÞq e d

Therefore,

∂g g ij ¼ g Γ m þ g Γ n ð2:231bÞ ij,k ∂uk mj ik in jk

The Kronecker delta is the product of the covariant and contravariant metric coefﬁcients:

δ j ij l ¼ glig

The partial derivative with respect to uk of the Kronecker delta (invariant) equals zero and can be written as 2.5 Tensor Calculus in General Curvilinear Coordinates 95

∂g ∂gij δ j ij li ij l, k ¼ ðÞglig , k ¼ g þ gli ∂uk ∂uk : ij ij ð2 232aÞ ¼ gli,kg þ glig,k ¼ 0

Multiplying Eq. (2.232a)byglm, one obtains

ij lm δ m ij g g gli,k þ i g,k ¼ 0 : mj ij lm ð2 232bÞ ) g,k ¼Àg g gli,k

Interchanging the indices, it results in

mj ij lm g,k ¼Àg g gli, k : ij mi nj ð2 232cÞ ) g, k ¼Àg g gmn,k

Using Eq. (2.211a), the covariant derivative of the metric contravariant coefﬁcient gij with respect to uk can be written as

∂gij gijj ¼ þ gmjΓ i þ gimΓ j k ∂uk km km ð2:233aÞ ij mjΓ i imΓ j ¼ g,k þ g km þ g km

Substituting Eqs. (2.231b), (2.232c) and (2.233a), one obtains after interchanging the indices gijj ¼ gij þ gmjΓ i þ gimΓ j k ,k km km ¼Àgmignjg þ gmjΓ i þ gimΓ j mn,k km km : ¼ÀgmiΓ j À gnjΓ i þ gmjΓ i þ gimΓ j ð2 233bÞ mk nk km km imΓ j mjΓ i mjΓ i imΓ j ¼Àg km À g km þ g km þ g km ¼ 0 ðÞq:e:d:

Note that Eqs. (2.231a) and (2.233b) are known as Ricci’s lemma.

2.5.12 Derivative of the Jacobian

In the following section, the derivative of the Jacobian J that is always positive in a right-handed-rule coordinate system can be calculated and its result is very useful in the Nabla operator (cf. [4, 7]). The determinant of the metric coefﬁcient tensor is given from Eq. (2.17): 96 2 Tensor Analysis g g : g 11 12 1N g g : g 2 det g ¼ 21 22 2N ¼ g ¼ J > 0 ð2:234Þ ij :::: : gN1 gN2 gNN

The contravariant metric coefﬁcient gij results from the cofactor Gij of the covariant metric coefﬁcient gij und the determinant g.

Gij gij ¼ ) Gij ¼ ggij ð2:235Þ g

Differentiating both sides of Eq. (2.234) with respect to uk, one obtains

∂g ∂g ¼ ij Gij for i, j ¼ 1, 2, ..., N ð2:236Þ ∂uk ∂uk

Prove Eq. (2.236): ∂ ∂ ∂ g g : g g11 g12 : g1N 11 12 1N ∂ k ∂ k ∂ k g g : g ∂g u u : u 21 22 2N ¼ g21 g22 g2N þ ...þ :::: ∂uk :::: ∂ ∂ ∂ gN1 gN2 gNN : : gN1 gN2 gNN ∂uk ∂uk ∂uk ð2:237Þ ∂g ∂g ∂g ¼ 11 G11 þ 12 G12 þ ...þ ...þ NN GNN ∂uk ∂uk ∂uk ∂g ¼ ij Gij for i, j ¼ 1, 2, ..., NqðÞ:e:d: ∂uk

Substituting Eq. (2.235) into Eq. (2.236), it gives

∂g ∂g ¼ ij Gij ∂uk ∂uk ∂g ð2:238Þ ¼ ij ggij ∂uk ij gij,kgg

Inserting Eq. (2.231b) into Eq. (2.238), one obtains 2.5 Tensor Calculus in General Curvilinear Coordinates 97

∂g ¼ g ggij ∂uk ij, k ¼ g Γ m þ g Γ n ggij mj ik in jk ð2:239Þ ¼ g δ i Γ m þ δ j Γ n m ik n jk Γ i Γ j Γ i ¼ g ik þ jk ¼ 2g ik

Using the chain rule of differentiation, the Christoffel symbol in Eq. (2.239) can be expressed in the Jacobian J > 0. ÀÁpffiffiffi 1 ∂g ∂ ln g Γ i ¼ ¼ ik ∂ k ∂ k 2g u u ð2:240Þ ∂ðÞlnJ 1 ∂J ¼ ¼ ∂uk J ∂uk

i Prove that Rijk ¼ 0 Using Eq. (2.220) for n ¼ i, the Riemann-Christoffel tensor can be written as

i Γ i Γ i Γ mΓ i Γ mΓ i Rijk ¼ ik,j À ij, k þ ik mj À ij mk

Interchanging j with k in the last term on the RHS of the equation, one obtains

i Γ i Γ i Γ mΓ i Γ mΓ i Rijk ¼ ik, j À ij,k þ ik mj À ik mj Γ i Γ i ¼ ik, j À ij,k

Using Eq. (2.240), the above Riemann-Christoffel tensor can be rewritten as

i Γ i Γ i Rijk ¼ ikÀÁ, j À ij,k ÀÁ 2 pffiffiffi 2 pffiffiffi ∂ ln g ∂ ln g : ¼ À ð2 241Þ ∂uj∂uk ∂uk∂uj ¼ 0 ðÞq:e:d:

2.5.13 Ricci Tensor

Both Ricci and Einstein tensors are very useful mathematical tools in the relativity theory. Note that tensors using in the relativity fields have been mostly written in the abstract index notation defined by Penrose [8]. This index notation uses the indices to express the tensor types, rather than their covariant components in the basis {gi}. The first-kind Ricci tensor results from the index contraction of k and n for n ¼ k of the Riemann-Christoffel tensor, as given in Eq. (2.220). 98 2 Tensor Analysis

k Rij Rijk ∂Γ k ∂Γ k ð2:242Þ ¼ ik À ij À Γ rΓ k þ Γ r Γ k ∂uj ∂uk ij rk ik rj

The second-kind Ricci tensor can be deﬁned as

R i gikR j kj ∂Γ m ∂Γ m ð2:243Þ ¼ gik km À kj À Γ mΓ n þ Γ m Γ n ∂uj ∂um kj mn kn mj

Using the Christoffel symbol in Eq. (2.240) pffiffiffi ÀÁpffiffiffi 1 ∂ g ∂ ln g Γ j ¼ ffiffiffi ¼ ij p ∂ i ∂ i g u u ð2:244Þ 1 ∂J ∂ðÞlnJ ¼ ¼ ; J ∂ui ∂ui the first-kind Ricci tensor can be rewritten as ÀÁpffiffiffi ÀÁpffiffiffi ∂2 ln g ∂Γ k ∂ ln g R ¼ À ij À Γ k þ Γ r Γ k ij ∂ui∂uj ∂uk ij ∂uk !ik rj ÀÁpffiffiffi ÀÁpffiffiffi ∂2 ln g ∂Γ k ∂ ln g ¼ À ij þ Γ k þ Γ r Γ k ∂ui∂uj ∂uk ij ∂uk ik rj ÀÁffiffiffi ffiffiffi! 2 p k p 2:245 ∂ ln g 1 pffiffiffi ∂Γ ∂ g ð Þ ¼ À ffiffiffi g ij þ Γ k þ Γ r Γ k ∂ i∂ j p ∂ k ij ∂ k ik rj u u g u u ÀÁ ffiffiffi 2 pffiffiffi ∂ p Γ k ∂ ln g 1 g ij ¼ À pffiffiffi þ Γ r Γ k ∂ui∂uj g ∂uk ik rj

Interchanging i with j in Eq. (2.245), one obtains ÀÁ ffiffiffi 2 pffiffiffi ∂ p Γ k ∂ ln g 1 g ji ffiffiffi Γ r Γ k Rji ¼ À p þ jk ri ∂uj∂ui g ∂uk ÀÁ ffiffiffi : 2 pffiffiffi ∂ p Γ k ð2 246Þ ∂ ln g 1 g ij ¼ À pffiffiffi þ Γ k Γ r ∂ui∂uj g ∂uk ir kj ¼ Rij

This result states that the ﬁrst-kind Ricci tensor is symmetric with respect to i and j. Substituting Eq. (2.245) with k ¼ m, r ¼ n, and i ¼ k into Eq. (2.243), the second- kind Ricci tensor results in 2.5 Tensor Calculus in General Curvilinear Coordinates 99

i ik Rj ¼ g 0Rkj 1 ÀÁffiffiffi pffiffiffi ∂2 p ∂ gΓ m : @ ln g 1 kj n mA ð2 247Þ ¼ gik À pffiffiffi þ Γ Γ ∂uk∂uj g ∂um km nj

The Ricci curvature R can be deﬁned as

i ij : R Ri ¼ g Rij ð2 248Þ

Substituting Eq. (2.245) into Eq. (2.248), the Ricci curvature results in 0 1 ÀÁffiffiffi pffiffiffi ∂2 p ∂ gΓ k ij@ ln g 1 ij r k A R ¼ g À pffiffiffi þ Γ Γ ð2:249Þ ∂ui∂uj g ∂uk ik rj

2.5.14 Einstein Tensor

The Einstein tensor is deﬁned by the second-kind Ricci tensor, Kronecker delta, and the Ricci curvature.

1 G i R i À δ iR ð2:250aÞ j j 2 j

The Einstein tensor is a mixed second-order tensor and can be written as

i ik : Gj ¼ g Gkj ð2 250bÞ

Using the tensor contraction rules, the covariant Einstein tensor results in 1 G ¼ g G k ¼ g R k À δ kR ij ik j ik j 2 j 1 ¼ R À g R : ij 2 ij ð2 251Þ 1 ¼ R À g R ji 2 ji ¼ Gji

This result proves that the covariant Einstein tensor is symmetric due to the symmetry of the Ricci tensor. The Bianchi ﬁrst identity in Eq. (2.226) gives 100 2 Tensor Analysis

Rlijk þ Rljki þ Rlkij ¼ 0; n n n ð2:252Þ Rijk þ Rjki þ Rkij ¼ 0

Differentiating covariantly Eq. (2.252) with respect to um, uk, and ul and then multiplying it by the covariant metric coefﬁcients gin, one obtains the Bianchi second identity, cf. [4, 7, 9, 10]. R n þ R n þ R n Á g ¼ 0 Á g jkl jlm jmk in in : m k l ð2 253Þ ) Rijkl m þ Rijlm k þ Rijmk l ¼ 0

Due to skew-symmetry of the covariant Riemann curvature tensors, as discussed in Eqs. (2.223) and (2.224), Eq. (2.253) can be rewritten as : Rijkl m À Rijml k À Rjimk l ¼ 0 ð2 254Þ

Multiplying Eq. (2.254)bygilgjk and using the tensor contraction rules (cf. Sect. 2.3.5), one obtains Rijkl m À Rijml k À Rjimk l ¼ 0 , il jk il jk il jk g g Rijkl m À g g R ijml k À g g Rjimk l jk jk il ¼ g Rjk m À g Rjm k À g Rim l k l ¼ Rjm À Rm kÀ Rm l ðÞl ! k k ¼ Rjm À 2Rm k ¼ 0

Thus, 1 R k ¼ Rj ð2:255Þ m k 2 m

Using Eq. (2.232a) and the symmetry of the Christoffel symbols, the covariant derivative of the Kronecker delta with respect to uk is equal to zero. δ i ¼ δ i þ Γ i δ m À Γ mδ i j j,k km j jk m k : δ i Γ i Γ i ð2 256Þ ¼ j,k þ kj À jk ¼ 0

Differentiating covariantly the Einstein tensor in Eq. (2.250a) with respect to uk and using Eq. (2.256), one obtains the covariant derivative References 101 i i 1δ i Gj ¼ Rj À j R k 2 k 1 i δ i δ i ¼ Rj À j R þ j R ð2:257Þ k 2 k k 1 i δ i ¼ Rj À j R k 2 k

Changing the index i into k in Eq. (2.257) and using Eq. (2.255), the divergence of the Einstein tensor equals zero. k k 1 δ k k 1 Gj ¼ Rj À j R ¼ Rj À Rjj ¼ 0 k k 2 k k 2 ∇ k j : : : : ) Div G Á G ¼ Gj g ¼ 0 ðÞq e d ð2 258Þ k

This result is very important and has been often used in the general relativity theories and other relativity ﬁelds.

References

1. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 2. Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York, NY (1982) 3. Klingbeil, E.: Tensorrechnung fur€ Ingenieure (in German). B.I.-Wissenschafts-Verlag, Mann- heim (1966) 4. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 5. Itskov, M.: Tensor Algebra and Tensor Analysis for Engineers – With Applications to Continuum Mechanics, 2nd edn. Springer, Berlin (2010) 6. Oeijord, N.K.: The Very Basics of Tensors. IUniverse, New York, NY (2005) 7. De, U.C., Shaikh, A., Sengupta, J.: Tensor Calculus, 2nd edn. Alpha Science, Oxford (2012) 8. Penrose, R.: The Road to Reality. Alfred A. Knopf, New York, NY (2005) 9. Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2000) 10. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. AMS, Oxford University Press, Rhode Island (1978) 11. Kay, D.C.: Tensor Calculus. Schaum’s Outline Series. McGraw-Hill, New York, NY (2011) 12. McConnell, A.J.: Applications of Tensor Analysis. Dover Publications, New York, NY (1960) 13. Brannon, R.M.: Curvilinear Analysis in a Euclidean Space. University of New Mexico, Albuquerque, NM (2004) Chapter 3 Elementary Differential Geometry

3.1 Introduction

We consider an N-dimensional Riemannian manifold M and let gi be a basis at the 1 N 1 N point Pi(u ,...,u ) and gj be another basis at the other point Pj(u ,...,u ). Note that each such basis may only exist in a local neighborhood of the respective points, and not necessarily for the whole space. For each such point we may construct an embedded afﬁne tangential manifold. The N-tuple of coordinates are invariant in any chosen basis; however, its components on the coordinates change as the coordinate system varies. Therefore, the relating components have to be taken into account by the coordinate transformations.

3.2 Arc Length and Surface in Curvilinear Coordinates

Consider two points P(u1,...,uN) and Q(u1,...,uN)ofanN-tuple of the coordinates (u1,...,uN) in the parameterized curve C2 RN. The coordinates (u1,...,uN) can be assumed to be a function of the parameter λ that varies from P(λ1)toQ(λ2), as shown in Fig. 3.1. The arc length ds between the points P and Q results from ds 2 dr dr ¼ Á ð3:1Þ dλ dλ dλ where the derivative of the vector r(u,v) can be calculated by

© Springer-Verlag Berlin Heidelberg 2017 103 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5_3 104 3 Elementary Differential Geometry

Fig. 3.1 Arc length and surface on the surface (S) dA

n dv v T g2 g ds Q C P 1 du (S)

r(u(ll),v( )) u

dr dðÞg ui ∂ui ¼ i ¼ g dλ dλ i ∂λ ð3:2Þ _ i λ ; giu ðÞ 8i ¼ 1, 2

Substituting Eq. (3.2) into Eq. (3.1), one obtains the arc length PQ. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε _ _ λ ε _ _ λ : ds ¼ ðÞÁgiu gjv d ¼ giju Á v d ð3 3Þ where ε(¼Æ1) is the functional indicator, which ensures that the square root always exists. Therefore, the arc length of PQ is given by integrating Eq. (3.3) from the parameter λ1 to the parameter λ2.

λ ð2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε _ λ _ λ λ : s ¼ gij u ðÞÁv ðÞd ð3 4Þ

λ1 where the covariant metric coefﬁcients gij are deﬁned by

gij ¼ gji ¼ gi Á gj ∂r ∂r ð3:5Þ ¼ Á ; i u, j v ∂u ∂v

Thus, the metric coefﬁcient tensor of the parameterized surface S is given in 3.2 Arc Length and Surface in Curvilinear Coordinates 105

! g g M ¼ g ¼ 11 12 ij g g 21!22 ! ð3:6Þ r r r r EF ¼ u u u v rvru rvrv FG

The area differential of the tangent plane T at the point P can be calculated by

dA ¼ jjdu Â dv ð3:7Þ ¼ jjg1du Â g2dv ¼ jjg1 Â g2 dudv

Using the Lagrange’s identity, Eq. (3.7) becomes

dA ¼ jjg Â g dudv q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : ¼ g11g22 À ðÞg12 dudv ¼ det gij dudv ð3 8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ EG À F2dudv

Integrating Eq. (3.8), the area of the surface S results in

λ λ ð2 ð2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ EG À F2dudv ð3:9Þ

λ1 λ1

In the following section, the circumference at the equator and surface area of a sphere with a radius R are calculated (see Fig. 3.2). The location vector of a given point P(u(λ),v(λ)) in the parameterized surface of the sphere (S) can be written as

ðÞS : x2 þ 0y2 þ z2 ¼ R2 )1 R sin φ cos θ ð3:10Þ rðÞ¼φ; θ @ R sin φ sin θ A; u φ 2 ½½0; π ; v θ 2 ½½0, 2π R cos φ

The covariant bases can be calculated in 0 1 φ θ ∂ R cos cos r @ φ θ A; gu ¼ ∂φ ¼ R cos sin φ 0 ÀR sin 1 : φ θ ð3 11Þ ∂ ÀR sin sin r @ φ θ A gv ¼ ∂θ ¼ R sin cos 0

Thus, the metric coefﬁcient tensor results from Eq. (3.11). 106 3 Elementary Differential Geometry

Fig. 3.2 Arc length and z R sinϕ (ϕ, θ) → (u, v): surface of a sphere (S) u ≡ ϕ ; v ≡ θ n As gv P

ϕ e3 R e gu 0 2 y C e1 eq θ R cosϕ

x (S)

! ! ! 2 g11 g12 EF R 0 : M ¼ ¼ 2 2φ ð3 12Þ g21 g22 FG 0 R sin

The circumference at the equator is given at u φ ¼ π/2 and v θ (λ) ¼ λ.

λ λ ð2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ λ _ λ λ _ 2 _ _ _ 2 λ : Ceq ¼ gij u ðÞÁv ðÞd ¼ g11u þ 2g12u v þ g22v d ð3 13Þ

λ1 λ1 where u_ ¼ 0 ; v_ ¼ 1. Therefore,

λð2 2ðπ pffiffiffiffiffiffiffiffi π C ¼ g dλ ¼ R sin dλ ¼ 2πR ð3:14Þ eq 22 2 λ1 0

The surface area of the sphere can be computed according to Eq. (3.9).

λ λ π π ð2 ð2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð ðpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 AS ¼ EG À F du dv ¼ EG À F dφ dθ λ λ 1 1 0 0 ð3:15Þ 2ðπðπ 2ðπ 2 φ φ θ 2 φ π θ π 2 ¼ R sin d d ¼À R cos 0 d ¼ 4 R 0 0 0 3.3 Unit Tangent and Normal Vector to Surface 107

3.3 Unit Tangent and Normal Vector to Surface

The unit tangent vectors to the parameterized curves u and v at the point P have the same direction as the covariant bases g1 and g2. Both tangent vectors generate the tangent plane T to the differentiable Riemannian surface (S) at the point P. The unit normal vector n is perpendicular to the tangent plane T at the point P, as shown in Fig. 3.1. The unit tangent vector t is defined, as given in Eq. (B.1a). g 1 ∂r t ¼ g* ffiffiffiffiffiffiffiffii ¼ ffiffiffiffiffiffiffiffi ; 8i ¼ 1, 2 i i pg pg ∂ui 8 ðÞii ðÞii ∂ > * 1 r < t1 ¼ g ¼ pffiffiffiffiffiffi ; ðÞ1 u ð3:16Þ 1 g ∂u ) 11 > 1 ∂r :> * ffiffiffiffiffiffi ; t2 ¼ g2 ¼ p ∂ ðÞ2 v g22 v

The unit normal vector is perpendicular to the unit tangent vectors at the point P and can be written as

∂ ∂ g Â g r Â r n ¼ ðÞ¼t Â t 1 2 ¼ ∂u ∂v ð3:17Þ 1 2 jjg Â g ∂r ∂r 1 2 ∂u Â ∂v

Using Eq. (3.8), the unit normal vector can be rewritten as

g1 Â g2 n ¼ ðÞ¼t1 Â t2 jjg1 Â g2 ru Â rv ru Â rv : ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3 18Þ EG À F2 det gij

in which the cross product of g1 and g2 can be calculated by e1 e2 e3 ∂ ∂ ÀÁ ÀÁ ÀÁ ε r r : g1 Â g2 ¼ ∂r ∂r ∂r ¼ ijk ∂ ∂ ek ð3 19Þ ÀÁ∂u 1 ÀÁ∂u 2 ÀÁ∂u 3 u i v j ∂r ∂r ∂r ∂v 1 ∂v 2 ∂v 3 where εijk is the permutation symbol given in Eq. (A.5) in Appendix A. The unit normal vector to the differentiable spherical surface (S) at the point P in Fig. 3.2 can be computed from Eq. (3.11). 108 3 Elementary Differential Geometry 0 1 0 1 φ θ φ θ ∂ R cos cos ∂ ÀR sin sin r @ φ θ A; r @ φ θ A : g1 ¼ ∂φ ¼ R cos sin g2 ¼ ∂θ ¼ R sin cos ð3 20Þ ÀR sin φ 0

Therefore, e1 e2 e3 φ θ φ θ φ g1 Â g2 ¼ R cos cos R cos sin ÀR sin ÀR sin φ sin θ R sin φ cos θ 0 0 1 ð3:21Þ R2 sin 2φ cos θ B C ¼ @ R2 sin 2φ sin θ A R2 sin φ cos φ

Thus, the unit normal vector results from Eqs. (3.12), (3.18), and (3.21). 0 1 0 1 R2 sin 2φ cos θ sin φ cos θ g1 Â g2 1 @ A @ A n ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ R2 sin 2φ sin θ ¼ sin φ sin θ ð3:22Þ 2 R2 sin φ EG À F R2 sin φ cos φ cos φ

Straightforwardly, the unit normal vector depends on each point P(φ, θ) on the spherical surface (S) and has a vector length of 1.

3.4 The First Fundamental Form

The first and second fundamental forms of surfaces are two important characteristics in differential geometry as they are used to measure arc lengths and areas of surfaces, to identify isometric surfaces, and to find the extrema of surfaces. The Gaussian and mean curvatures of surfaces are based on both fundamental forms. Initially, the first fundamental form is examined in the following section. Figure 3.3 displays the unit tangent vector t to the parameterized curve C at the point P in the differentiable surface (S). The unit normal vector n to the surface (S) at the point P is perpendicular to t and (S) at the point P. Both unit tangent and normal vectors generate a Frenet orthonormal frame {t, n, (n Â t)} in which three unit vectors are orthogonal to each other, as shown in Fig. 3.3. The curvature vector k to the curve C can be rewritten as a linear combination of the normal curvature vector kn and the geodesic curvature vector kg at the point P in the Frenet frame. κ κ κ k ¼ kn þqkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig , nC ¼ nn þ gðÞn Â t : κ κ2 λ κ2 λ ð3 23Þ ) ¼ nðÞþ gðÞ 3.4 The First Fundamental Form 109

Fig. 3.3 Normal and k = κn geodesic curvatures of the C k = κ n surface S n n

θ T = κ × n k g g (n t) n×t P t C 1 II dr 2 Q

r + dr S r(u(λ),v(λ))

O where κ is the curvature of the curve C at P; κn is the normal curvature of the surface (S)atP in the direction t; κg is the geodesic curvature of the surface (S)atP. The ﬁrst fundamental form I of the surface (S) is deﬁned by the arc length on the curve C in the surface (S).

I ds2 ¼ dr Á dr ∂r ∂r ð3:24Þ ¼ du j Á du j ∂ui ∂u j

The ﬁrst term on the RHS of Eq. (3.24) can be rewritten in the parameterized coordinate ui(λ).

∂r dui ¼ g dui ∂ui i ð3:25Þ duiðÞλ ¼ g dλ ¼ g u_ iðÞλ dλ i dλ i

i in which gi is the covariant basis of the curvilinear coordinate u , as shown in Fig. 3.1. Inserting Eq. (3.25) into Eq. (3.24), the ﬁrst fundamental form results in

_ i λ _ j λ λ2 I ¼ gi Á gju ðÞu ðÞd i j : ¼ gijdu du ð3 26Þ 2 2 ¼ g11du þ 2g12dudv þ g22dv

Using Eq. (3.6), the ﬁrst fundamental form of the surface (S) can be rewritten as 110 3 Elementary Differential Geometry

I ¼ Edu2 þ 2Fdudv þ Gdv2 ð3:27aÞ

Therefore, the arc length ds can be rewritten as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ Eu_ 2 þ 2Fu_ v_ þ Gv_ 2dλ ð3:27bÞ where E, F, and G are the covariant metric coefficients of the metric tensor M,as given in ! ! EF ru Á ru ru Á rv : M ¼ gij ¼ ð3 28Þ FG rv Á ru rv Á rv

3.5 The Second Fundamental Form

The second fundamental form II is deﬁned as twice of the projection of the arc length vector dr on the unit normal vector n of the parameterized surface (S) at the point P, as demonstrated in Fig. 3.3.

1 II dr Á n ð3:29Þ 2

Using the Taylor’s series for a vectorial function with two variables u and v, the differential of the arc length vector dr(u,v) can be written in the second order. ∂r ∂r dr ¼ ∂ du þ ∂ dv u v ! 1 ∂2r ∂2r ∂2r ÀÁ þ du2 þ 2 dudv þ dv2 þ Odr3 ð3:30Þ 2 ∂u2 ∂u∂v ∂v2 1ÀÁÀÁ r du þ r dv þ r du2 þ 2r dudv þ r dv2 þ Odr3 u v 2 uu uv vv

Therefore, the second fundamental form can be computed as ÀÁ 2 2 II 2ðÞþru Á ndu þ rv Á ndv ruu Á ndu þ 2ruv Á ndudv þ rvv Á ndv ð3:31Þ

Due to the orthogonality of (ru, rv) and n, one obtains the inner products

ru Á n ¼ rv Á n ¼ 0 ð3:32Þ where the covariant bases of the curvilinear coordinate (u,v) are shown in Fig. 3.1. 3.5 The Second Fundamental Form 111

∂r ∂r r ¼ ¼ g ; r ¼ ¼ g ð3:33Þ u ∂u 1 v ∂v 2

Substituting Eqs. (3.31) and (3.32), the second fundamental form results in

2 2 II ¼ ruu Á ndu þ 2ruv Á ndudv þ rvv Á ndv ð3:34Þ Ldu2 þ 2Mdudv þ Ndv2 in which L, M, and N are the elements of the Hessian tensor [1, 2] ! ! ÀÁ LM ruu Á nruv Á n H ¼ hij ¼ ð3:35aÞ MN ruv Á nrvv Á n with

ru Â rv ru Â rv n ¼ ðÞ¼t1 Â t2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:35bÞ EG À F2 det gij

In case of the projection equals zero, the second fundamental form II is also equal to zero. It gives the quadratic equation of du according to Eq. (3.34).

Ldu2 þ 2Mdudv þ Ndv2 ¼ 0 ð3:36Þ

Resolving Eq. (3.36) for du, one obtains the solution 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ1 ÀM Æ M2 À LN du ¼ @ A dv ð3:37Þ L

There are three cases for Eq. (3.37) with L 6¼ 0[3, 4]: (M 2 À LN) > 0: two different solutions of du. The surface (S) cuts the tangent plane T with two lines that intersect each other at the point P (hyperbolic point); (M 2 À LN) ¼ 0: two identical solutions of du. The surface (S) cuts the tangent plane T with one line that passes through the point P (parabolic point); (M 2 À LN) < 0: no solution of du. The surface (S) does not cut the tangent plane T except at the point P (elliptic point). In another way, the second fundamental form II can be derived by the change rate of the differential of the arc length ds when the surface (S) moves along the unit normal vector n with a parameterized variable α according to [2]. The location vector of the point P can be written in the parameterized variable α. 112 3 Elementary Differential Geometry

RuðÞ¼u; v; α rðÞÀu; v αnðÞu; v ð3:38Þ

The second fundamental form II can be calculated at α ¼ 0 using Eq. (3.27a):

∂ðÞds 1 ∂ðÞds2 1 ∂I II¼ ds: j ¼ j ¼ jα ∂α α¼0 2 ∂α α¼0 2 ∂α ¼0 ∂ ÀÁ 1 2 2 ¼ Edu þ 2Fdudv þ Gdv jα 2 ∂α ¼0 ∂ ∂ ∂ 1 E 2 F 1 G 2 ¼ jα du þ jα dudv þ jα dv ð3:39Þ 2 ∂α ¼0 ∂α ¼0 2 ∂α ¼0 in which the ﬁrst term on the RHS of Eq. (3.39) can be calculated as

E ¼ RuðÞÁu; v; α RuðÞu; v; α ¼ ðÞÁru À α nu ðÞru À α nu ð3:40Þ 2α2 α 2 ¼ nu À 2ru Á nu þ ru

Thus,

∂ ÀÁ 1 E 2 jα ¼ n α À r n jα ¼Àr Á n ð3:41Þ 2 ∂α ¼0 u u u ¼0 u u

Further calculations deliver the second and third terms on the RHS of Eq. (3.39):

∂ F : ∂α jα¼0 ¼ÀðÞru Á nv þ rv Á nu ; ð3 42Þ 1 ∂G jα ¼Àr Á n ð3:43Þ 2 ∂α ¼0 v v

Using the orthogonality of ru and n, one obtains

∂ðÞr Á n u ¼ r Á n þ r Á n ¼ 0 ) r Á n ¼Àr Á n ð3:44Þ ∂u uu u u uu u u

Similarly, one obtains using the orthogonality of ru and n; rv and n ∂ ∂ ðÞru Á n ðÞrv Á n ; ∂ ¼ ∂ ¼ 0 ) 2ruv Á n ¼ÀðÞru Á nv þ rv Á nu v u ð3:45Þ ∂ðÞr Á n v ¼ 0 ) r Á n ¼Àr Á n ∂v vv v v

Substituting Eqs. (3.41)–(3.45) into Eq. (3.39), the second fundamental form II can be written as 3.6 Gaussian and Mean Curvatures 113

∂ ∂ ∂ 1 E 2 F 1 G 2 II ¼ jα du þ jα dudv þ jα dv 2 ∂α ¼0 ∂α ¼0 2 ∂α ¼0 2 2 ¼Àru Á nudu À ðÞru Á nv þ rv Á nu dudv À rv Á nvdv ð3:46Þ 2 2 ¼ ruu Á ndu þ 2ruv Á ndudv þ rvv Á ndv Ldu2 þ 2Mdudv þ Ndv2 where L, M, and N are the components of the Hessian tensor ! ! LM r Á nrÁ n H ¼ ¼ uu uv ð3:47Þ MN ruv Á nrvv Á n

3.6 Gaussian and Mean Curvatures

The Gaussian and mean curvatures are based on the principal normal curvatures κ1 and κ2 in the direction t1 and t2 of the surface (S) at a given point P, as shown in Fig. 3.4. The unit tangent vectors t1 and t2 and the unit normal vector n at the point P generate two principal curvature planes that are perpendicular to each other. The normal curvature κ1 of the surface (S) in the principal direction t1 at the point P is defined as the maximum normal curvature in the curvature plane P1; the normal curvature κ2 in the principal direction t2 is the minimum normal curvature in the curvature plane P2. The maximum and minimum normal curvatures κ1 and κ2 of the surface (S)at the point P are the eigenvalues of the corresponding eigenvectors t1 and t2 [1, 2, 4, 5]. These eigenvalues are given from the characteristic equation that can be derived from the first and second fundamental forms in Eqs. (3.27a) and (3.46). The Gaussian curvature of the surface (S) at the point P is defined by

K ¼ κ1κ2 ð3:48Þ

The mean curvature of the surface (S) at the point P is deﬁned by

1 H ¼ ðÞκ þ κ ð3:49Þ 2 1 2

The covariant metric tensor related to the ﬁrst fundamental form can be written as

2 2; I ¼ Eduþ 2Fdudv þ!Gdv ! EF ru Á ru ru Á rv ð3:50Þ M ¼ gij ¼ FG rv Á ru rv Á rv

The Hessian tensor related to the second fundamental form can be written as 114 3 Elementary Differential Geometry

Fig. 3.4 Gaussian and principal curvature planes mean curvatures of the P2 surface S tangent plane

unit normal vector P1

unit tangent vectors

ll S r(u( ),v( )) C2

II ¼ Ldu2 þ 2 Mdudv þ!Ndv 2; ! ÀÁ LM ruu Á nruv Á n ð3:51Þ H ¼ hij ¼ MN ruv Á nrvv Á n

The characteristic equation of the principal curvatures results in ðÞL À κE ðÞM À κF detðÞ¼H À κM ¼ 0 ð3:52Þ ðÞM À κF ðÞN À κG

Therefore,

ðÞL À κE :ðÞÀN À κG ðÞM À κF 2 ¼ 0 , ÀÁ ÀÁ ð3:53Þ EG À F2 κ2 þ ðÞEN À 2MF þ LG κ þ LN À M2 ¼ 0

The Gaussian curvature results from Eq. (3.53) ÀÁ 2 LN À M dethij K ¼ κ1κ2 ¼ ¼ ð3:54Þ EG À F2 det gij

Note that the Gaussian curvature K at a point in the surface is the product of two principal curvatures at this point. According to Gauss’s Theorema Egregium (remarkable theorem) in [1, 2, 4, 5], the Gaussian curvature depends only on the ﬁrst fundamental form I. 3.6 Gaussian and Mean Curvatures 115

Similarly, the mean curvature results in

1 ðÞEN ÀÁÀ 2MF þ LG H ¼ ðÞ¼Àκ1 þ κ2 ð3:55Þ 2 2 EG À F2

The maximum and minimum principal curvatures κ1 and κ2 of the surface S at the point P result from Eqs. (3.54) and (3.55). & pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ κ 2 1 ¼ max ¼ H þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH À K : 2 ð3 56Þ κ2 ¼ κmin ¼ H À H À K

In the following section, the Gaussian and mean curvatures of the rotational paraboloid surface (S)inR3 are computed as follows:

ðÞS : z ¼ x2 þ y2; pﬃﬃ ð3:57Þ htðÞ¼ t; t > 0

The location vector of the curvilinear surface (S) can be written as 0 1 htðÞcos ϕ rðÞ¼t; h @ htðÞsin ϕ A ð3:58Þ h2ðÞt

The bases of the curvilinear coordinate (t, ϕ) result from Eq. (3.58)in 0 1 0 1 _ ϕ ∂ htðÞcos ϕ ∂ Àh sin r @ _ A r @ A rt ¼ ¼ htðÞsin ϕ ; rϕ ¼ ¼ h cos ϕ ð3:59Þ ∂t ∂ϕ 2hht_ ðÞ 0

Thus, the covariant metric tensor can be further computed as ! ! r Á r r Á rϕ EF M ¼ t t t ¼ rϕ Á rt rϕ Á rϕ FG ÀÁ! ð3:60Þ h_ 2 1 þ 4h2 0 ¼ 0 h2

The unit normal vector can be calculated as e1 e2 e3 gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Â g2 _ ϕ _ ϕ _ n ¼ p ¼ h cos h sin 2hh EG À F2 Àh sin ϕ h cos ϕ 0 0 1 0 1 ð3:61Þ À2hh_ 2 cos ϕ 2h cos ϕ 1 B C À1 B C ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@ À2hh_ 2 sin ϕ A ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ 2h sin ϕ A hh_ 1 þ 4h2 1 þ 4h2 hh_ À1 116 3 Elementary Differential Geometry

The components of the Hessian tensor are calculated by differentiating Eq. (3.59) with respect to t and ϕ. 0 1 0 1 € _ ∂2 h cos ϕ ∂2 Àh sin ϕ r @ € A r @ _ A r ¼ ¼ h sin ϕ ; r ϕ ¼ ¼ h cos ϕ ; tt ∂t2 t ∂t∂ϕ 2hh€ þ 2h_ 2 0 0 1 ð3:62Þ 2 Àh cos ϕ ∂ r @ A rϕϕ ¼ ¼ Àh sin ϕ ∂ϕ2 0

The Hessian tensor results from Eqs. (3.61) and (3.62)in ! ! r Á nrϕ Á n LM H ¼ tt t ¼ rϕ Á nrϕϕ Á n MN t : _ 2 ð3 63Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 h 0 ¼ 2 1 þ 4h2 0 h

Therefore, the Gaussian curvature can be calculated from Eqs. (3.60) and (3.63). 2 ffiffiffiffiffiffiffiffiffiffi2 _ 2 2 2 p h h LN À M 1þ4h2 K ¼ κ1κ2 ¼ ¼ ÀÁ EG À F2 h_ 2h2 1 þ 4h2 ð3:64Þ 4 4 ÀÁ > ¼ 2 ¼ 2 0 1 þ 4h2 ðÞ1 þ 4t

In case of (M2 À LN) < 0, no solution of du exists. Thus, the surface (S) does not cut the tangent plane T except at the point P that is called the elliptic point. Analogously, the mean curvature results from Eqs. (3.60) and (3.63)in

1 EN ÀÁÀ 2MF þ LG H ¼ ðÞ¼κ1 þ κ2 2 2 EG À F2 ! 2h_ 2h2 ÂÃÀÁ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Á 1 þ 1 þ 4h2 2 1 þ 4h ÀÁ 21ðÞþ 2t : ¼ _ 2 2 2 ¼ 3 ð3 65Þ 2h h 1 þ 4h ðÞ1 þ 4t 2 in which h2 ¼ t. 3.7 Riemann Curvature 117

3.7 Riemann Curvature

The Riemann curvature (also Riemann curvature tensor) is closely related to the Gaussian curvature of the surface in differential geometry [1–3]. At ﬁrst, let us look into the second covariant derivative of an arbitrary ﬁrst-order tensor. The covariant derivative of the tensor with respect to uj was derived in Eq. (2.208). Γ k : Tij¼ Ti,j À ijTk ð3 66Þ

Obviously, the covariant derivative Ti|j is a second-order tensor component. k Differentiating Ti|j with respect to u , the covariant derivative of the second- order tensor (component) Ti|j is the second covariant derivative of an arbitrary ﬁrst- order tensor (component) Ti. This second covariant derivative has been given from Eq. (2.211a) [3]. À Á T T j ijk À ij Á k m m ¼ Tij , k À Γ TmjÀ Γ Timj ð3:67Þ ik jk Γ m Γ m ¼ Tij, k À ik TmjÀ jk Timj

Equation (3.66) delivers the relations of Γ m Γ m : Tij, k ¼ Ti, jk À ij,kTm þ ij Tm, k ð3 68aÞ Γ m Γ m Γ n : ik Tmj¼ ik Tm, j À mjTn ð3 68bÞ ÀÁ Γ m Γ m Γ n : jk Timj ¼ jk Ti,m À imTn ð3 68cÞ

Inserting Eqs. (3.68a)–(3.68c) into Eq. (3.67), one obtains the second covariant derivative of Ti. T ¼ T À Γ mT À Γ mT j ijk ij, k ik mj jk im Γ m Γ m ¼ Ti,jk À ij,kTm þ ij Tm,k ÀÁ Γ m Γ n Γ m Γ n ð3:69Þ À ik Tm,j À mjTn À jk Ti, m À imTn Γ m Γ m ¼ Ti,jk À ij,kTm À ij Tm, k Γ m Γ mΓ n Γ m Γ mΓ n À ik Tm,j þ ik mjTn À jk Ti,m þ jk imTn where the second partial derivative of Ti is symmetric with respect to j and k:

∂2T ∂2T T i ¼ i T ð3:70Þ i, jk ∂uj∂uk ∂uk∂uj i,kj

Interchanging the indices j with k in Eq. (3.69), one obtains 118 3 Elementary Differential Geometry Γ m Γ m Tikj¼ Ti,kj À ik,jTm À ik Tm,j ð3:71Þ Γ m Γ mΓ n Γ m Γ mΓ n À ij Tm,k þ ij mkTn À kj Ti,m þ kj imTn

Using the symmetry properties given in Eq. (3.70), Eq. (3.71) can be rewritten as Γ m Γ m Tikj¼ Ti,jk À ik,jTm À ik Tm, j ð3:72Þ Γ m Γ mΓ n Γ m Γ mΓ n À ij Tm,k þ ij mkTn À jk Ti,m þ jk imTn

In a ﬂat space, the second covariant derivatives in Eqs. (3.69) and (3.72) are identical. On the contrary, they are not equal in a curved space because of its surface curvature. The difference of both second covariant derivatives is proportional to the curvature tensor. Subtracting Eq. (3.69) from Eq. (3.72), the curvature tensor results in Γ n Γ n Γ mΓ n Γ mΓ n TijkÀ Tikj¼ ik,j À ij, k þ ik mj À ij mk Tn ð3:73Þ n RijkTn

Thus, the Riemann curvature (also Riemann-Christoffel tensor) can be expressed as

n Γ n Γ n Γ mΓ n Γ mΓ n : Rijk ik, j À ij,k þ ik mj À ij mk ð3 74Þ

It is straightforward that the Riemann-Christoffel tensor is a fourth-order tensor with respect to the indices of i, j, k, and n. They contain 81 (¼34) components in a three-dimensional space. In Eq. (3.74), the partial derivatives of the Christoffel symbols are deﬁned by

∂Γ n ∂Γ n Γ n ¼ ik ; Γ n ¼ ij ð3:75Þ ik, j ∂uj ij, k ∂uk

According to Eq. (2.172), the second-kind Christoffel symbol is given 1 Γ k ¼ g þ g À g gkp ð3:76Þ ij 2 ip,j jp,i ij,p

Therefore, the Riemann curvature tensor in Eq. (3.74) only depends on the covariant and contravariant metric coefficients of the metric tensor M, as given in Eq. (3.28). Furthermore, the covariant Riemann curvature tensor of fourth order is defined by the Riemann-Christoffel tensor and covariant metric coefficients.

n n ln : Rlijk gln Rijk , Rijk ¼ g Rlijk ð3 77Þ 3.7 Riemann Curvature 119

For a differentiable two-dimensional manifold of the curvilinear coordinates (u,v), the Bianchi ﬁrst identity gives the relation between the Riemann curvature tensors R and Gaussian curvature K, cf. Eq. (3.117b). : Rlijk K Á gljgki À glkgji ð3 78Þ

Equation (3.78) indicates that the Gaussian curvature K of the two-dimensional surface only depends on the metric coefﬁcients of E, F, and G. Therefore, the Gaussian curvature is only a function of the ﬁrst fundamental form I. This result was proved by Gauss Theorema Egregium [1, 4–6]. The Riemann curvature tensor has the following properties: • First skew-symmetry with respect to l and i:

Rlijk ¼ÀRiljk ð3:79Þ

• Second skew-symmetry with respect to j and k:

; n n : Rlijk ¼ÀRlikj Rijk ¼ÀRikj ð3 80Þ

• Block symmetry with respect to two pairs (l, i) and ( j, k):

Rlijk ¼ Rjkli ð3:81Þ

• Cyclic property in i, j, k:

Rlijk þ Rljki þ Rlkij ¼ 0; n n n ð3:82Þ Rijk þ Rjki þ Rkij ¼ 0

Resulting from these properties, there are six components of Rlijk in the three- dimensional space as follows [2]:

Rlijk ¼ R3131, R3132, R3232, R1212, R3112, R3212 ð3:83Þ

In Cartesian coordinates, all second-kind Christoffel symbols equal zero according to Eq. (2.184). Therefore, the Riemann-Christoffel tensor, as given in Eq. (3.74) must be equal to zero.

n Γ n Γ n Γ mΓ n Γ mΓ n : Rijk ¼ ik, j À ij,k þ ik mj À ij mk 0 ð3 84Þ

Therefore, the Riemann curvature tensor in Cartesian coordinates becomes 120 3 Elementary Differential Geometry

n : Rlijk glnRijk ¼ 0 ð3 85Þ

3.8 Gauss-Bonnet Theorem

The Gauss-Bonnet theorem in differential geometry connects the Gaussian and geodesic curvatures of the surface to the surface topology by means of the Euler’s characteristic. Figure 3.5 displays a differentiable Riemannian surface (S) surrounded by a closed boundary curve Γ. The Gaussian curvature vector K is perpendicular to the manifold surface at the point P lying in the curve C and has the direction of the unit normal vector n. The geodesic curvature vector kg has the amplitude of the geodesic curvature κg; its direction of (n Â t) is perpendicular to the unit normal and tangent vectors in the Frenet orthonormal frame. The Gauss-Bonnet theorem is formulated for a simple closed boundary curve Γ [1, 2, 4]. ðð þ

KdA þ κgdΓ ¼ 2π ð3:86Þ S Γ

The compact curvilinear surface S is triangulated into a ﬁnite number of curvilinear triangles. Each triangle contains a point P on the surface. This procedure is called the surface triangulation where two neighboring curvilinear triangles have one common vertex and one common edge (see Fig. 3.6). Therefore, the integral of the geodesic curvature over all triangles on the compact curvilinear surface S equals zero [2]. þ

κgdΓ ¼ 0 ð3:87Þ Γ

In case of a compact triangulated surface, the Gauss-Bonnet theorem can be written in Euler’s characteristic χ of the compact triangulated surface ST [1, 2, 4]. ðð

KdA ¼ 2πχðÞST ð3:88Þ

The Euler’s characteristic of the compact triangulated surface can be deﬁned by

χðÞST V À E þ F ð3:89Þ 3.8 Gauss-Bonnet Theorem 121

Fig. 3.5 Gaussian and K = Kn geodesic curvatures for a simple closed curve Γ

t = κ × k g g (n t) dA C P S dΓ Γ

Fig. 3.6 Gaussian and K = Kn geodesic curvatures for a compact triangulated F (face) V (vertex) surface

dA P E (edge)

(ST ) in which V, E, and F are the numbers of vertices, edges, and faces of the considered compact triangulated surface, respectively. Substituting Eqs. (3.8), (3.54), and (3.89) into Eq. (3.88), the Gauss-Bonnet theorem can be written for a compact triangulated surface. ðð LN À M2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dudv ¼ 2πðÞV À E þ F ð3:90Þ EG À F2 ST 122 3 Elementary Differential Geometry

3.9 Gauss Derivative Equations

Gauss derivative equations were derived from the second-kind Christoffel symbols and the basis g3 that denotes the normal unit vector n (¼g3) of the curvilinear surface with the covariant bases (gi, gj) for i, j ¼ 1, 2 at any point P(u,v) in the surface. Note that all indices i, j, k in the curvature surface S vary from 1 to 2. j The partial derivatives of the covariant basis gi with respect to u can be written according to Eq. (2.166) as

Γ k Γ3 : gi, j ¼ ijgk þ ijg3 for i, j, k ¼ 1, 2 ð3 91Þ

The Christoffel symbols in the normal direction n result from Eq. (2.164).

Γ3 : ij ¼ n Á gi, j ¼ g3 Á gi, j ð3 92Þ

Differentiating gi.g3 and using the orthogonality of gi and g3, the partial derivative j of the covariant basis gi with respect to u can be calculated as

ðÞgi Á g3 , j ¼ gi, j Á g3 þ gi Á g3,j ¼ 0 ð3:93Þ ) gi,j ¼Àg3 gi Á g3, j ¼À g3 Á g3, j gi

Substituting Eq. (3.93) into Eq. (3.92), the Christoffel symbols in the normal direction n can be expressed as

Γ3 ij ¼ g3 Á gi,j ¼ÀðÞg3 Á g3 g3,j Á gi : ¼Àg3, j Á gi ¼Ànj Á gi ð3 94aÞ Γ3 hij ¼ hji ¼ ji for i, j ¼ 1, 2

Thus,

i nj ¼Àhijg for i, j ¼ 1, 2 ð3:94bÞ where hij is the symmetric covariant components of the Hessian tensor H, as given in Eq. (3.104b). Inserting Eq. (3.94a) into Eq. (3.91), the covariant derivative of the basis gi with respect to uj results in

Γ k gi,j ¼ ijgk þ hijg3 ð3:95Þ Γ k ¼ ijgk þ hijn for i, j, k ¼ 1, 2

This equation is called Gauss derivative equations in which the second-kind Christoffel symbol is deﬁned as 3.10 Weingarten’s Equations 123 1 Γ k ¼ gkp g þ g À g ð3:96Þ ij 2 jp,i pi,j ij,p

3.10 Weingarten’s Equations

The Weingarten’s equations deal with the derivatives of the normal unit vector i n (¼g3) of the surface at the point P(u,v) in the curvilinear coordinates {u }. The partial derivative of the normal unit vector n results from Eq. (3.91).

Γ k Γ3 : ni g3,i ¼ 3igk þ 3ig3 for i, k ¼ 1, 2 ð3 97Þ

i Differentiating g3Ág3 ¼ 1 with respect to u , one obtains : ðÞg3 Á g3 ,i ¼ 2g3 Á g3,i ¼ 0 ) g3 Á g3,i ¼ 0 ð3 98Þ

Using Eqs. (3.92) and (3.98), one obtains

Γ3 Γ3 : ji ¼ g3 Á gj,i ) 3i ¼ g3 Á g3,i ¼ 0fori ¼ 1, 2 ð3 99Þ

Inserting Eq. (3.99) into Eq. (3.97) and using Eq. (3.94b), it gives ÀÁ Γ k k kj j ni ¼ 3igk ¼Àhikg ¼À hikg gj ¼Àhi gj ∂n ð3:100Þ ) n ¼Àh jg for i, j ¼ 1, 2 ∂ui i i j

j The mixed components hi are calculated from the Hessian tensor H and metric tensor M, as shown in Eq. (3.105). ÀÁ j kj À1 : hi ¼ hikg ¼ HM ð3 101Þ

Using Eq. (3.101) for i, j ¼ u, v, the Weingarten’s equations (3.100) can be written as FM À LG FL À EM nu ¼ ru þ rv EG À F2 EG À F2 ð3:102Þ ∂n FM À LG ∂r FL À EM ∂r , ¼ þ ∂u EG À F2 ∂u EG À F2 ∂v 124 3 Elementary Differential Geometry FN À GM FM À EN nv ¼ ru þ rv EG À F2 EG À F2 ð3:103Þ ∂n FN À GM ∂r FM À EN ∂r , ¼ þ ∂v EG À F2 ∂u EG À F2 ∂v where the covariant metric and Hessian tensors result from the coefﬁcients of the ﬁrst and second fundamental forms I and II. ! ! EF r Á r r Á r M ¼ g ¼ u u u v ij FG r Á r r Á r v u v!v ð3:104aÞ 1 G ÀF ) MÀ1 ¼ ðÞ¼gij EG À F2 ÀFE ! ! ÀÁ LM ruu Á nruv Á n H ¼ hij ¼ ð3:104bÞ MN ruv Á nrvv Á n

Therefore, ! À1 ðÞFM À LG ðÞFL À EM HMÀ1 ¼ h j ¼ ð3:105Þ i EG À F2 ðÞFN À GM ðÞFM À EN

3.11 Gauss-Codazzi Equations

The Gauss-Codazzi equations are based on the Gauss derivative and Weingarten’s equations. The Gauss derivative equation (3.95) can be written as

Γ k Γ3 gi,j ¼ ijgk þ ijn ð3:106Þ Γ k ¼ ijgk þ Kijn for i, j, k ¼ 1, 2 in which the symmetric covariant components Kij (¼hij) of the surface curvature tensor K are given in Eq. (3.94a).

Γ3 ij ¼ Kij 2 K ðÞ H ð3:107Þ Γ3 ¼ Kji ¼ ji

j The covariant derivative of gi with respect to u results from Eq. (3.106). Γ k gi j ¼ gi,j À gk ij ð3:108Þ ¼ Kijn for i, j ¼ 1, 2 3.11 Gauss-Codazzi Equations 125

The Weingarten’s equation (3.100) can also be written in the mixed components of the surface curvature tensor K.

j : ni g3, i ¼ÀKi gj for i, j ¼ 1, 2 ð3 109Þ in which the mixed components of the surface curvature tensor K is deﬁned according to Eq. (3.101)as

j kj À1 : Ki ¼ Kikg 2 KM for i, j, k ¼ 1, 2 ð3 110Þ

Differentiating Eq. (3.108) with respect to uk and using Eq. (3.109), the covariant second derivatives of the basis gi can be calculated as l : gi jk ¼ Kij, kn þ Kijnk ¼ Kij, kn À KijKkgl ð3 111Þ

k j Similarly, the covariant second derivatives of the basis gi with respect to u and u result from interchanging k with j in Eq. (3.111). l : gi kj ¼ Kik,jn þ Kiknj ¼ Kik,jn À KikKj gl ð3 112Þ

The Riemann curvature tensor R results from the difference of the covariant second derivatives of Eqs. (3.111) and (3.112) according to Eq. (3.73). l g jk À g kj ¼ R g i ÀÁi ijk l : l l ð3 113Þ ¼ Kij,k À Kik,j n þ KikKj À KijKk gl

Multiplying Eq. (3.113) by the normal unit vector n and using the orthogonality between gl and n, one obtains ÀÁ l l l Kij,k À Kik,j n Á n þ KikK À KijK gl Á n ¼ R gl Á n ÀÁj k ijk l l l ) Kij,k À Kik,j Á 1 þ KikKj À KijKk Á 0 ¼ Rijk Á 0

Thus,

Kij, k À Kik,j ¼ 0fori, j, k ¼ 1, 2 ð3:114Þ

Equation (3.114) is called the Codazzi’s equation. Multiplying both sides of Eq. (3.113)bygm, using the Codazzi’s equation, and employing the tensor contraction rules, one obtains 126 3 Elementary Differential Geometry R l g Á g ¼ K K l À K K l g Á g ijk l m ik j ij k l m m m m ) Rijkglm ¼ KikKj À KijKk glm

Thus, the Riemann curvature tensors can be calculated as

Rlijk ¼ KikKlj À KijKlk ð3:115Þ

Equation (3.115) is called the Gauss equation. As a result, both Eqs. (3.114) and (3.115) are deﬁned as the Gauss-Codazzi equations. The Codazzi’s equation (3.114) gives only two independent nontrivial terms [3, 6]:

Kij, k ¼ Kik,j ) ðÞðK11,2 ¼ K12,1 ; K21,2 ¼ K22,1 3:116Þ

On the contrary, the Gauss equation delivers only one independent nontrivial term [3, 6]:

2 : R1212 ¼ K11K22 À K12 ð3 117aÞ

Therefore, the Gaussian or total curvature K in Eq. (3.54) can be rewritten as ÀÁ j 1 2 1 2 K ¼ det Ki ¼ K1K2 À K2K1 2 ð3:117bÞ K11K22 À K12 R1212 ¼ 2 ¼ g11g22 À g12 g in which j À1 Ki is the mixed components of the mixed tensor KM , as shown in Eq. (3.105); Kij is the covariant components of the surface curvature tensor K; R1212 is the covariant component of the Riemann curvature tensor; g is the determinant of the covariant metric tensor (gij).

3.12 Lie Derivatives

The Lie derivatives (pronouncing/Li:/) named after the Norwegian mathematician Sophus Lie (1842–1899) are very useful geometrical tools in Lie algebras and Lie groups in differential geometry of curved manifolds. The Lie derivatives are based on vector ﬁelds that are tangent to the set of curves (also called congruence) of the curved manifold. 3.12 Lie Derivatives 127

Fig. 3.7 Vector ﬁeld on the Tp M tangent space TpM of the manifold M u j (β) (B) ≡ d Y dβ P21 Δβ ≡ d M X dα P11

f (P11) Δα P12 ui (α) f f (P12 ) (A)

3.12.1 Vector Fields in Riemannian Manifold

The vector field tangent to the curve (A) parameterized by the geodesic parameter α can be written in the coordinate ui of the N-dimensional manifold M, as displayed in Fig. 3.7. The vector field lies on the tangent space TpM (tangent surface) that is tangent to the manifold M at the contact point P. As the contact point moves along the curve on the manifold M, a tangent bundle TM of the manifold M is generated. Therefore, the tangent bundle consists of all tangent spaces of the manifold M. Using Einstein summation convention (cf. Sect. 2.2.2), the vector field X on the ∂ i tangent space TpM can be written with the basis vectors ∂ui of the coordinates u .

d dui ∂ X ¼ dα dα ∂ui : ∂ ð3 118Þ Xi for i ¼ 1, 2, ..., N ∂ui where Xi is the vector component in the coordinate ui. Similarly, the vector ﬁeld tangent to the curve (B) parameterized by another geodesic parameter β can be written in the coordinate uj.

d duj ∂ Y ¼ dβ dβ ∂uj : ∂ ð3 119Þ Yj for j ¼ 1, 2, ..., N ∂uj where Yj is the vector component in the coordinate uj. 128 3 Elementary Differential Geometry

3.12.2 Lie Bracket

Let X and Y be the vector fields of the congruence in the N-dimensional manifold M and f be a mapping function of the coordinate ui in the curve. The commutator of a vector field is called the Lie bracket and can be defined by

½X; Y XYðÞÀðÞf YXðÞðÞf ð3:120Þ

The ﬁrst mapping operator on the RHS of Eq. (3.120) can be calculated using the chain rule of differentiation. ∂ ∂ XYðÞ¼ðÞf Xj Yi ∂ j ∂ui u ð3:121Þ ∂Yi ∂ ∂2 ¼ Xj þ XjYi ∂uj ∂ui ∂ui∂uj

Analogously, the second mapping operator on the RHS of Eq. (3.120) results in ∂ ∂ YXðÞ¼ðÞf Yj Xi ∂ j ∂ui u ð3:122Þ ∂Xi ∂ ∂2 ¼ Yj þ XiYj ∂uj ∂ui ∂ui∂uj

Interchanging the indices i with j in the second term on the RHS of Eq. (3.122), one obtains

∂Xi ∂ ∂2 YXðÞ¼ðÞf Yj þ XjYi ð3:123Þ ∂uj ∂ui ∂uj∂ui

Subtracting Eq. (3.121) from Eq. (3.123), the Lie bracket is given. ∂Yi ∂Xi ∂ ½¼X; Y Xj À Yj ∂ j ∂ j ∂ i u u u 3:124 ∂ ð Þ ½X; Y i for i ¼ 1, 2, ..., N ∂ui

Thus, the component i in the coordinate ui of the Lie bracket is deﬁned by

∂Yi ∂Xi ½X; Y i Xj À Yj for i, j ¼ 1, 2, ..., N ð3:125Þ ∂uj ∂uj

The vectors X and Y commute if its Lie bracket equals zero.

½¼X; Y 0 ð3:126Þ

According to Eq. (3.125), the Lie bracket is skew-symmetric (anti-symmetric). 3.12 Lie Derivatives 129

½¼ÀX; Y ½Y; X ∂Xi ∂Yi ∂ ð3:127Þ ¼À Yj À Xj ∂uj ∂uj ∂ui

The Lie bracket (commutator) of the vector ﬁeld (X, Y) can be expressed in another way as ∂ ∂ ∂ ∂ ½¼X; Y Xj Yi À Yj Xi ∂ j ∂ i ∂ j ∂ui u u u ð3:128Þ d d ¼ X À Y dβ dα

Therefore, the Lie bracket of the vector ﬁeld can be written as ! d d ½X; Y ; dα dβ ð3:129Þ d d d d ¼ À dαdβ dβdα

The Lie bracket of a vector ﬁeld is generally not equal to zero in a curved manifold due to space torsions and Riemann surface curvatures that will be discussed later in Sect. 3.12.5. The Lie bracket has some cyclic permutation properties of the vector ﬁeld (X, Y, and Z) in the N-dimensional manifold M.

½¼X; ½Y; Z XYZ À XZY À YZX þ ZYX ½¼Y; ½Z; X YZX À YXZ À ZXY þ XZY ð3:130Þ ½Z; ½X; Y ¼ ZXY À ZYZ À XYZ þ YXZ

The Jacobi identity written in the Lie brackets results from substituting the properties of Eq. (3.130):

½þX; ½Y; Z ½þY; ½Z; X ½¼Z; ½X; Y 0 ð3:131Þ

3.12.3 Lie Dragging

3.12.3.1 Lie Dragging of a Function

Let f be a function that maps a point P11 to another point P12 on the curve (A)bya geodesic parameter distance Δα. The function f(P12) at the point P12 is called the image of f(P11) at the point P11 carrying by the mapping function f at Δα on the same curve (A) 2 M (see Fig. 3.7). If the image function f(P12) equals the function f(P11), the function f is invariant under the mapping. Furthermore, the mapping function f can be deﬁned as Lie 130 3 Elementary Differential Geometry dragged if f is invariant for any geodesic parameter distance Δα along any congruence on the manifold M.

df ¼ 0 , f is Lie dragged: ð3:132Þ dα

3.12.3.2 Lie Dragging of a Vector Field

Let X and Y be vector fields on the tangent space TpM of the tangent bundle TM in an N-dimensional manifold M, as shown in Fig. 3.8. The congruence consists of α- i j and β-curves with the coordinates u and u . The tangent vector X to the curve (A1) at the point P11 is dragged to the curve (A2) at the point P21 by a geodesic parameter distance Δβ. The image vector X* of the original vector X is dragged by Δβ from (A1)to(A2) and tangent to the curve (A2) at the point P21. Generally, both tangent vectors X and X* are different to each other under the Lie dragging by Δβ. However, if they are equal for every geodesic parameter distance Δβ, the Lie dragging is invariant. In this case, the vector field is called Lie dragged on the manifold M [7, 8]. Similarly, the tangent vector Y to the curve (B1) at the point P11 is dragged to the image vector Y* tangent the curve (B2) at the point P12 by a geodesic parameter distance Δα in the tangent bundle TM. Thus, the vector fields of X and Y along the congruence are generated on the manifold M. The vector field is defined as Lie dragged if its Lie bracket or the commutator given in Eq. (3.129) equals zero. ! d d ½X; Y ; dα dβ d d d d ¼ À ¼ 0 ð3:133Þ dαdβ dβdα d d d d ) ¼ dαdβ dβdα

In this case, the vector ﬁeld is commute under the Lie-dragged procedure on the manifold M. In general, the Lie bracket of a vector ﬁeld is not always equal to zero due to space torsions besides the Riemann surface curvature on the manifold M.

3.12.4 Lie Derivatives

The Lie derivatives of a function with respect to the vector ﬁeld X are deﬁned by the change rate of the function f between two different points P11 and P12 in the same curve under the Lie dragging by a geodesic parameter distance Δα. 3.12 Lie Derivatives 131

Fig. 3.8 Lie dragging β uj -curves ( ) Tp M vector ﬁelds on the TM manifold M

X* ≡ d (A2 ) dαΔβ

≡ d P21 Y dβ P Δβ 22

d M (A1 ) ≡ X dα P11 Δα Y* ≡ d dβΔα (B1 ) P12

(B2 ) α -curves (ui)

f ðÞÀα þ Δα f ðÞα £ f lim 0 0 X Δα Δα !0 i i dfðÞ u du ∂f ¼ ¼ ð3:134Þ dα dα ∂ui α0 α0 ∂f ¼ Xi ∂ui

Thus, the Lie derivative of a function f with respect to the vector ﬁeld X can be simply expressed in ∂ £ f ¼ Xi f ¼ Xf ð3:135Þ X ∂ui

Analogously, the Lie derivative of a vector Y with respect to the vector ﬁeld ∂ i X results from the Lie bracket with the basis vectors ∂ui of the coordinates u [8, 9]. ∂ i £XY ¼ ½¼X; Y ½X; Y ∂ui : ∂ ð3 136Þ ¼ ðÞ£ Y i for i ¼ 1, 2, ..., N X ∂ui

According to Eq. (3.125), the component i in the coordinate ui of the Lie derivative of the vector Y with respect to the vector ﬁeld X can be calculated as

ðÞ£ Y i ¼ ½X; Y i X ∂ ∂ ¼ Xj Yi À Yj Xi ∂uj ∂uj ð3:137Þ d d ¼ Yi À Xi for i ¼ 1, 2, ..., N dα dβ 132 3 Elementary Differential Geometry

Using Eqs. (3.136) and (3.137) the Lie derivative of a vector ﬁeld is skew- symmetric because of the skew-symmetry of the Lie bracket, as shown in Eq. (3.127). ∂ ∂Yi ∂Xi ∂ £ Y ¼ ðÞ£ Y i ¼ Xj À Yj X X ∂ui ∂uj ∂uj ∂ui ð3:138Þ ¼À£YX

In the following section, the properties of the Lie derivatives are proved.

3.12.4.1 Lie Derivative of a Function Product

Let f and g be functions. The Lie derivative of product of two functions is given as

£XðÞ¼fg ðÞ£X f g þ f ðÞ£Xg ð3:139Þ

Proof ∂ £ ðÞ¼fg XðÞ¼fg Xi ðÞfg X ∂ui ∂f ∂g ¼ Xi g þ fXi ∂ui ∂ui ¼ ðÞ£X f g þ f ðÞ£Xg ðÞq:e:d:

3.12.4.2 Lie Derivative of a Tensor Product

Let S and T be tensor ﬁelds and X a vector ﬁeld. The Lie derivative of tensor product is given as

£XðÞ¼S T ðÞ £XS T þ S ðÞ£XT ð3:140Þ

Proof £XðÞ¼S T ½X, S T ¼ ½ X; S T þ S ½X; T ¼ ðÞ £XS T þ S ðÞ£XT

The Lie derivative of mixed tensors T with respect to a vector ﬁeld X can be written in the Lie derivative components and their bases of the coordinates.

; i...k ∂ ... ∂ l ... n £XT ½¼X T ðÞ£XT l...n i k du du 3.12 Lie Derivatives 133

l n where ∂i,..., ∂k and du ,..., du are the covariant and contravariant bases of the local coordinates {ui}, respectively (cf. Sect. 3.14), is the tensor product, and i ... k Tl ... n are the components of the mixed tensor T. The Lie derivative components of the tensor ﬁeld T with respect to a vector ﬁeld X are computed in [9]:

i...k p i...k p i...k ... k i...p ... : ðÞ£XT l...n ¼ X Tl...n,p þ X, lTp...n þ À X, pTl...n À ð3 141Þ in which the partial derivatives are deﬁned as

i...k p k ... ∂T ∂X ∂X Ti k l...n ; X p ; X k l...n, p ∂up ,l ∂ul ,p ∂up

3.12.4.3 Lie Derivative of a Differential Form

Let dω be a differential k-form that is dual to its k-order covariant tensor ﬁeld on a differentiable manifold M and X be a vector ﬁeld. The Lie derivative of the differential k-form dω with respect to X is given as

£XðÞ¼dω dðÞ£Xω ð3:142Þ

Proof At ﬁrst, Eq. (3.142) is derived for a differential zero-form on a differentiable N- dimensional manifold M. The zero-form f is deﬁned as a smooth function on the manifold M. Its differential zero-form df can be written in the dual bases duj using Einstein’s summation convention.

∂fuðÞj df ¼ duj for j ¼ 1, 2, ..., N ∂uj

Using the chain rule of differentiation, the LHS of Eq. (3.142) for ω f can be calculated as ∂ f j £XðÞ¼df £X du ∂uj ∂f ∂f ÀÁ ¼ £ duj þ £ duj X ∂uj ∂uj X

Interchanging the index i with j in the second term in Eq. (3.143a), one obtains 134 3 Elementary Differential Geometry ∂ ∂ ∂ ∂ j i f j f X i £XðÞ¼df X du þ du ∂ui∂uj ∂u j∂ui ∂ ∂ ∂ ∂ i i f j f X j ¼ X du þ du ð3:143aÞ ∂ui ∂uj ∂ui!∂u j ∂2f ∂f ∂Xi ¼ Xi þ duj ∂ui∂u j ∂ui ∂u j

Using the chain rule of differentiation, the RHS of Eq. (3.142) for ω f is computed as ∂f dðÞ¼£ f dðÞ¼Xf dXi X ∂ i u ∂f ∂f ¼ Xid þ dXi ∂ i ∂ i ð3:143bÞ u u ! ∂2f ∂f ∂Xi ¼ Xi þ duj ∂ui∂uj ∂ui ∂uj

Subtracting Eq. (3.143a) from Eq. (3.143b), Eq. (3.142) is proved for a differential zero-form.

£XðÞÀdf dðÞ¼£Xf 0 ) £XðÞ¼df dðÞ£Xf

This equation is called the Cartan’s formula in the special case for a differential zero-form df. In the following step, Eq. (3.142) is derived for a differential one-form on a differentiable N-dimensional manifold M. The one-form ω on the manifold M can be generally written in terms of the dual bases duj using Einstein summation convention, cf. Eq. (3.179).

j ω ¼ ωjdu

Thus, the differential one-form results in

∂ω dω ¼ duj ∂uj

The LHS of Eq. (3.142) can be calculated as ∂ω £ ðÞ¼dω £ duj X X ∂ j u ∂ω ∂ω ÀÁ ¼ £ duj þ £ duj X ∂uj ∂uj X 3.12 Lie Derivatives 135 within ∂uj £ ðÞ¼duj dðÞ¼£ uj dXi ¼ dXiδ j X X ∂ui i ∂Xj ¼ dXj ¼ duifor i ¼ 1, 2, ..., N ∂ui

Further calculating the LHS of Eq. (3.142), one obtains interchanging i with j in the second term on the RHS of Eq. (3.143c) ∂ ∂ω ∂ω ∂ j i j X i £XðÞ¼dω X du þ du ∂ui ∂uj ∂uj ∂ui ∂ ∂ω ∂ω ∂ i i j X j ¼ X du þ du ð3:143cÞ ∂ui ∂uj ∂u!i ∂uj ∂2ω ∂ω ∂Xi ¼ Xi þ duj ∂ui∂uj ∂ui ∂uj

Next, the RHS of Eq. (3.142) is computed using the chain rule of differentiation. ∂ dðÞ¼£ ω dXi ω X ∂ i u ∂ω ∂ω ¼ Xid þ dXi ∂ i ∂ i ð3:143dÞ u u ! ∂2ω ∂ω ∂Xi ¼ Xi þ duj ∂ui∂uj ∂ui ∂uj

Comparing Eqs. (3.143c) and (3.143d), Eq. (3.142) is proved for a differential one-form.

£XðÞ¼dω dðÞ£Xω

Finally, Eq. (3.142) is derived for an arbitrary differential k-form (k > 0) on a differentiable N-dimensional manifold M using the Cartan’s formula. Let C be a ﬁber bundle of the manifold M and a point p 2 C.Ak-form at the point p on C 2 M is an element of the cotangent bundle T*M M consisting of k cotangent spaces (cf. Sect. 3.14). Generally, the k-form on the manifold M is deﬁned in local coordinates with the dual bases dui as 136 3 Elementary Differential Geometry

XN ÀÁ ω ω i i1 i2 ik ¼ i1i2...ik u du ^ du ^ÁÁÁ^du i1 0) is derived in [8].

£Xω ¼ iXdω þ diðÞXω ð3:144aÞ where the notation iXω is called the interior product of the k-form ω with respect to the vector ﬁeld X on the differentiable N-dimensional manifold M and deﬁned as

iXω ðÞ¼ω; X ωðÞX ∂ω ¼ Xj ∂uj

From the Cartan’s formula, one obtains the interior product of the differential k- form.

iXdω ¼ £Xω À diðÞXω ð3:144bÞ

Changing the k-form ω into its exterior derivative dω in the Cartan’s formula in Eq. (3.144a), using Eq. (3.144b) and dd ¼ 0, Eq. (3.142) has been derived.

£XðÞ¼dω iXddðÞþω diðÞXdω

¼ iXddðÞþω dðÞ£Xω À diðÞXω

¼ iXddω þ dðÞÀ£Xω ddðÞ iXω

¼ dðÞ£Xω : q:e:d:

3.12.4.4 Lie Derivative of a One-Form and Vector Product

Let X and Y be vector ﬁelds, and ω a one-form. The Lie derivative of product between the one-form and vector is given as

£XðÞÀωY ðÞ£Xω Y ¼ ω½X; Y ð3:145Þ

Proof £XðÞ¼ωY ½X, ωY ¼ ½X; ω Y þ ω½X; Y ¼ ðÞXω Y þ ω½X; Y

¼ ðÞ£Xω Y þ ωðÞ£XY

Therefore, 3.12 Lie Derivatives 137 À Á £XðÞÀωY £Xω Y ¼ ω½X; Y ðÞq:e:d:

3.12.4.5 Lie Derivative of a One-Form

Let ω be a one-form and X a vector ﬁeld. The Lie derivative of one-form is given as

i i £Xω ¼ ðÞXωi du þ ωidX ð3:146Þ

Proof From Eqs. (3.137) and (3.145) one obtains interchanging the index i with j. ÀÁ ðÞ£ ω Yi ¼ £ ω Yi À ω ðÞ£ Y i X i X ÀÁi i X i i i ∂ ωiY ∂Y ∂X ¼ Xj À ω Xj À Yj ∂uj i ∂uj ∂uj i i i ∂Y ∂ωi ∂Y ∂X ¼ Xjω þ XjYi À ω Xj þ ω Yj : i ∂uj ∂uj i ∂uj i ∂uj ð3 147Þ ∂ω ∂Xi ¼ XjYi i þ ω Yj ðÞi $ j ∂ j i ∂ j u u ∂ω ∂Xj ¼ Xj i þ ω Yi ∂uj j ∂ui

Equation (3.147) gives

∂ω ∂Xj ðÞ£ ω ¼ Xj i þ ω ð3:148Þ X i ∂uj j ∂ui

Multiplying Eq. (3.148)bydui and interchanging i with j in the second term on the RHS of Eq. (3.149), one obtains

∂ω ∂Xj ðÞ£ ω dui ¼ Xj i dui þ ω dui X i ∂uj j ∂ui ð3:149Þ i i ¼ ðÞXωi du þ ωidX

Using the chain rule of differentiation, the RHS of Eq. (3.149) can be written as ÀÁ ðÞXω dui þ ω dXi ¼ ðÞ£ ω dui þ ω £ dui i i XÀÁi i X i : ¼ £X ωidu ð3 150Þ

£Xω where the one-form ω can be expressed in the contravariant basis vector dui using Einstein summation convention, cf. Eq. (3.179). 138 3 Elementary Differential Geometry

i ω ¼ ωidu

Thus, one obtains the Lie derivative of the one-form in the direction of the vector ﬁeld X.

ω ω i £X ¼ ðÞ£X idu i i ¼ ðÞXωi du þ ωidX ðÞq:e:d:

3.12.5 Torsion and Curvature in a Distorted and Curved Manifold

The normal vector field N perpendicular to the surface of the manifold M is dragged in two different paths from the same point P11 via P21 to Q in the one path; and via P12 to S in the other path, as shown in Fig. 3.9. Due to the effect of space torsions and surface curvatures, the vector field N does not close the connection loop at the path ends Q and S of the dragging paths. The gap of the path ends is O(ε2)ina distorted and curved manifold and is reduced to the order of O(ε3) in an only curved manifold [7]. The Lie derivative of the vector Y with respect to the vector field X is induced by the space torsion of the distorted manifold. As a consequence, the torsion tensor ε2 [X,Y] generates the open connection gap QR (see Fig. 3.9). The Riemann surface curvature is to blame for the other open connection gap RS on the order of O(ε3)in the curved manifold. Therefore, the connection loop is always closed in a torsion-free and flat space.

½ÀY; X ½¼X; Y 0 , £YX ¼ £XY ð3:151Þ

The curvature equation of a distorted and curved manifold can be written by means of the Lie formulations, Riemann curvature tensors, and covariant metric coefﬁcients [7].

; ; ε2 ; ε2 ½ÀY X ½¼X Y ½þX Y Rlijk : ε2 ε2 n ð3 152Þ , £YX À £XY ¼ £XY þ gnlRijk where Rlijk is the Riemann curvature tensors of the curved manifold M.

3.12.6 Killing Vector Fields

The Killing vector field K is defined as a vector field in an N-dimensional manifold in which the Lie derivative of the metric tensor g with respect to the vector field K along the congruence equals zero. 3.12 Lie Derivatives 139

Fig. 3.9 Connection loop ε 2 £Y = ε 2 []X,Y : torsion tensor of vector ﬁelds in a distorted X ε 2 R : curvature tensor and curved manifold M lijk εX* P εY 21 Q N R ε S M X εY* P11

P12

£Kg ¼ 0 ð3:153Þ

Equation (3.153) shows that the metric tensor g is invariant on the manifold with respect to the Killing vector ﬁeld K. The covariant tensor components of the Lie derivative of the metric tensor with respect to the Killing vector ﬁeld K can be expressed in [8].

∂g ∂Kk ∂Kk ðÞ£ g ¼ Kk ij þ g þ g ¼ 0 ð3:154Þ K ij ∂uk ik ∂uj kj ∂ui

Equation (3.154) can be written in one-dimensional coordinate uk with respect to the Killing vector ﬁeld K.

∂g ðÞ£ g ¼ ij ¼ 0 ð3:155Þ K ij ∂uk

Therefore, if the covariant metric coefficient is independent of any coordinate, the basis of the coordinate is a Killing vector. As an example for the Killing vector field, the covariant metric coefficients of the spherical coordinates (r,φ,θ) are given in

∂ ∂ g ¼ g Á g Á ¼ 1 rr r r ∂r ∂r ∂ ∂ 2 gφφ ¼ gφ Á gφ ∂φ Á ∂φ ¼ r ∂ ∂ 2 2φ : gθθ ¼ gθ Á gθ ∂θ Á ∂θ ¼ r sin ð3 156Þ

Equation (3.156) shows that the metric coefﬁcients are independent of the coordinates (r,φ,θ). Hence, the basis vectors gr, gφ, and gθ are the Killing vectors. 140 3 Elementary Differential Geometry

Fig. 3.10 Invariant ﬁelds V i on a moving surface S(t) P

α V α u gα N T(t + Δt,S′) Δt ′ + Δ S (t t) T (t, S)

S(t) xi

3.13 Invariant Time Derivatives on Moving Surfaces

In the following section, the invariant time derivatives of tensors are applied to a surface S(t) moving with a velocity vector V in the ambient coordinate system. For this case, the invariant time derivative of an invariant field T(t, S) parameterized by the time t and moving surface S can be calculated in the surface coordinate. Generally, two coordinates of the unchanged ambient coordinate xi with the α covariant basis gi and the moving surface coordinate u with the covariant basis gα are used in the moving surface S(t), as shown in Fig. 3.10. The surface S at the time t moves to the new surface position S’ at the time t + Δt at which the invariant field T(t, S) at the time t is changed into T(t + Δt, S’) in a very short time interval Δt. The time-dependent surface S moves with a coordinate velocity Vi in the ambient coordinate xi [10]. The ambient coordinate velocity of the moving surface S(t) in the coordinate xi can be defined by

∂xiðÞt; S Vi ð3:157Þ ∂t

The tangential coordinate velocity Vα results from projecting the ambient coordinate velocity Vi onto the surface along the surface coordinate uα. To calculate the tangential coordinate velocity, the surface velocity vector V can be formulated in both ambient and surface coordinates using the chain rule of differentiation. 3.13 Invariant Time Derivatives on Moving Surfaces 141

α V ¼ V gα ∂ i i r ¼ V gi ¼ V ∂xi : ∂ ∂ α ∂ α ð3 158Þ i r u i u ¼ V α ¼ V gα ÀÁ∂u ∂xi ∂xi i α V x, i gα

α where the derivative x,i is called the shift tensor between the ambient and surface coordinates. α Thus, the tangential coordinate velocity V results from the ambient coordinate velocity and shift tensor.

α α ∂u α V ¼ Vi ¼ Vix ð3:159aÞ ∂xi , i

Analogously, one obtains

i α ∂x α Vi ¼ V ¼ V x i ð3:159bÞ ∂uα ,α

3.13.1 Invariant Time Derivative of an Invariant Field

The invariant time derivative of an invariant field T(t, S) can be defined as the time change rate of the invariant field itself and its change rates along the surface coordinates between the old and new surface positions [10].

_ ∂TtðÞ; S α ∇ T À V ∇αT ð3:160Þ ∂t

At ﬁrst, the covariant surface derivative of a ﬁrst-order tensor T can be written as ÀÁ ∂ ∂ i T T gi ∇αT ¼ ¼ ∂uα ∂uα ∂Ti ∂g ¼ g þ Ti i ð3:161aÞ ∂uα i ∂uα ∂Ti ∂g ∂xj ¼ g þ Ti i ∂uα i ∂xj ∂uα

Using Eq. (2.158), the partial derivative of the basis gi results in 142 3 Elementary Differential Geometry

∂g i ¼ Γ kg ð3:161bÞ ∂xj ij k

Inserting Eq. (3.161b) into Eq. (3.161a), one obtains

∂ i ∂ j ∇ T x Γ k i αT ¼ ∂ α gi þ ∂ α ijT gk u u ∂Tk ∂xj ð3:161cÞ ¼ þ Γ kTi g ∂uα ∂uα ij k ÀÁ ∇ k αT gk

Thus,

∂ k ∂ j k T x k i ∇αT ¼ þ Γ T ∂uα ∂uα ij : ∂ i ∂ j ð3 161dÞ i T x i m , ∇αT ¼ þ Γ T ∂uα ∂uα jm

The covariant surface derivative of a contravariant ﬁrst-order tensor results in using Eq. (3.161d) and chain rule of coordinates.

∂ i ∂ j i i T x i m ∇αT T ¼ þ Γ T α ∂uα ∂uα jm ∂Ti ∂xj ∂xj ¼ þ Γ i Tm ∂ j ∂ α ∂ α jm x u u ð3:161eÞ ∂xj ∂Ti ¼ þ Γ i Tm ∂ α ∂ j jm u x j i j ∇ i ¼ x,αT j x,α jT

Analogously, the covariant surface derivative of a mixed second-order tensor results using Eq. (2.211a) and chain rule of coordinates in ∂ i ∂ k i i Tj x i m n i ∇αT T ¼ þ Γ T À Γ T j j α ∂uα ∂uα km j kj n ∂T i ∂xk ∂xk ¼ j þ Γ i T m À Γ n T i ∂xk ∂uα ∂uα km j kj n !ð3:161fÞ ∂xk ∂T i ¼ j þ Γ i T m À Γ n T i ∂uα ∂xk km j kj n k i k ∇ i ¼ x, αTj x, α kTj k

The ambient coordinate is dependent on time and the surface coordinate of the moving surface S. Thus, the invariant ﬁeld T can be expressed as 3.13 Invariant Time Derivatives on Moving Surfaces 143

TtðÞ¼; S TtðÞ, xtðÞ; S ð3:162Þ

Using the Taylor series and the chain rule of differentiation, the partial time derivative of T with two independent variables of t and S can be calculated as

∂TtðÞ; S ∂TtðÞ; x ∂TtðÞ; x ∂xiðÞt; S ¼ þ ∂ ∂ ∂ i ∂ t t x t ð3:163Þ ∂TtðÞ; x ¼ þ ðÞ∇ T Vi ∂t i because

∂TtðÞ; x ∂xiðÞt; S ∇ T ¼ ; Vi ¼ ð3:164Þ i ∂xi ∂t

The invariant time derivative in Eq. (3.160) can be rewritten using Eqs. (3.161e) and (3.163).

_ ∂TtðÞ; S α ∇ T À V ∇αT ∂t ∂ ; TtðÞx i∇ α k ∇ ¼ ∂ þ V iT À V x,α kT t ð3:165aÞ ∂TtðÞ; x α ¼ þ Vi∇ T À V x i ∇ T ∂t i ,α i ÀÁ ∂TtðÞ; x α ¼ þ Vi À x i V ∇ T ∂t , α i

The second term on the RHS in Eq. (3.165a) can be further calculated using Eq. (3.159a). i α α ∂x ∂u Vi À x i V ¼ Vi À Vj , α ∂uα ∂xj α : ¼ Vi À x i x Vj ð3 165bÞ , α , j j δ i i α ¼ V j À x,αx, j

The useful relation between the contra- and covariant normal vector components and shift tensors of coordinates is derived by [10]:

i i α δ i : N Nj þ x,αx,j ¼ j ð3 165cÞ

i in which δj is the Kronecker delta. Substituting Eq. (3.165c) into Eq. (3.165b), the invariant time derivative of T given in Eq. (3.165a) can be rewritten as 144 3 Elementary Differential Geometry

∂ ; ÀÁ ∇_ TtðÞx j i∇ T ¼ ∂ þ V Nj N iT t ð3:165dÞ ∂TtðÞ; x ¼ þ PNi∇ T ∂t i where P is the normal velocity at a given point on the moving surface S,as displayed in Fig. 3.10. In fact, the normal velocity is the projection of the ambient i coordinate velocity V on the surface normal Ni.

i P ¼ V Ni ) P ¼ PN 3:165e ÀÁ ð Þ i i j ¼ V Ni N ¼ V NiN gj

3.13.2 Invariant Time Derivative of Tensors

Analogously, the invariant time derivative of tensors can be derived from the invariant ﬁeld. The contravariant tensor can be written in the covariant basis gi.

i : T ¼ T gi ð3 166Þ

The invariant time derivative of the tensor T can be expressed on the moving surface S according to Eq. (3.160)[10].

_ ∂TðÞt; S α ∇ T ¼ À V ∇αT ð3:167Þ ∂t

Substituting Eq. (3.166) into Eq. (3.167), one obtains ÀÁ ∂ Tig ÀÁ ∇_ i α∇ i T ¼ ∂ À V α T gi t ð3:168Þ ∂ i ∂ ÀÁ T i gi α i ¼ g þ T À V ∇αT g ∂t i ∂t i

Using Eqs. (2.158) and (3.157), the time derivative of the coordinate basis gi in Eq. (3.168) can be calculated.

∂g ∂g ∂xj ∂xj g_ i ¼ i ¼ g i ∂ ∂ j ∂ i, j ∂ t x t t ð3:169Þ ∂xj ¼ Γ k g ¼ Γ kVjg ij ∂t k ij k

Therefore, the invariant time derivative of the tensor T can be rewritten as 3.13 Invariant Time Derivatives on Moving Surfaces 145

∂ i ÀÁ ∇_ T jΓ k i α ∇ i T ¼ ∂ gi þ V ijT gk À V αT gi t ∂ k T j k i α k ð3:170Þ ¼ þ V Γ T À V ∇αT g ∂t ij k ∇_ k T gk

The invariant time derivative of a contravariant tensor Ti is given from Eq. (3.170).

∂ i _ i T j i k α i ∇ T ¼ þ V Γ T À V ∇αT ð3:171Þ ∂t jk

Similarly, one obtains the invariant time derivative of a covariant tensor Ti ∂ _ Ti j k α ∇ T ¼ À V Γ T À V ∇αT ð3:172Þ i ∂t ji k i

i The invariant time derivative of a mixed tensor Tj can be derived in

∂T i _ i j k i m k n i α i ∇ T ¼ þ V Γ T À V Γ T À V ∇αT j ∂t km j kj n j : ∂ i ð3 173Þ Tj k i m n i α i ¼ þ V Γ T À Γ T À V ∇αT : ∂t km j kj n j

The general invariant time derivative of a mixed fourth-order tensor can be derived in [10].

∂Tiα _ iα jβ γ iα p i qα p q iα ∇ T ¼ À V ∇γT þ V Γ T À V Γ T jβ ∂t jβ pq jβ pj qβ ð3:174Þ Γ_ α iδ Γ_ δ iα þ δ Tjβ À β Tjδ where the time derivative of the Christoffel symbols for a moving surface is deﬁned as

_ α α α Γ β ¼ ∇βV À PRβ ð3:175Þ

α in which P is the normal velocity in Eq. (3.165e) and Rβ is the mean curvature of the moving surface. 146 3 Elementary Differential Geometry

Fig. 3.11 Transport of a tangent space tangent bundle TM vector ﬁeld Y in the tangent bundle TM

Tp M manifold Y

α Y 0 ∞ X C M P1 Δα α + Δα 0 P2 ui (α) X

Tp M ∞ congruence C ∈ M

3.14 Tangent, Cotangent Bundles and Manifolds

Some deﬁnitions of spaces and manifolds are discussed before dealing with the Levi-Civita connection. Let C1 be a set of inﬁnitely differentiable curves (congru- 1 ence) on the manifold M. The vector X is tangent to the curve C at the point P1 and moves along it as the parameter α varies in the local coordinate ui(α), as shown in Fig. 3.11. The tangent vector X lies on the tangent space TpM (tangent surface) that is 1 tangent to the curve C at the point P1 on the manifold M. As the vector X moves 1 along C , the tangent space also changes from P1 to P2. Thus, the tangent spaces TpM generate a tangent bundle TM that consists of all tangent spaces TpM of the manifold M. i The covariant basis of the local coordinate u (α) on the tangent space TpM is written as

∂ ∂ ð3:176Þ i ∂ui

Therefore, the tangent vector X on the tangent space TpM is expressed in the covariant basis.

∂ X ¼ Xi ∂ i u ð3:177Þ ∂uiðÞα ∂ d ¼ ¼ ∂α ∂ui dα

Furthermore, the dual space of the tangent space TpM is called the cotangent space * i i Tp M in which the contravariant basis vectors du of the local coordinates u (α) are used for covariant tensors, covariant vectors, and differential forms. Analogously, * * the cotangent bundle T M covers all cotangent spaces Tp M of the manifold M. 3.14 Tangent, Cotangent Bundles and Manifolds 147

Due to orthonormality of the covariant and contravariant bases, their inner product (dot or scalar product) vanishes for i 6¼ j. Using the Kronecker delta, the j * contravariant basis (dual basis) du on the cotangent space Tp M is deﬁned as ÀÁ ∂ j δ j : i, du ¼ i ð3 178Þ

In differential geometry, a differential one-form dω on a differentiable manifold M is defined as a smooth cross section of the cotangent bundle T*M of the manifold M. As a result, the differential one-form dω is naturally dual to its first-order covariant tensor field of type (0,1) in the dual cotangent space. Sometimes, differential one-forms are called covariant vector fields or dual vector fields within physics. Using Einstein summation convention, the one-form ωp at the point P is generally used on the dual space (cotangent space) with the contravariant basis dui.

i ωp ¼ ωidu ð3:179Þ

i where ωi is a function of u . The differential one-form results from Eq. (3.179)in ∂ω ∂ω p dω ¼ dui ¼ dui ð3:180Þ p ∂ i ∂ i u p u

Using the Leibniz’s rule, the product of two differential one-forms fp and gp is calculated.

: dfgðÞp ¼ f pdgp þ gpdf p ð3 181Þ

Differential 0-forms, 1-forms, and 2-forms are the special cases of the k-form ω that can be generally expressed in terms of the dual bases on a differentiable N- dimensional manifold M as (cf. Chap. 4)

XN ÀÁ ω ¼ hui1 ; ...; uik dui1 ^ dui2 ^ÁÁÁ^duik

3.15 Levi-Civita Connection on Manifolds

Levi-Civita connection (LC connection) describes the process of transporting (dragging) a vector field Y with respect to another vector field X on a smooth differentiable N-dimensional manifold M (affine connection). This connection is related to the Riemann connection; therefore, it is sometime called the Riemann Levi-Civita connection (RLC connection). The covariant derivative of the vector field Y with respect to the vector X moving along to a differentiable curve C1 on the manifold M is generally used to formulate the transport of the vector field Y on the N-dimensional manifold M,as displayed in Fig. 3.12. As a result, the Levi-Civita connection on (M, g) associates the covariant derivative (absolute derivative) with respect to the geodesic parameter α of the coordinates ui(α) on the manifold M. The Levi-Civita connection ∇ is defined as the covariant derivative of Y with respect to the vector field X.

α DYðÞ _ i j k i ∇XY ¼ Y þ Y u_ Γ ∂i Dα jk ð3:182Þ i ðÞ∇XY ∂i where

d dYiðÞα dukðÞα X ¼ ; Y_ i ; u_ k dα dα dα

The component i of the Levi-Civita connection of the vector ﬁeld Y with respect to the vector ﬁeld X results from Eq. (3.182)in

∇ i _ i j _ kΓ i : ðÞXY ¼ Y þ Y u jk ð3 183Þ

Proof Using Leibniz rule of differentiation, one obtains ÀÁ ∇ Y ¼ ∇ Yi∂ X ÀÁX i : i i ð3 184Þ ¼ ∇XY ∂i þ Y ðÞ∇X∂i

The covariant derivative of a function f ¼ Yi(α) is calculated as

∇Xf ¼ Xf d ð3:185Þ ) ∇ YiðÞ¼α XYiðÞ¼α YiðÞα Y_ iðÞα X dα

The covariant derivative of the coordinate basis with respect to vector X is expressed as 3.15 Levi-Civita Connection on Manifolds 149

Fig. 3.12 Levi-Civita connection of vector ﬁeld Y with respect to X Y

∞ X C M P1 Y

P2 ui (α) X

j ∇X∂i ¼ X ∇∂ ðÞ∂i j ð3:186Þ jΓ k∂ ¼ X ij k

j in which the covariant derivative of the basis ∂j of the coordinate u with respect to ∂i is deﬁned as

∇ ∂ Γ k∂ Γ k∂ : ∂j ðÞ¼i ji k ¼ ij k ð3 187Þ

Substituting Eqs. (3.185) and (3.186) into Eq. (3.184), one obtains interchanging k with i in the second term on the RHS. ∇ _ i∂ i jΓ k∂ XY ¼ Y i þ Y X ij k

_ i k j i ¼ Y ∂i þ Y X Γ ∂i kj _ i k jΓ i ∂ : ¼ Y þ Y X kj i ð3 188Þ

The vector X is written in the bases of the coordinates ui.

d dui ∂ X ¼ ¼ dα dα ∂ui ∂ ¼ Xi ¼ Xi∂ ∂ui i

Thus,

duj Xj ¼ u_ jðÞα ð3:189Þ dα

Inserting Eq. (3.189) into Eq. (3.188) and interchanging k with j, the Levi-Civita connection ∇ in Eq. (3.182) is derived. 150 3 Elementary Differential Geometry ∇ _ i k _ jΓ i ∂ _ i j _ kΓ i ∂ XY ¼ Y þ Y u kj i ¼ Y þ Y u jk i

An afﬁne connection ∇ is called the Levi-Civita connection on a smooth differentiable geometric manifold (M, g)if It preserves the metric: ∇Xg ¼ 0 for all vector ﬁelds on M (Riemann connection) and is torsion-free:

∇XY À ∇YX ¼ ½X; Y where [X,Y] is defined as the Lie bracket. Generally, the covariant derivative of a vector field Y with respect to X at a non-parallel transport does not equal zero (cf. Fig. 3.12). In the case of a parallel transport, the direction of the vector field Y does not change as the vector X moves along the curve C1 on M. Then, the covariant derivative of Y with respect to X vanishes: ∇ _ i j _ kΓ i ∂ : XY ¼ Y þ Y u jk i ¼ 0 ð3 190Þ

Therefore, the ﬁrst-order differential equation of Y for a parallel transport results in

i α k α dY ðÞ Γ i du ðÞ j α α þ jk α Y ðÞ¼0 d d 3:191 duk ∂Yi ð Þ , þ Γ i Yj ¼ 0 dα ∂uk jk

Thus,

_ i α Γ i _ k j i α j α Y ðÞ¼À jku Y Sj ðÞY ðÞ

A geodesic is the shortest distance between two any points on the manifold M. The curve C1 is geodesic (or auto-parallel) if the covariant derivative of vector ﬁelds X with respect to itself vanishes:

∇XX ¼ 0 ð3:192Þ

Substituting X into Y in Eq. (3.191), the second-order differential equation of a geodesic results in 3.15 Levi-Civita Connection on Manifolds 151

i i α du i α X ðÞ¼ α ! Y ðÞ d ð3:193Þ d2ui duj duk ) þ Γ i ¼ 0for8i ¼ 1, 2, ::, N dα2 jk dα dα

Furthermore, let X, Y, Z be vector ﬁelds, f and h be linear functions. The Levi- Civita connection ∇ on a smooth differentiable geometric manifold (M, g) satisﬁes the following properties: – Linearity:

∇XðÞ¼Y þ Z ∇XY þ ∇XZ ð3:194aÞ

– Leibniz rule of differentiation: ∇ ðÞ¼f Y f ∇ Y þ ðÞ∇ f Y X X X ð3:194bÞ ¼ f ∇XY þ ðÞXf Y

– Function linearity:

∇ ∇ ∇ : ðÞf XþhY Z ¼ f XZ þ h YZ ð3 194cÞ

– Free torsion: TXðÞ; Y ∇ Y À ∇ X À ½¼X; Y 0 X Y ð3:194dÞ ) ∇XY À ∇YX ¼ ½X; Y where T(X,Y) is the torsion tensor; [X,Y] is the Lie bracket. – Metric compatibility:

∇XgðÞ¼Y; Z gðÞþ∇XY, Z gðÞY, ∇XZ ð3:194eÞ where g is the bilinear function g(.,.) relating to the metric tensor on the geometric manifold (M, g). The metric tensor g, a covariant second-order tensor of type (0,2) on the manifold M is deﬁned as

i j; g ¼ gijÀÁdu du : ∂ ; ∂ ∂ ∂ ð3 195Þ gij g i j ¼ i Á j

The differential distance between two points can be written in a two-form of the metric tensor g: 152 3 Elementary Differential Geometry qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i j ds ¼ gijdu du

The metric tensor g generates a bilinear function g(.,.) on the tangent space TpM at any point p on the manifold M. In fact, the metric operator g(.,.) is a bilinear mapping of two arbitrary vectors at any point p to a real function.

: g TpM Â TpM ! R ÀÁ : ; ; i j ð3 196Þ gðÞX Y p ¼ gp Xp Yp ¼ gijX Y p where Xp and Yp are tangent vectors at the point p on TpM. The vector ﬁelds X and Y are written on the manifold M as

i j X ¼ X ∂i; Y ¼ Y ∂j:

Using Eq. (3.196), one obtains for any point p on the manifold M

g : TpM Â TÀÁpM ! R ÀÁ i j i j gðÞ¼X; Y gX∂i, Y ∂j ¼ g ∂i; ∂j X Y ð3:197Þ ; i j ... ) gðÞ¼X Y gijX Y for 8i, j ¼ 1, 2, , N

The metric operator g(.,.) has the following properties:

gðÞ¼X; Y gðÞY; X : symmetric; ð3:198Þ gðÞ¼αX þ βY, Z αgðÞþX; Z βgðÞY; Z : bilinear in which α and β are scalars. The Levi-Civita connection satisﬁes the Koszul formula for vector ﬁelds X, Y, and Z on the tangent space TpM.

2gðÞ¼∇XY, Z ∇XgðÞþY; Z ∇YgðÞÀX; Z ∇ZgðÞX; Y ð3:199Þ þ gðÞÀ½X; Y ; Z gðÞÀ½X; Z ; Y gðÞ½Y; Z ; X

Proof Using the metric compatibility in Eq. (3.194e), one obtains

∇XgðÞ¼Y; Z gðÞþ∇XY, Z gðÞY, ∇XZ ; ∇YgðÞ¼X; Z gðÞþ∇YX, Z gðÞX, ∇YZ ; ð3:200Þ ∇ZgðÞ¼X; Y gðÞþ∇ZX, Y gðÞX, ∇ZY :

Substituting Eq. (3.200) into the RHS of Eq. (3.199), its new RHS becomes References 153

RHS gðÞ∇XY þ ∇YX þ ½X; Y , Z

þ gðÞ∇XZ À ∇ZX À ½X; Z , Y ð3:201Þ

þ gðÞ∇YZ À ∇ZY À ½Y; Z , X

Levi-Civita connection is torsion free and therefore satisﬁes using Eq. (3.194d)

∇XY À ∇YX ¼ ½X; Y ; ∇XZ À ∇ZX ¼ ½X; Z ; ð3:202Þ ∇YZ À ∇ZY ¼ ½Y; Z :

Inserting Eq. (3.202) into Eq. (3.201), the Koszul formula has been proved.

RHS gðÞ¼2∇XY, Z 2gðÞ∇XY, Z LHS

In general, the Levi-Civita connection of tensor ﬁelds T with respect to a vector ﬁeld X on the manifold M is calculated in [9]

∇ ∇ i...k ∂ ... ∂ l ... n : XT ¼ ðÞXT l...n i k du du ð3 203Þ in which

∇ i...k p i...k pΓ k i...q ... pΓ q i...k ... : ðÞXT l...n ¼ X Tl...n,p þ X pqTl...n þ À X lpTq...n À ð3 204Þ

Let S and T be arbitrary tensor fields, and X be a vector field. The Levi-Civita connection ∇ on a smooth differentiable geometric manifold (M, g) fulfills the following properties: – Linearity:

∇XðÞ¼S þ T ∇XS þ ∇XT ð3:205aÞ

– Leibniz rule for tensor product:

∇XðÞ¼S T ðÞ ∇XS T þ S ðÞ∇XT ð3:205bÞ

References

1. Ba¨r, C.: Elementare Differentialgeometrie (in German), Zweiteth edn. De Gruyter, Berlin (2001) 2. Chase, H.S.: Fundamental Forms of Surfaces and the Gauss-Bonnet Theorem. University of Chicago, Chicago, IL (2012) 154 3 Elementary Differential Geometry

3. Klingbeil, E.: Tensorrechnung fur€ Ingenieure (in German). B.I.-Wissenschafts-verlag, Mann- heim (1966) 4. Kuhnel,€ W.: Differentialgeometrie – Kurven, Fla¨chen, Mannigfaltigkeit (in German), 6th edn. Springer-Spektrum, Wiesbaden (2013) 5. Lang, S.: Fundamentals of Differential Geometry, 2nd edn. Springer, New York, NY (2001) 6. Danielson, D.A.: Vectors and Tensors in Engineering and Physics, 2nd edn. ABP, Westview Press, CO (2003) 7. Penrose, R.: The Road to Reality. Alfred A. Knopf, New York, NY (2005) 8. Schutz, B.: Geometrical Methods of Mathematical Physics. Cambridge University Press (CUP), Cambridge (1980) 9. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 10. Grinfeld, P.: Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer Science + Business Media, New York, NY (2013) Chapter 4 Differential Forms

4.1 Introduction

Alternative to tensors, differential forms are very useful in differential geometry without considering the coordinates compared to tensors. Differential forms are based on exterior algebra in which the coordinates are not taken into account. The exterior algebra was developed by E´ lie Cartan (1869–1951) and Henri Poincare´ (1854–1912). Before going into details in differential forms in exterior algebra, we look into multi-dimensional spaces and shapes of the objects under points of view between topology and differential geometry. Geometers are concerned with the exact shape, size, and curvature of the object in all possible dimensions. On the contrary, topologists generally look into the overall shape of the object as a whole without considering it in detail as the geometers did. In topology, there are usually two basic kinds of spaces with one-dimensional and two-dimensional shapes of any object [1]. The one-dimensional spaces are a straight line or a circle; however, both are fundamentally different from each other. The difference is that the circle can be transformed into any shape of loops, but a circle cannot be made in a line without cutting it. The two-dimensional spaces are classiﬁed into two basic types: either a sphere or a donut (torus) with two-dimensional surfaces. The big difference between them is that the sphere has no hole in it and the donut with a hole through it. Whatever what you do, it is deﬁnitely that a donut cannot be transformed into a sphere without cutting a hole through the middle. Cubes, hexahedrons, pyramids, and tetrahedrons are topolog- ically homeomorphic (similar shape) to a sphere with genus 0. Note that the genus denotes the number of holes in the object. Obviously, a donut (torus) has genus 1, which denotes one hole in it. Pretzels with two holes have genus 2; pretzels with three holes, genus 3 (see Fig. 4.1).

© Springer-Verlag Berlin Heidelberg 2017 155 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5_4 156 4 Differential Forms

Fig. 4.1 Topological manifolds with various genera of 0, 1, 2, and 3

If different objects have the same genus, they are homeomorphisms or topological isomorphisms in the mathematical field of topology. Any cube with a hole through it can be somehow reshaped into the shape of a torus (genus 1) by squeezing and stretching. However, an object with any genus other than one (genus 6¼ 1); e.g., bun (genus 0), American pretzel (genus 2), and German pretzel (genus 3) cannot be molded into a donut (genus 1) without punching a hole through it or cutting the holes through the middles. Objects with surfaces at topological dimensions equal to or higher than three are quite difficult to display. They are called hypersurfaces of N-manifolds (N 3). In fact, a sphere has a topological dimension of two because only its 2-D surface is considered. Therefore, it is called 2-sphere S2 or a sphere. Any surface on which every loop can be shrinked to a point is defined as a sphere (genus 0). On the contrary, loops on the torus (genus 1) surface cannot be tightened to a point, as shown in Fig. 4.1. In differential geometry, the dimension of a space (manifold) is the necessary numbers of coordinates to determine the characteristic of a given point (location and physical states) on the manifold. Therefore, the manifold of the object in differential geometry is studied in more detail in higher-dimensional Riemannian spaces, different to topology in which the object is considered as a whole. Hence, the sphere S2 is considered as a three-dimensional space in differential geometry. The computations in N-dimensional spaces are carried out using tensors and differential forms without drawing them on the paper. In order to maintain that the maximum velocity of any object must be less than or equal to the light speed (v c), the four-dimensional Minkowski spacetime (t, x, y, z) is generally used in the Maxwell’s equations, special relativity theory, and cosmology (cf. Chap. 5). 4.3 Differential k-Forms 157

4.2 Deﬁnitions of Spaces on the Manifold

Differential forms deal with calculations in differential geometry without considering any coordinate contrary to vector and tensor calculus. However, they use many mapping operators in various spaces, vector and differential form bundles using exterior algebra that is quite different to linear algebra. Unfortunately, they are very confused in the applications of differential forms. In order to comprehend the differential forms, some necessary spaces and bundles on the manifold are deﬁned in the following section [2]. Let M be a smooth and differentiable N-dimensional manifold, M* be a dual manifold of M, and p be a point on the manifold M.

– The tangent space TpM at the point p on M is defined as the space that contains the tangent vector to the manifold M at the point p. The basis vectors in the tangent space TpM have been discussed in Sect. 3.14. * – The cotangent space Tp M at the point p on M is defined as the dual space of i the manifold M. The dual bases dx for i ¼ i1,...,ik are used in the cotangent space * Tp M. – The tangent bundle TM is a set of all tangent spaces TpM as the point p moves on the manifold M. * * – The cotangent (dual) bundle T M is a set of all dual spaces Tp M as the point p moves on the manifold M. k – ^ (TpM) is defined as the subset of the k-tangent-vector space at the point p on the tangent space TpM. k * – ^ (Tp M) is defined as the subset of the k-form space at the point p on the * cotangent (dual) space Tp M. – ^k(M) is defined as the exterior k-vector-bundle space on M that consists of all k subsets of the tangent vector spaces ^ (TpM). – ^k(M*) is defined as the exterior k-form-bundle space on M that consists of all k * subsets of the k-form spaces ^ (Tp M). – Ak(M) denotes the space of smooth sections of the exterior k-form bundle of ^k (M*).

4.3 Differential k-Forms

The differential k-form ωp at the point p on M is an alternating (skew-symmetric) k * multilinear map from p into the subset of the k-form space ^ (Tp M). ω : T*M Â ...Â T*M ! Λk T*M ΛkðÞM p |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}p p p k times X À Á ω : i1 ... ik ω ... i1 ... ik : , p ðÞ!dx ^ ^ dx p ¼ f I xp1 , , xpn dx ^ ^ dx ð4 1Þ XI À Á ... I Λk * f I xp1 , , xpn dx 2 TpM I 158 4 Differential Forms

ω p k k cotangent space −form space exterior −form bundle

Λk (M * ) ⊆ R N dx I * Λk (T *M ) Tp M p p manifold ω = I p ∑ f I dx I

M ⊆ R N

Fig. 4.2 A differential k-form ωp of a point p on M

The elementary k-form of ωp is deﬁned as

dxI dxi1 ^ dxi2 ^ ...^ dxik ð4:2Þ for I ¼ ðÞi1; ...; ik ; i1 < ...< ik

i in which the dual bases dx for i ¼ i1,...,ik are used in the cotangent (dual) space * Tp M on the manifold M. The k-form ωp is a subset of the exterior form bundle on M (see Fig. 4.2). ÀÁ ω Λk * Λk * Λk * : p 2 TpM M [ TpM ð4 3Þ p2M

Some examples of k-forms are given as – Zero-form (k ¼ 0): ω ¼ 1 – One-form (k ¼ 1): ω ¼ yzdx þ xydy À x2zdz – Two-form (k ¼ 2): ω ¼ x2y3ðÞþdx ^ dy xy3ðÞÀdy ^ dz xyz2ðÞdx ^ dz – Three-form (k ¼ 3): ω ¼ xy2z3ðÞdx ^ dy ^ dz The wedge product (exterior product) of two any differential one-forms ω and η is deﬁned as

ω ^ ηω η À η ω ð4:4Þ where is the tensor product (cf. Chaps. 2 and 6). Let ω and η be one-forms for k ¼ 1. If the order of two dual bases is interchanged, one permutation is carried out. As a result, the sign of the k-form is changed. In the case of an even number of permutations, the sign of the k-form is +1 and the sign is À1 for an odd number of permutations (see Table 4.1). 4.3 Differential k-Forms 159

Table 4.1 Permutations of I i1 i2 i3 3-forms dx σ the dual bases in 3-forms x yzdx ^ dy ^ dz ¼þdx ^ dy ^ dz 0 x zydx ^ dz ^ dy ¼Àdx ^ dy ^ dz 1 y xzdy ^ dx ^ dz ¼Àdx ^ dy ^ dz 1 y zxdy ^ dz ^ dx ¼þdx ^ dy ^ dz 2 z xydz ^ dx ^ dy ¼þdx ^ dy ^ dz 2 z yxdz ^ dy ^ dx ¼Àdx ^ dy ^ dz 3

Generally, the elementary k-forms result after σ permutations as

σ σ σ dx ðÞi1 ^ dx ðÞi2 ^ ...^ dx ðÞik ¼ÀðÞ1 mσ dxi1 ^ dxi2 ^ ...^ dxik ð4:5Þ where mσ is the modulo 2 of σ that is the residual of the permutation number σ divided by 2.

mσ ¼ σmod2 ð4:6Þ

Some examples of the permutations of k-forms are given in the following section. – If no permutation (σ ¼ 0) is carried out, one obtains

m0 ¼ 0mod2 ¼ 0 þ 0 ) m0 ¼ 0

Thus,

dxi1 ^ dxi2 ^ ...^ dxik ¼þðÞ1 dxi1 ^ dxi2 ^ ...^ dxik

–Ifσ equals one permutation (σ ¼ 1), one obtains

m1 ¼ 1mod2 ¼ 0 þ 1 ) m1 ¼ 1

Thus,

dxi2 ^ dxi1 ^ ...^ dxik ¼ÀðÞ1 dxi1 ^ dxi2 ^ ...^ dxik

–Ifσ permutations are carried out, one obtains

mσ ¼ &σmod2; σ > 1 1 if σ is an odd number ¼ 0 if σ is an even number

Therefore,

dxik ^ dxi1 ^ dxi2 ^ ...^ dxikÀ1 ¼ÀðÞ1 mσ dxi1 ^ dxi2 ^ ...^ dxik 160 4 Differential Forms

Table 4.1 displays the permutations of the dual bases in six 3-forms (k! ¼ 3! ¼ 6) with the number of permutations σ. The first 3-form of the dual bases is called the elementary form without any permutation (σ ¼ 0). Moving the dual bases in the first 3-form, there are fives possibilities of the 3-forms that are transformed into the elementary form using Eq. (4.5). Interchanging dz with dy in the second 3-form, one obtains the elementary form at one permutation (σ ¼ 1); thus, its sign is changed. The fourth 3-form needs two permutations (σ ¼ 2) of dz with dx and dy with dx to become the elementary form; therefore, its sign is unchanged. Similarly, other remaining 3-forms are easily transformed into the elementary form using Table 4.1. In general, let ω be a k-form, η be an l-form, and v be a m-form. The wedge product of two differential forms has the following properties:

ω ^ ðÞ¼η þ v ðÞþω ^ η ðÞω ^ v : left À distributive ðÞ^η þ v ω ¼ ðÞþη ^ ω ðÞv ^ ω : right À distributive ð4:7Þ ω ^ ðÞ¼η ^ v ðÞ^ω ^ η v : associative ω ^ η ¼ÀðÞ1 klη ^ ω : graded À anticommutative

Using the graded-anticommutative law in Eq. (4.7), one obtains for ω ¼ η

ω ^ ω ¼ 0 ) dω ^ dω ¼ 0 ð4:8Þ

Moving k dual bases of the k-form behind the l dual bases of the l-form, the wedge product of two differential forms in Eq. (4.7) is proved. X ÀÁÀÁ i i j j ω ^ η ¼ ω ... η ... dx 1 ^ ...^ dx k ^ dx 1 ^ ...^ dx l i1 ik j1 jl Xi, j ÀÁ l i i ¼ ðÞÀ1 ω ... η ... dx 1 ^ ...^ dx kÀ1 ^ i1 ik j1 jl i, j : j j j i ð4 9Þ X ðÞ^dx 1 ^ dx 2 ^ ...ÀÁ^ dx l dx k ÀÁ lk j j i i ¼ ðÞÀ1 η ... ω ... dx 1 ^ ...^ dx l ^ dx 1 ^ ...^ dx k j1 jl i1 ik i, j ¼ÀðÞ1 klη ^ ω ðÞq:e:d:

According to Eq. (4.9), the wedge product of two differential k- and l-forms has the order of (k + l) and belongs to the exterior (k + l)-form space in exterior algebra. ÀÁ ÀÁ ω 2 Λk T*M Λk M* ; η 2 Λl T*M Λl M* p p ÀÁ ω η klη ω Λkþl * Λkþl * ) ^ ¼ÀðÞ1 ^ 2 TpM M 4.4 The Notation ω Á X 161

The differential forms of ω and η are written in their dual bases as X X ω ω i1 i2 ... ik ω I; ¼ i1...ik dx ^ dx ^ ^ dx Idx <...< i1X ik XI j j j J η ¼ η ... dx 1 ^ dx 2 ^ ...^ dx l η dx : j1 jl J <...< j1 jl J

I J where ωI and ηJ are smooth differentiable functions of x and x , respectively. k * N The dimension of the k-form space ^ (Tp M)inanN-dimensional space R is defined as the number of choosing k-element subsets of an N-element set that consists of N distinct elements. The dimension of the k-form space is defined as the possible number of k permutations from N dimensions disregarding the permutation order. As a result, the k-form dimension is calculated as the binomial coefficient in combinatorics. N N! ¼ ¼ dim Λk T*M ; 0 k N ð4:10Þ k k! ðÞN À k ! p

The factorial of N is deﬁned by

YN N! ¼ k ¼ 1 Á 2 ÁÁÁðÞÁN À 1 Nfor8N > 0; 0! 1 k¼1

Thus, the factorial of N can be written in the recurrence relation as & 1 if N ¼ 0; N! ¼ ðÞN À 1 !NifN> 0:

Using Eq. (4.10), the dimensions of k-form spaces in a three-dimensional space R3 are calculated and shown in Table 4.2.

4.4 The Notation ω Á X

1 * Let ω be a one-form in the 1-form space ^ (Tp M); X be a vector ﬁeld in the N tangent space TpM 2 R . A linear mapping of y 2 M to ωÁX( y) by the mapping function ωÁX is written as

ω Á X : y 2 M ! ω Á XyðÞ2R 162 4 Differential Forms

Table 4.2 Dimensions of the k-form spaces in R3 ω (0 k 3) Form space Dual bases of k-forms Dimensions of form space ÀÁ 0–form ^0 R3 ¼ R 11 ÀÁ 1 2 3 1–form ^1 R3 dx , dx , dx 3 ÀÁ 2–form ^2 R3 dx1 ^ dx2, dx2 ^ dx3, dx1 ^ dx3 3 ÀÁ 3–form ^3 R3 dx1 ^ dx2 ^ dx3 1

The notation ωÁX linking the differential forms to vector ﬁelds is deﬁned as

y 2 M ! ω Á XyðÞωðÞÁy XyðÞ2R

Using Einstein’s summation convention, the one-form ω in the 1-form space ^1 * (Tp M) and the vector ﬁeld X in the tangent space TpM are written as ω i Λ1 * ; ... ¼ hidx 2 TpM i ¼ 1, , N ∂ X ¼ ξj 2 T M; j ¼ 1, ..., N ∂xj p

Due to orthonormality of the bases of the tangent and cotangent spaces according to Eq. (3.178), the notation ωÁX( y) is written using Kronecker delta as

ω ω ξj δ j Á XyðÞ¼ ðÞÁy XyðÞ¼hiðÞÁy ðÞy i i ¼ hiðÞÁy ξ ðÞ2y R; i ¼ 1, ..., N

p * Analogously, the notation is extended to p-forms ω in the p-form space ^ (Tp M) p and vector bundles (X1,..., Xp) in the p-vector-bundle space ^ (M)as ÀÁÀÁ y 2 M ! ω Á X1 ^ ...^ Xp ðÞ¼y ωðÞÁy X1ðÞ^y ...^ XpðÞy 2 R

Due to linearity of the p-forms, an element (monomial) of the p-form is used in this case. The monomial ω of the p-form for y 2 M is written as ω i1 ... ip Λp * ðÞ¼y hyðÞdx ^ ^ dx 2 TpM

Using orthonormality of the bases, the extended notation ωÁX results as [3] ÀÁX ÀÁ ÀÁ mσ i1 ip ω Á X1 ^ ...^ Xp ¼ ðÞÀ1 h Á Xσ i x ÁÁÁXσ x ðÞ1 ðÞip σ2Σp where σ is the number of permutations; Σp is a set of permutations {1,2,...,p}; 4.5 Exterior Derivatives 163

mσ ¼ σ mod 2, cf. Eq. (4.6); i * x are the local coordinates in the cotangent space Tp M.

4.5 Exterior Derivatives

Let ω be any k-form; η be any l-form in any N-dimensional space RN. The exterior derivative dω of the k-form ω is a (k + 1)-form [3–5]. ÀÁ ÀÁ ÀÁ ω 2 Λk RN Λk M* ) dω 2 Λkþ1 RN

The k-form ω is written in the dual bases as X X ω i1 i2 ... ik I ¼ f i1...ik dx ^ dx ^ ^ dx f Idx i1<...

Generally, the exterior derivative upgrades the differential form by one order. The exterior derivative has the following properties:

dðÞ¼ω þ η dω þ dη ð4:12aÞ X ∂f df ¼ Idxi ð4:12bÞ I ∂ i i x dðÞ¼ω ^ η dω ^ η þÀðÞ1 kω ^ dη ð4:12cÞ ddðÞ¼ω 0 : The Poincare Lemma ð4:12dÞ

Proof of Eq. (4.12c) Using Eq. (4.11) and the chain rule of differentiation, one obtains the exterior derivative of the wedge product of two differential forms. 164 4 Differential Forms X X À Á ω η ω η I J ω η I J dðÞ¼^ d ðÞI J dx ^ dx ¼ d I Jdx ^ dx I, J I, J X X ∂ðÞω η ¼ I J dxi ^ dxI ^ dxJ ∂xi I, J i X X ∂ω ∂η ¼ I η þ J ω dxi ^ ðÞdxI ^ dxJ ∂xi J ∂xi I I, J i X X ∂ω X ¼ I dxi ^ dxI ^ η dxJþ ∂xi J I i J X X X ∂η ðÞÀ1 k ω dxI ^ J dxi ^ dxJ I ∂ i I J i x ¼ ðÞþÀdω ^ η ðÞ1 kðÞω ^ dη ðÞq:e:d:

Proof of the Poincare´ Lemma Eq. (4.12d) Using Eqs. (4.11) and (4.12a–c), the exterior derivative of any differential k-form ω is written as ! X X ÀÁX I I I dω ¼ d ωIdx ¼ d ωIdx ¼ dωI ^ dx I !I I : X X ð4 13Þ ∂ωI ¼ dxi ^ ðÞdxi1 ^ ...^ dxik ∂ i I i x

Using Eq. (4.13) and the chain rule of differentiation, the exterior derivative of dω is calculated as ! ! X X ∂ω X X ∂2ω ddðÞ¼ω d Idxi ^ dxI ¼ I dxj ^ dxi ^ dxI ∂xi ∂xj∂xi I i !I i, j X X ∂2ω ∂2ω ÀÁ ¼ I À I dxj ^ dxi ^ dxI ¼ 0 ðÞq:e:d: ∂xj∂xi ∂xi∂xj I i

This equation is called the Poincare´ Lemma. If f is any zero-form (a smooth function with k ¼ 0) and ω is any one-form, one obtains using Eqs. (4.12c) and (4.12d)

dfðÞ^ dω dfdðÞω ω 0 ω ω : ¼ df ^ d þÀðÞ1 f ^ |fflffl{zfflffl}ddðÞ¼ df ^ d ð4 14Þ ¼0 4.5 Exterior Derivatives 165

Using Eq. (4.12b), Eq. (4.14) can be expressed as

X ∂f dfðÞ¼^ dω df ^ dω ¼ dxi ^ dω ð4:15Þ ∂ i i x

Some examples of differential forms and their exterior derivatives are given in the following section. 1. One-form (k ¼ 1) A one-form ω is given in R3 as ÀÁ ω ¼ xydx þ xzdy þ xyzdz 2 Λ1 R3

Using Eq. (4.13), the exterior derivative of ω is calculated as

dω ¼ dxyðÞ^dx þ dxzðÞ^dy þ dðÞ^ xyz dz ¼ xdy ^ dx þ ðÞþzdx ^ dy þ xdz ^ dy ðÞyzdx ^ dz þ xzdy ^ dz ¼Àxdx ^ dy þ zdx ^ dy À xdy ^ dz þ yzdx ^ dz þ xzdy ^ dz ÀÁ ¼ÀðÞx þ z dx ^ dy þ xzðÞÀ 1 dy ^ dz þ yzdx ^ dz 2 Λ2 R3

The exterior derivative of dω is calculated further using Eq. (4.5)as

ddðÞ¼ω ddωd2ω ¼ dz ^ dx ^ dy þ ðÞz À 1 dx ^ dy ^ dz þ zdy ^ dx ^ dz ÀÁ ¼ ðÞ1 þ z À 1 À z dx ^ dy ^ dz 2 Λ3 R3 ¼ 0 ðÞq:e:d:

2. Two-form (k ¼2) A two-form ω is given in R3 as ÀÁ ÀÁ ω ¼ x2ðÞy þ z dx ^ dy þ yx2 þ 2z dy ^ dz þ 2xy2dx ^ dz 2 Λ2 R3

Using Eq. (4.13), the exterior derivative of ω is calculated as ÀÁ ÀÁÀÁ ÀÁ dω ¼ dx2ðÞy þ z dx ^ dy þ dyx2 þ 2z dy ^ dz þ d 2xy2 dx ^ dz ¼ x2dz ^ dx ^ dy þ 2xydx ^ dy ^ dz þ 4xydy ^ dx ^ dz ÀÁ ¼ x2 þ 2xy À 4xy dx ^ dy ^ dz ÀÁ ¼ xxðÞÀ 2y dx ^ dy ^ dz 2 Λ3 R3

Due to (dxi^ dxi) ¼ 0 and using Eq. (4.13), one obtains Eq. (4.12d). 166 4 Differential Forms

ddðÞ¼ω dxx½ðÞÀ 2y dx ^ dy ^ dz ¼ dxx½^ðÞÀ 2y dx ^ dy ^ dz X ∂½xxðÞÀ 2y ¼ dxi ^ dx ^ dy ^ dz ∂ i i x ¼ 0 ðÞq:e:d:

3. Three-form (k ¼3) A three-form ω is given in R3 as

ω ¼ fxðÞ; y; z dx ^ dy ^ dz ÀÁ ¼ x2yz3dx ^ dy ^ dz 2 Λ3 R3

Using (dxi^ dxi) ¼ 0 and Eq. (4.13), the exterior derivative of ω is calculated as ÀÁ dω ¼ dx2yz3dx ^ dy ^ dz ÀÁ ¼ dx2yz3 ^ dx ^ dy ^ dz X ∂ðÞx2yz3 ¼ dxi ^ dx ^ dy ^ dz ∂ i i x ¼ 0 fordxi ¼ dx, dy, dz

Thus,

ddω ¼ dðÞ¼0 0 ðÞq:e:d:

4.6 Interior Product

The interior product or interior derivative degrades the differential form by one order. It is usually applied to exterior algebra of k-differential forms on a smooth N- dimensional manifold M. In fact, the interior product is a linear map of a k-form ω to the (k À 1)-form iXω ... in the vector ﬁelds (X1, ,XkÀ1) on the manifold M. In exterior algebra, the interior product is an operator for contracting vectors and differential forms to one-order lower forms. The interior product of any differential k-form ω on the manifold M is deﬁned as : ω Λk * ω ΛkÀ1 * iX 2 TpM ! iX 2 TpM ω ; ...; ω ; ; ...; ... ω ðÞiX ðÞ¼X1 XkÀ1 ðÞXX1 X XkÀ1 iXkÀ1 iX1 iX ð4:16Þ iω i1 ... ikÀ1 ¼ X ii1ÁÁÁikÀ1 dx ^ ^ dx i1<::::

Using Einstein summation convention, the vector ﬁeld X is written as

∂ X ¼ Xi Xi∂ ∂xi i

Let f be a smooth function (a zero-form), X be a vector ﬁeld on M, ω be a one-form, η and ξ be k- and l-forms, respectively. The interior product has the following properties:

iX f ¼ 0 iXω ¼ ωðÞX hiω; X : inner product k ð4:17Þ iXðÞ¼η ^ ξ ðÞ^iXη ξ þÀðÞ1 η ^ ðÞiXξ iXiYη ¼ÀiYiXη

Furthermore, if α and β are one-forms; X and Y are the vector ﬁelds on the manifold M, one obtains from Eqs. (4.16) and (4.17)

α ; α α : ðÞ¼X Y iYiX ¼ iY |fflfflffl{zfflfflffl}ðÞðÞX ¼ 0 ð4 18aÞ 0Àform and hi 1 ðÞα ^ β ðÞ¼X; Y iY ½¼iXðÞα ^ β iY iXα ^ β þÀðÞ1 α ^ iXβ

¼ iY ½αðÞ^X β À α ^ βðÞX ¼ iY ½αðÞX β À βðÞX α ¼ αðÞX ðÞÀiYβ βðÞX ðÞiY α ¼ αðÞX βðÞÀY βðÞX αðÞY ð4:18bÞ

Therefore,

ðÞα ^ β ðÞ¼X; Y αðÞX βðÞÀY βðÞX αðÞY ¼ hiα; X Á hiβ; Y À hiβ; X Á hiα; Y ð4:19Þ

The Cartan’s formula for any differential form ω is given in [4, 6]

£Xω ¼ iXdω þ diðÞXω ð4:20Þ where ω is the k-form on M;£Xω is the Lie derivative of ω, cf. Eq. (3.144a). Changing ω into a zero-form f and using the property of iX f ¼ 0, one obtains the relation between the Lie derivative and interior product of any function f

£ f Xf X ð4:21Þ ¼ iXdf þ diðÞ¼X f iXdf 168 4 Differential Forms

In general, the Cartan’s formula (known as Weil’s formula) in differential calculus is written as

£X ¼ diX þ iXd ¼ ½d; iX ð4:22Þ

4.7 Pullback Operator of Differential Forms

The pullback operator of differential forms is used in the transformation of coordinate variables from one manifold to another manifold. It is a smooth linear map of two mapping functions f and g.

f : XxðÞ2M RN ! fXðÞ¼yxðÞ2N RN g : fXðÞ2N RN ! gfXðÞ¼ðÞ gyxðÞ2ðÞ P R ) f *g : XxðÞ2M RN ! f *gXðÞ¼gfXðÞ2ðÞ P R

Let X (x) be a vector field on the manifold M. The pullback operator f*g of g by f in the vector field X(x) 2 M is defined as

f *g : XxðÞ2M ! f *gXðÞ¼gfXðÞ¼ðÞ gyxðÞ2ðÞ P ð4:23Þ

The operator f*gis called the pullback g by f since the mapping function g is pulled from N backwards to M (see Fig. 4.3). The function f maps the vector ﬁeld X to f(X) ¼ y(x) that is further mapped by the function g to g( f(X)). In fact, the pullback operator f*g maps the vector ﬁeld X 2 M to go f(X) 2 P directly. An example of the pullback is given in the following section. Let ω be a one-form of y that is written as ÀÁ ω 2 Λ1 3 ¼ y1y2dy1 þ y2y3dy2 þ y1y2y3dy3 2 R

* ≡ = f g go f (X ) g( f (X ))

f g

X (x)∈ M f (X ) = y(x)∈ N g( f (X )) = g(y(x))∈ P

Fig. 4.3 Pullback operator f*g of g by f over M 4.7 Pullback Operator of Differential Forms 169

Let X(t) be a vector ﬁeld (t,2t,3t) 2 R3. The linear function f maps X to f(X)as 8 < y1 ¼ t : f XtðÞ!fXðÞ¼ytðÞ¼: y2 ¼ 2t 8 y3 ¼ 3t < dy1 ¼ dt ) dfðÞ¼ X dyðÞ¼ t : dy2 ¼ 2dt dy3 ¼ 3dt

Using the above equation, the pullback of the one-form ω by f results as a one-form

f *ωðÞ¼X ωðÞ¼fXðÞ ωðÞytðÞ ¼ tðÞ2t ðÞþdt ðÞ2t 2ðÞ3t ðÞþ2dt tðÞ2t ðÞ3t ðÞ3dt ÀÁ ¼ 2t2ðÞ21t þ 1 dt 2 Λ1 R3

The pullback of differential forms has the following properties:

f *ðÞ¼ω þ η f *ω þ f *η; f *ÀÁðÞ¼ω ^ η f *ω ^ f *η; ð4:24Þ df*ω ¼ f *ðÞdω : where ω and η are the arbitrary k-forms; and f is the linear mapping function. The function g in the pullback mapping is replaced by a k-form ω that is written in RN as X X ÀÁ J j j k N ω ¼ aJdy ¼ aJdy 1 ^ ...^ dy k 2 Λ R J J

The pullback of the k-form ω by the mapping function f results as ! X X À Á * * J * * j j f ω ¼ f aJdy ¼ f aJ Á f dy 1 ^ ...^ dy k J J X À Á ÀÁ ÀÁ * j1 ... * jk ¼ aJo f Á f dy ^ ^ f dy J X À Á ÀÁ ÀÁ j1 i ... jk i ¼ aJo f Á df x ^ ^ df x J ! ! X X X À Á ∂f j1 ∂f jk i1 ... ik ¼ aJo f dx ^ ^ dx ∂xi1 ∂xik J i1 ik X X À Á ∂f j1 ∂f jk i1 ... ik ¼ aJo f ÁÁÁ dx ^ ^ dx ∂xi1 ∂xik J i1, ..., ik 170 4 Differential Forms

The second ∑ term on the RHS of the above equation is in fact the Jacobian J. X j j ∂f 1 ∂f k ∂ðÞyj1 ; ...; yjk J ÁÁÁ ¼ ∂xi1 ∂xik ∂ðÞxi1 ; ...; xik i1, ..., ik

Thus, X À Á *ω ω i1 ... ik f ¼ o f ¼ J aJo f dx ^ ^ dx XJ ÀÁ : I k N ð4 25Þ J aIdx 2 Λ R I

4.8 Pushforward Operator of Differential Forms

Figure 4.4 shows the mapping scheme of the pushforward operator in an N- dimensional space RN. The linear smooth mapping function f creates the vector ﬁeld y 2 V from any vector ﬁeld x 2 U:

f : x 2 U RN ! y ¼ fxðÞ2V RN ð4:26Þ ) x ¼ f À1ðÞy

* N * N The linear function f* maps the dual space TX R of X to the dual space Ty R of the vector ﬁeld y:

: * N * N : dfðÞ x f * XxðÞ2TXR ! f *X 2 TyR ð4 27Þ

f y = f (x) x ∈U ⊆ R N ∈V ⊆ R N

X f* X

≡ df (x) f* ∈ * N ∈ * N X (x) TX R f* X Ty R

Fig. 4.4 Pushforward operator f*X of X by f 4.9 The Hodge Star Operator 171

The operator f*X is called the pushforward operator of X by the mapping function f that is deﬁned as ÀÁ À1 * N : f *XxðÞdfðÞ x Xf ðÞy 2 Ty R ð4 28Þ

* N The pullback of a one-form ω by f for any vector ﬁeld X 2 TX R is deﬁned as [3] ÀÁ *ω ω : f Á XxðÞ¼ Á f *XyðÞ ð4 29aÞ

Therefore, the pullback of a k-form ω (k > 1) by f for the bundle of vector ﬁelds * (Xi1,..., Xik) 2 T M results from Eq. (4.29a).

*ω ... ω ... : f Á ðÞ¼Xi1 ^ ^ Xik Á ðÞðf *Xi1 ^ ^ f *Xik 4 29bÞ

4.9 The Hodge Star Operator

The Hodge star operator (star operator) is used to carry out various operations, such as gradient, div, and curl in exterior algebra, in which the coordinates are not taken into account. The Hodge * operator is a counterpart of Nabla operator in vector calculus in linear algebra that considers the coordinates in calculations. The star operator is a linear map from an exterior k-form bundle to another exterior (N À k)-form bundle on the N-dimensional manifold M [5, 6] ÀÁ ÀÁ * : Λk RN ! ΛNÀk RN ; 0 k N ð4:30Þ where ^k(RN) is the differential k-form bundle that consists of all differential k- form spaces. Let ω be any k-form that is written for I ¼ (i1,...,ik) with 1 i1 < ...< ik N as X X ÀÁ ω I i1 ... ik Λk N : ¼ f Idx ¼ f Idx ^ ^ dx 2 R ð4 31Þ I I

The Hodge * operator maps the k-form ω to the Hodge dual *ω in the exterior (N À k)-form bundle. The Hodge dual *ω is a pseudo (N À k)-form that is written as ÀÁ ÀÁ : I Λk N I ΛNÀk N * dx 2 X R ! *dx 2ÀÁR ω I ΛNÀk N ð4:32Þ ) * ¼ f I*dx 2 R I

Similarly, the Hodge dual elementary (N À k)-form is deﬁned as the Hodge * operator of the elementary form dxI as 172 4 Differential Forms

I ikþ1 ikþ2 ... iN : *dx ¼ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}dx ^ dx ^ ^ dx ð4 33Þ ðÞNÀk times

Substituting Eq. (4.33) into Eq. (4.32), the Hodge dual of the k-form ω is written as X X ω I ikþ1 ikþ2 ... iN * ¼ f I*dx ¼ f Idx ^ dx ^ ^ dx XI I ÀÁ 4:34 J ΛNÀk N ð Þ f Idx 2 R I, J where I ¼ (i1,...,ik) with 1 i1 < ...< ik N and J ¼ (ik+1,...,iN) with 1 ik+1 < ...< iN N. Obviously, the wedge product of the elementary k-form and its Hodge dual is the elementary N-form.

dxI ^ *dxI ¼ ðÞ^dxi1 ^ ...^ dxik ðÞdxikþ1 ^ ...^ dxiN ð4:35Þ ¼ dxi1 ^ dxi2 ^ ...^ dxiN dxN

Some examples of the Hodge * operators of the dual bases (dx,dy,dz)inR3 are given in the following section.

*dx ¼ dy ^ dz ) *dyðÞ¼^ dz dx *dy ¼ dz ^ dx ) *dzðÞ¼^ dx dy ð4:36aÞ *dz ¼ dx ^ dy ) *dxðÞ¼^ dy dz *1 ¼ dx ^ dy ^ dz ) *dxðÞ¼^ dy ^ dz 1

Proof Using Eqs. (4.5) and (4.35), one writes ÀÁ dx ^ ðÞ¼*dx dx ^ ðÞ2dy ^ dz Λ3 R3 ) *dx ¼ dy ^ dz; ÀÁ dy ^ ðÞ¼*dy dx ^ dy ^ dz ¼Àdy ^ ðÞ2dx ^ dz Λ3 R3 ) *dy ¼Àdx ^ dz ¼ dz ^ dx; ÀÁ dz ^ ðÞ¼*dz dx ^ dy ^ dz ¼ dz ^ ðÞ2dx ^ dy Λ3 R3 ) *dz ¼ dx ^ dy; ðÞ^dx ^ dy ^ dz *dxðÞ¼^ dy ^ dz dx ^ dy ^ dz ) *dxðÞ¼^ dy ^ dz 1:

In general, Eq. (4.36a) can be expressed in an N-dimensional space RN.

*dxi1 ¼ dxi2 ^ ...^ dxiN ; *dxðÞ¼i1 ^ dxi2 ^ ...^ dxiN 1; ð4:36bÞ *1 ¼ dxi1 ^ dxi2 ^ ...^ dxiN 4.9 The Hodge Star Operator 173

4.9.1 Star Operator in Vector Calculus and Differential Forms

Let f be a function of x, y, and z in R3. The gradient of a function f(x,y,z) is written in a one-form as

∂f ∂f ∂f ∂f df ¼ dxi ¼ dx þ dy þ dz ¼ ∇f Á dr ð4:37aÞ ∂xi ∂x ∂y ∂z

Using the Hodge duals, the Laplacian of f(x,y,z) is calculated as

Δf ¼ ∇ Á ∇f ¼ *dðÞ *df ð4:37bÞ

3 Similarly, let v be any vector (one-form) in R . The vector v (v1,v2,v3) corresponds to a one-form ωv that can be expressed as

v $ ωvv1dx þ v2dy þ v3dz ¼ v Á dr ð4:38Þ

Using Eq. (4.14), the exterior derivative of the one-form ωv is written as

dωv ¼ dvðÞ1dx þ v2dy þ v3dz ¼ ðÞþdv1 ^ dx ðÞþdv2 ^ dy ðÞdv3 ^ dz

The ﬁrst term on the RHS of dωv is calculated as ∂v ∂v ∂v dv ^ dx ¼ 1 dx þ 1 dy þ 1 dz ^ dx 1 ∂x ∂y ∂z ∂v ∂v ¼ 1 dy ^ dx þ 1 dz ^ dx ∂y ∂z ∂v ∂v ¼À 1 dx ^ dy þ 1 dz ^ dx ∂y ∂z

Analogously, the second and third terms on the RHS of dωv result as ∂v2 ∂v2 dv2 ^ dy ¼ dx ^ dy À dy ^ dz ∂x ∂z ∂v ∂v dv ^ dz ¼À 3 dz ^ dx þ 3 dy ^ dz 3 ∂x ∂y

Hence, the exterior derivative of the one-form ωv is calculated as

dωv ¼ dvðÞ1dx þ v2dy þ v3dz ∂v ∂v ∂v ∂v ∂v ∂v ¼ 3 À 2 dy ^ dz þ 1 À 3 dz ^ dx þ 2 À 1 dx ^ dy ∂y ∂z ∂z ∂x ∂x ∂y 174 4 Differential Forms

Thus, the Hodge dual of the exterior derivative dωv results as ∂v ∂v ∂v ∂v ∂v ∂v *dω ¼ 3 À 2 dx þ 1 À 3 dy þ 2 À 1 dz ð4:39aÞ v ∂y ∂z ∂z ∂x ∂x ∂y

The curl of v is calculated in R3 as hi ∂ ∂ ∂ ∂ ∂ ∂ T ∇ v3 v2 v1 v3 v2 v1 : Â v ¼ ∂y À ∂z ∂z À ∂x ∂x À ∂y ð4 39bÞ

Using Eqs. (4.39a) and (4.39b), one obtains the relation between the Hodge dual *dωv and curl v

*dωv ¼ ðÞÁ∇ Â v dr ð4:40Þ

To calculate the divergence of a vector, a two-form ηv is deﬁned using the vector components as

η vv1dy ^ dz þ v2dz ^ dx þ v3dx ^ dy

The Hodge dual of the one-form ωv in Eq. (4.38) results as

*ωvv1ðÞþ*dx v2ðÞþ*dy v3ðÞ*dz

Using Eq. (4.36a), the two-form ηv equals the Hodge dual *ωv. η ω : v ¼ * v ð4 41Þ

The exterior derivative dηv is calculated as η : d v ¼ dvðÞð1dy ^ dz þ v2dz ^ dx þ v3dx ^ dy 4 42Þ

Using Eq. (4.14), the RHS of Eq. (4.42) is rewritten as

RHSdvðÞdy ^ dz þ v dz ^ dx þ v dx ^ dy 1 2 3 ð4:43Þ ¼ dv1 ^ ðÞþdy ^ dz dv2 ^ ðÞþdz ^ dx dv3 ^ ðÞdx ^ dy

Using the property of the wedge product (dx ^ dx) ¼ 0, the ﬁrst term on the RHS in Eq. (4.43) is calculated as ∂v ∂v ∂v dv ^ ðÞ¼dy ^ dz 1 dx þ 1 dy þ 1 dz ^ ðÞdy ^ dz 1 ∂x ∂y ∂z ∂v ¼ 1 dx ^ dy ^ dz ∂x 4.9 The Hodge Star Operator 175

Analogously, the second and third terms on the RHS in Eq. (4.43) result as

∂v dv ^ ðÞ¼dz ^ dx 2 dx ^ dy ^ dz; 2 ∂y ∂v dv ^ ðÞ¼dx ^ dy 3 dx ^ dy ^ dz: 3 ∂z

Therefore, the exterior derivative dηv results as ∂v ∂v ∂v dη ¼ 1 þ 2 þ 3 dx ^ dy ^ dz v ∂x ∂y ∂z

Using Eqs. (4.36a) and (4.41), the Hodge dual dηv is written as ∂v ∂v ∂v *dη ¼ 1 þ 2 þ 3 *dxðÞ^ dy ^ dz v ∂x ∂y ∂z ∂v ∂v ∂v ¼ 1 þ 2 þ 3 ¼ *dðÞ *ω ∂x ∂y ∂z v

Furthermore, the divergence of v is calculated as

∂v ∂v ∂v ∇ Á v ¼ 1 þ 2 þ 3 ∂x ∂y ∂z

Therefore, the relation between *dωv and div v results as

*dðÞ¼ *ωv ∇ Á v ð4:44Þ

Table 4.3 shows the overview of gradient, curl, and divergence operators that are displayed in vector calculus and differential forms.

4.9.2 Star Operator and Inner Product

Let ω be any k-form 2^k(RN)inanN-dimensional space RN. The Hodge * operator *ω is the (N À k)-form 2^NÀk(RN).

Table 4.3 Gradient, curl, and divergence in differential forms Operators Variable Grad Curl Divergence Vector calculus grad Laplacian f ! ∇f ! Δf v curl! ∇ Â v !div ∇ Á v Differential forms f !d df !*d Δf ¼ *dðÞ *df *d *d ωv ! *dωv ! *dðÞ *ωv 176 4 Differential Forms

If η is any k-form 2^k(RN), the inner product of two k-forms ω and η is deﬁned so that [6, 7]

η ^ *ω ¼ hiω; η dxN ÀÁ ¼ hiω; η dxi1 ^ ...^ dxiN 2 ΛN RN ð4:45Þ

The inner product of two orthonormal dual bases is written in Kronecker delta as & 0 fori 6¼ j dxi, dxj ¼ δ j ¼ ð4:46Þ i 1 fori ¼ j

Let dxI and dxJ be two k-forms of the dual bases. The inner product of dxI and dxJ is deﬁned as X DEDE mσ σ σ hidxI, dxJ ¼ ðÞÀ1 dxi1 , dx ðÞj1 dxi2 , dx ðÞj2 ÁÁÁ σ ð4:47Þ σ dxik , dx ðÞjk 2 R where mσ ¼ σ mod2; σ is the number of permutations on k elements. For any i 6¼ j in an N-dimensional space RN, one obtains using Eqs. (4.45) and (4.46)

dxi ^ *dxj ¼ hidxj, dxi dxN j δ i1 ... iN ¼ i ðÞdx ^ ^ dx ¼ 0

Some examples of elementary two- and three-forms are given here to demonstrate the inner product of the elementary k-forms. For elementary two-forms (k ¼ 2), there are two terms of the inner product; they result from k!. Using Eq. (4.47), the inner product is written as 0 dxi1 ^ dxi2 , dxj1 ^ dxj2 ¼ÀðÞ1 dxi1 , dxj1 dxi2 , dxj2 1 þÀðÞ1 dxi1 , dxj2 dxi2 , dxj1 ¼ dxi1 , dxj1 dxi2 , dxj2 À dxi1 , dxj2 dxi2 , dxj1 ð4:48Þ

The terms of the inner product of two elementary 2-forms result from Table 4.4. For elementary three-forms (k ¼ 3), there are six terms of the inner product; they result from k!. These terms with their signs are given in Table 4.5. The contravariant indices of any term of the elementary forms are displayed in each line, in which the same indices of i and j (¼1, 2, 3) could occur only two times. The number of permutations σ of the following terms results from the order of the ﬁrst term of the 4.9 The Hodge Star Operator 177

Table 4.4 Inner product of ijij2! ¼ 2 terms Permutation σ the elementary 2-forms < >< > σ 1 1 2 2 þ i1, j1 i2, j2 ¼ 0 ! mσ ¼ 0 < >< > σ 1 2 2 1 À i1, j2 i2, j1 ¼ 1 ! mσ ¼ 1

Table 4.5 Inner product of the elementary 3-forms ijijij3! ¼ 6 terms Permutation σ < >< >< > σ 1 1 2 2 3 3 þ i1, j1 i2, j2 i3, j3 ¼ 0 ! mσ ¼ 0 < >< >< > σ 1 2 2 1 3 3 À i1, j2 i2, j1 i3, j3 ¼ 1 ! mσ ¼ 1 < >< >< > σ 1 3 2 1 3 2 þ i1, j3 i2, j1 i3, j2 ¼ 2 ! mσ ¼ 0 < >< >< > σ 1 3 2 2 3 1 À i1, j3 i2, j2 i3, j1 ¼ 1 ! mσ ¼ 1 < >< >< > σ 1 2 2 3 3 1 þ i1, j2 i2, j3 i3, j1 ¼ 2 ! mσ ¼ 0 < >< >< > σ 1 1 2 3 3 2 À i1, j1 i2, j3 i3, j2 ¼ 1 ! mσ ¼ 1

Table 4.6 Summary of the operators using in differential forms Operators in RN Symbols Forms Results Orders Wedge product (exterior product) ^ k-form ω; p-form ηω^ η k + p

Interior product (Interior derivative) ix k-form ω ixω k À 1 Exterior derivative dk-form ω dω k +1

Pullback operator f* function f; k-form ω f * ω; ω0 fk Hodge star operator *k-form ω * ω N À k elementary forms. Namely, the second term of the elementary forms is given by one interchange of the indices 1 and 2; therefore, the number of permutation σ ¼ 1. Using Eq. (4.47), the inner product of two elementary 3-forms is written from Table 4.5 as

hidxi1 ^ dxi2 ^ dxi3 , dxj1 ^ dxj2 ^ dxj3 ¼ þhidxi1 , dxj1 hidxi2 , dxj2 hidxi3 , dxj3 Àhidxi1 , dxj2 hidxi2 , dxj1 hidxi3 , dxj3 þ hidxi1 , dxj3 hidxi2 , dxj1 hidxi3 , dxj2 ð4:49Þ Àhidxi1 , dxj3 hidxi2 , dxj2 hidxi3 , dxj1 þ hidxi1 , dxj2 hidxi2 , dxj3 hidxi3 , dxj1 Àhidxi1 , dxj1 hidxi2 , dxj3 hidxi3 , dxj2

Table 4.6 gives the overview of the discussed operators in differential forms of exterior algebra.

4.9.3 Star Operator in the Minkowski Spacetime

The Minkowski spacetime is applied to the special relativity, in which time t is an extra dimension besides the space coordinates of x, y, and z. As a result, the Minkowski is called the four-dimensional spacetime (t, x, y, z) 2 R4 (cf. Chap. 5). 178 4 Differential Forms

In the following section, the Hodge * operator will be used in the Minkowski spacetime [6, 7]. In gravitation, relativity theory, and cosmology, the light speed c is deﬁned as c2 1, in which c is considered as a constant. The distance ds between two arbitrary points in the Minkowski spacetime for special relativity is written as

ds2 ¼ g dxμdxv μv ð4:50Þ ¼ dt2 À dx2 À dy2 À dz2

The Minkowski metric with four spacetime coordinates (t, x, y, z) is expressed as 0 1 10 0 0 B 0 À10 0C g ¼ g ¼ B C ð4:51Þ μv @ 00À10A 00 0À1

The elementary form of the Minkowski spacetime is deﬁned as ÀÁ dxI ¼ dt ^ dx ^ dy ^ dz 2 Λ4 R4 ð4:52Þ

Note that the inner products of the orthonormal dual bases of the four-dimensional Minkowski spacetime result from Eqs. (4.50) and (4.51)

hidt; dt ¼þ1; hidx; dx ¼ hidy; dy ¼ hidz; dz ¼À1; ð4:53aÞ hidt, dxi ¼ hidxi, dxj ¼ 0 if i 6¼ j:

Using Eqs. (4.48) and (4.53a), the inner products of 2-forms of the dual bases result as

hi¼dt ^ dx, dt ^ dx hidt ^ dt hiÀdx ^ dx hidt ^ dx hidx ^ dx ¼þðÞÁÀ1 ðÞÀ1 ðÞÁ0 ðÞ¼À0 1 ) hidt ^ dy, dt ^ dy ¼ hidt ^ dz, dt ^ dz ¼À1 ð4:53bÞ hidx ^ dy, dx ^ dy ¼ hidx ^ dx hidy ^ dy À hidx ^ dy hidy ^ dx ¼ÀðÞÁÀ1 ðÞÀ1 ðÞÁ0 ðÞ¼þ0 1 ) hidx ^ dz, dx ^ dz ¼ hidy ^ dz, dy ^ dz ¼þ1

Using Eqs. (4.49), (4.53a) and (4.53b), the inner products of 3-forms of the dual bases result as

hi¼þdt ^ dx ^ dy, dt ^ dx ^ dy 1; hidt ^ dy ^ dz, dt ^ dy ^ dz ¼þ1; ð4:53cÞ hidt ^ dx ^ dz, dt ^ dx ^ dz ¼þ1; hidx ^ dy ^ dz, dx ^ dy ^ dz ¼À1: 4.9 The Hodge Star Operator 179

Using the inner product of two k-forms in Eq. (4.45), the Hodge dual *dt is calculated as

dt ^ *dt ¼ hidt; dt dxN ¼ hidt; dt dt ^ dx ^ dy ^ dz *dtðÞ ) dt ^ ¼ dt ^ dx ^ dy ^ dz hidt; dt ð4:54Þ *dtðÞ ) dt ^ ¼ dt ^ ðÞdx ^ dy ^ dz ðÞþ1

Therefore, the Hodge dual of the 1-form dt results as

*dt ¼ dx ^ dy ^ dz ð4:55Þ

Using Eq. (4.54), one obtains three Hodge duals of 1-forms of the dual bases.

*dx ¼ dt ^ dy ^ dz *dy ¼ dt ^ dz ^ dx ð4:56Þ *dz ¼ dt ^ dx ^ dy

Six Hodge duals of 2-forms of the dual bases are computed as [7, 8]

*dtðÞ¼À^ dx dy ^ dz; *dtðÞ¼À^ dy dz ^ dx; *dtðÞ¼À^ dz dx ^ dy; ð4:57Þ *dxðÞ¼À^ dy dz ^ dt; *dxðÞ¼À^ dz dt ^ dy; *dyðÞ¼À^ dz dx ^ dt:

Four Hodge duals of 3-forms of the dual bases are computed as [7, 8]

*dxðÞ¼^ dy ^ dz dt; *dtðÞ¼^ dx ^ dy dz; ð4:58Þ *dtðÞ¼^ dz ^ dx dy; *dtðÞ¼^ dy ^ dz dx:

According to Eq. (4.45), the Hodge dual of the 4-form of the dual bases is computed as

*dtðÞ¼^ dx ^ dy ^ dz 1 ð4:59Þ ) *1 ¼ dt ^ dx ^ dy ^ dz 180 4 Differential Forms

References

1. Yau, S.T., Nadis, S.: The Shape of Inner Space. Basic Books, London (2012) 2. Chern, S.S., Chen, W.H., Lam, K.S.: Lectures on Differential Geometry. World Scientiﬁc Publishing, Singapore (2000) 3. Darling, R.W.R.: Differential Forms and Connections. Cambridge University Press, Cambridge (2011) 4. Cartan, H.: Differential Forms. Dover Publications, London (2006) 5. Dray, T.: Differential Forms and the Geometry of General Relativity. CRC Press, Taylor & Francis Group, Boca Raton, FL (2015) 6. Renteln, P.: Manifolds, Tensors, and Forms. Cambridge University Press, Cambridge (2014) 7. Garrity, T.A.: Electricity and Magnetism for Mathematicians. Cambridge University Press, Cambridge (2015) 8. Bachman, D.: A Geometric Approach Differential Forms, 2nd edn. Birkha¨user, Basel (2012) Chapter 5 Applications of Tensors and Differential Geometry

5.1 Nabla Operator in Curvilinear Coordinates

Nabla operator is a linear map of an arbitrary tensor into an image tensor in N- dimensional curvilinear coordinates. The Nabla operator can be usually deﬁned in N-dimensional Cartesian coordinates {xi} using Einstein summation convention as

∂ ∇ ei for i ¼ 1, 2, ..., N ð5:1Þ ∂xi

According to Eq. (2.12), the relation between the bases of Cartesian and general curvilinear coordinates can be written as

∂r ∂r ∂xj ∂xj g ¼ ¼ ¼ e ð5:2Þ i ∂ui ∂xj ∂ui j ∂ui

Multiplying Eq. (5.2)bygie j, one obtains the basis of Cartesian coordinates expressed in the curvilinear coordinate basis.

∂xj ej ¼ gi ð5:3Þ ∂ui

Using chain rule of coordinate transformation, the Nabla operator in the general curvilinear coordinates {ui} results from Eq. (5.3)[1, 2].

© Springer-Verlag Berlin Heidelberg 2017 181 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5_5 182 5 Applications of Tensors and Differential Geometry ∂ ∂uj ∂xi ∂ ∂uj ∇ ei ¼ gk ∂uj ∂xi ∂uk ∂uj ∂xi ∂ ∂xi ∂uj ∂ ∂uj ¼ gk ¼ gk ∂uj ∂uk ∂xi ∂uj ∂uk ∂ ∂ ¼ gk δ j ¼ gk ð5:4Þ ∂uj k ∂uk

Thus, the Nabla operator can be written in the curvilinear coordinates {ui}using Einstein summation convention.

∂ ∇ gi ¼ gi∇ for i ¼ 1, 2, ..., N ð5:5Þ ∂ui i

5.2 Gradient, Divergence, and Curl

Let φ be a velocity potential that exists only in a vortex-free ﬂow. The velocity potential can be deﬁned as ð φ ¼ vdx ð5:6Þ

Differentiating Eq. (5.6) with respect to x, the velocity component results in

∂φ v ¼ ð5:7Þ ∂x

The velocity vector v can be written in the general curvilinear coordinates {ui}with the contravariant basis.

∂φ v ¼ v gi ¼ gi ð5:8Þ i ∂ui

5.2.1 Gradient of an Invariant

The gradient of an invariant φ (function, zero-order tensor) can be deﬁned by 5.2 Gradient, Divergence, and Curl 183

Table 5.1 Essential Nabla operators Operator Curl Laplacian Operand () Grad ∇ðÞ Div ∇ ÁðÞ ∇ ÂðÞ Δ () Function f 2 R (0th order Vector 2 RN – – Scalar 2 R tensor) Vector v 2 RN (1st order 2nd order tensor Scalar 2 R Vector Vector tensor) 2 RN Â RN 2 RN 2 RN Tensor T 2 RN Â RN (2nd 3rd order tensor Vector –– order tensor) 2 RN Â RN Â RN 2 RN Order of results One order higher One order Order Order lower unchanged unchanged

∂φ Grad φ ¼ ∇φ ¼ gi φ gi ¼ v gi ¼ v ∂ui ,i i ∂φ ∂φ ∂φ ð5:9Þ ) ∇φ ¼ v ¼ g1 þ g2 þ g3 ∂u1 ∂u2 ∂u3

Obviously, gradient of a function is a vector (cf. Table 5.1).

5.2.2 Gradient of a Vector

The gradient of a contravariant vector v can be calculated using the derivative of the covariant basis gj, as given in Eq. (2.158). ∂ j ∂ v gj Grad v ¼ ∇v ¼ gi v jg ¼ gi ∂ui j ∂ui i j j k jΓ k i ¼ g v,igj þ v gj,i ¼ v,igk þ v ijgk g k jΓ k i ¼ v, i þ v ij gkg k i : v igkg ð5 10Þ

Analogously, the gradient of a covariant vector v can be written using the derivative of the contravariant basis gj in Eq. (2.189). 184 5 Applications of Tensors and Differential Geometry ÀÁ j ∂ vjg Grad v ¼ ∇v ¼ gi ∂ i u i j j k Γ j k i ¼ g vj, ig þ vjg,i ¼ vk,ig À vj ikg g Γ j k i ¼ vk,i À vj ik g g

k i : vkjig g ð5 11Þ

Obviously, grad of a vector is a second-order tensor; grad of a second-order tensor is a third-order tensor (cf. Table 5.1).

5.2.3 Divergence of a Vector

Let v be a vector in the curvilinear coordinates {ui}; it can be written in the covariant basis gj.

j : v ¼ v gj ð5 12Þ

The divergence of v can be deﬁned by ∂ j ∂ v gj ∇ i i Div v ¼ Á v ¼ g i Á v ¼ g Á i : ∂u ∂u ð5 13Þ i j j i ∇ ¼ g Á v,igj þ v gj, i g Á iv

Using Eq. (2.158), the derivative of the covariant basis gj results in

Γ k Γ k : gj, i ¼ jigk ¼ ijgk ð5 14Þ

Substituting Eq. (5.14) into Eq. (5.13), one obtains the divergence of v. ∇ i j jΓ k Á v ¼ g Á v, igj þ v ijgk i k jΓ k i k jΓ k ¼ g Á v, igk þ v ijgk ¼ g Á v,i þ v ij gk ð5:15aÞ k jΓ k i k δ i ¼ v,i þ v ij g Á gk v ji k i jΓ i i ¼ v,i þ v ij ¼ v ji

According to Eq. (2.240), the second-kind Christoffel symbol can be rewritten as 5.2 Gradient, Divergence, and Curl 185

pffiffiffi ÀÁpffiffiffi 1 ∂ g ∂ ln g Γ i ¼ pffiffiffi ¼ ð5:15bÞ ij g ∂uj ∂uj

Substituting Eq. (5.15b) into Eq. (5.15a), the divergence of v can be expressed in 1 pffiffiffi ∂vi ∂J ∇ Á v ¼ pffiffiffi g þ vi g ∂ui ∂ui ÀÁpffiffiffi 1 ∂ gvi 1 ÀÁpffiffiffi ¼ pffiffiffi ¼ pffiffiffi gvi ð5:15cÞ g ∂ui g ,i

Analogously, the covariant vector v can be written in the contravariant basis gj.

j v ¼ vjg ð5:16Þ

The divergence of v can be derived using the contraction law and derivative of the basis g j in Eq. (2.189). ÀÁ ∂ v gj ∇ i j i j j Á v ¼ g Á ¼ g Á vj,ig þ vjg, i ∂ui i k j k ¼ g Á vk,ig À vjΓ g ik ð5:17Þ Γ j i k ¼ vk,i À vj ik g Á g i k ik m ik vkij g Á g ¼ vkij g ¼ v jigmkg m δ i i ¼ v ji m ¼ v ji

Obviously, divergence of a vector is a scalar (cf. Table 5.1). Some useful abbreviations are listed as follows: • Divergence of the contravariant vector v: ∇ i ∇ i ∇ j Á v g Á iv ¼ g Á i v gj i jΓ i i ¼ v, i þ v ij v ji ð5:18aÞ ÀÁpffiffiffi 1 ∂ gvi 1 ÀÁpffiffiffi ¼ pffiffiffi ¼ pffiffiffi gvi g ∂ui g , i

• Divergence of the covariant vector v: ÀÁ ∇ Á v ¼ gi Á ∇ v ¼ gi Á ∇ v gj i i j ð5:18bÞ Γ j ki ki ¼ vk, i À vj ik g ¼ vkij g 186 5 Applications of Tensors and Differential Geometry

• Covariant derivative of the contravariant vector component:

∂vk vkj v k þ v jΓ k ¼ þ v jΓ k ð5:19aÞ i , i ij ∂ui ij

• Covariant derivative of the covariant vector component: ∂v v j v À v Γ j ¼ k À v Γ j ð5:19bÞ ki k, i j ki ∂ui j ki

• Covariant derivative of the contravariant vector v with respect to ui: ∇ k jΓ k k : iv ¼ v,i þ v ij gk ¼ v jigk ð5 20aÞ

• Covariant derivative of the covariant vector v with respect to ui: ∇ Γ j k k m k : iv ¼ vk,i À vj ik g ¼ vkij g ¼ v jigmk g ð5 20bÞ

5.2.4 Divergence of a Second-Order Tensor

Let T be a contravariant tensor in the curvilinear coordinates {ui}; it can be written in the covariant bases gi and gj.

ij : T ¼ T gigj ð5 21Þ

The divergence of T can be calculated from ∂ ij ∂ T gigj ∇ Á T ¼ gk Á T ¼ gk Á ∂uk ∂uk ð5:22Þ k ij ij ij ¼ g Á T,kgigj þ T gi, kgj þ T gigj,k

Using Eq. (2.158), the derivative of the covariant basis gi results in

Γ m Γ m ; gi,k ¼ ik gm ¼ ki gm Γ n Γ n gj,k ¼ jkgn ¼ kjgn

Interchanging the indices, the divergence of a contravariant second-order tensor T becomes 5.2 Gradient, Divergence, and Curl 187 ∇ ij Γ i mj Γ j im k Á T ¼ T,kgigj þ kmT gigj þ kmT gigj Á g ð5:23Þ ij δ k Γ i mjδ k Γ j imδ k ¼ T, k i þ kmT i þ kmT i gj

Equation (5.23) can be written in the covariant basis gj at k ¼ i. ∇ ij Γ i mj Γ j im δ k Á T ¼ T þ T þ T gj , k km km i ij Γ i mj Γ j im ð5:24aÞ ¼ T, i þ imT þ imT gj ij T jigj

Using Eq. (2.240), the covariant derivative of the tensor component Tij with respect ui on the RHS of Eq. (5.24a) can be expressed in

ij ij Γ i mj Γ j im T ji ¼ T, i þ imT þ imT pffiffiffi ∂Tij 1 ∂ g ¼ þ Tmj pffiffiffi þ TimΓ j ∂ui g ∂um im pffiffiffi 1 pffiffiffi ∂Tij ∂ g ð5:24bÞ ¼ TimΓ j þ pffiffiffi g þ Tij im g ∂ui ∂ui ÀÁpffiffiffi 1 ∂ gTij 1 ÀÁpffiffiffi ¼ TikΓ j þ pffiffiffi ¼ TikΓ j þ pffiffiffi gTij ik g ∂ui ik g , i

Therefore, ÀÁffiffiffi ij ik j 1 p ij ∇ Á T ¼ T jig ¼ T Γ þ pffiffiffi gT g ð5:24cÞ j ik g , i j

Interchanging the indices, the divergence of a covariant second-order tensor T can be written as ÀÁ i j ∂ Tijg g ∇ Á T ¼ gk Á ∂uk i j i j i j k ¼ Tij,kg g þ Tijg,kg þ Tijg g Á g , k : i j Γ i m j Γ j i m k ð5 25Þ ¼ Tij,kg g À Tij kmg g À Tij kmg g Á g ÀÁ Γ m Γ m i j k ¼ Tij,k À Tmj ki À Tim kj g g Á g jk i ¼ Tijj kg g

Furthermore, the divergence of a mixed second-order tensor T results as the same way at k ¼ i. 188 5 Applications of Tensors and Differential Geometry ∂ i j Tj g gi ∇ Á T ¼ gk Á ∂uk i Γ i m Γ m i δ k j ¼ Tj, k þ kmTj À jk Tm i g ð5:26aÞ i Γ i m Γ m i j ¼ Tj, i þ imTj À ji Tm g i j Tj jig i kj ¼ Tj jig gk

Using Eq. (2.240), the covariant derivative of the mixed tensor component with respect to ui on the RHS of Eq. (5.26a) can be written in

i i Γ i m Γ m i Tj ji ¼ Tj,i þ imTj À ij Tm ffiffiffi! ∂ i p 1 pffiffiffi Tj ∂ g ¼ pffiffiffi g þ T i À Γ mT i g ∂ui j ∂ui ij m ð5:26bÞ 1 ffiffiffi ffiffiffi p i iΓ k ¼ p gTj À Tk ij g ,i

Therefore,

∇ Á T ¼ T ij gj ¼ T ij gkjg j i j i k 1 pffiffiffi 5:26c ffiffiffi i iΓ k kj ð Þ ¼ p gTj À Tk ij g gk g ,i

These results prove that the divergence of a second-order tensor T, such as the stress tensor Π or deformation tensor D results in a ﬁrst-order tensor which is a vector in the curvilinear coordinates {ui}. Obviously, divergence of a second-order tensor is a vector (cf. Table 5.1).

5.2.5 Curl of a Covariant Vector

Let v be a covariant vector in the curvilinear coordinates {ui}; it can be written in the contravariant basis gj.

j v ¼ vjg ð5:27Þ

The curl (rotation) of v can be deﬁned by 5.2 Gradient, Divergence, and Curl 189

Curl v Rot v ∇ Â v ÀÁ ∂ ÀÁ ∂ j i j i vjg ¼ g Â vjg ¼ g Â ∂ui ∂ui ð5:28Þ i j j ¼ g Â vj,ig þ vjg,i i j i j ¼ vj, iðÞþg Â g vj g Â g,i

Using Eq. (2.189), the derivative of the contravariant basis g j results in

j Γ j k : g,i ¼À ikg ð5 29Þ

Substituting Eq. (5.29) into Eq. (5.28), one obtains the curl of v. ÀÁ ∇ i j i j Â v ¼ vj,i g Â g þ vj g Â g,i ÀÁ ^ε ijk Γ j i k ð5:30Þ ¼ vj,igk À vj ik g Â g ^ε ijk ^ε ikm Γ j ¼ vj,igk À vj ikgm where the contravariant permutation symbols can be defined as (cf. Appendix A). 8 > 1 > þpffiffiffi if ðÞi, j, k is an even permutation < g ^ε ijk 1 : ¼ > ffiffiffi ð5 31Þ > Àp if ðÞi, j, k is an odd permutation :> g 0ifi ¼ j,ori ¼ k; or j ¼ k

However, the second term in RHS of Eq. (5.30) vanishes due to the symmetric Christoffel symbols with respect to the indices of i and k, and the anticyclic permutation property with respect to i, k, and m. ikm j vj j j ^ε vjΓ g ¼ pffiffiffi Γ À Γ g ik m g ik ki m ð5:32Þ vj j j ¼ pffiffiffi Γ À Γ g ¼ 0 g ik ik m

Therefore, the curl of v in Eq. (5.30) becomes

∂v ∇ Â v ¼ ^ε ijkv g ¼ ^ε ijk j g ð5:33Þ j, i k ∂ui k

Obviously, curl of a vector is a vector (cf. Table 5.1). 190 5 Applications of Tensors and Differential Geometry

5.3 Laplacian Operator

Laplacian operator is a linear map of an arbitrary tensor into an image tensor in N- dimensional curvilinear coordinates.

5.3.1 Laplacian of an Invariant

Laplacian of an invariant φ (function, zeroth-order tensor) is the divergence of grad φ. Using Eq. (5.9), this expression can be written in

DivðÞ Grad φ ∇ Á ∇φ ¼ ∇2φ Δφ ð5:34Þ

Substituting the gradient ∇φ of Eq. (5.9) into Eq. (5.34), one obtains the Laplacian Δφ. ÀÁ ÀÁ Δφ ∇ ∇φ ∇ φ k ∇ k Á ¼ Á , kg ¼ Á vkg ∂ ÀÁ ÀÁð5:35Þ ¼ gl Á φ gk ¼ gl Á φ gk þ φ g k ∂ul ,k ,kl , k ,l

Using Eq. (2.189), the derivative of the contravariant basis g j results in

k Γ k m : g, l ¼À lmg ð5 36Þ

Inserting Eq. (5.36) into Eq. (5.35) and using Eq. (5.19b), the Laplacian of φ can be computed as

Δφ ¼ ∇2φ ÀÁÀÁ ¼ gl Á φ gk þ φ g k ¼ φ gk À φ Γ k gm Á gl ÀÁ,kl , k ,l , kl ,k lm ¼ φ gk À φ Γ mgk Á gl ð5:37Þ ÀÁ, kl ,m lk ¼ φ À φ Γ m gk Á gl ÀÁ, kl ,m lk φ φ Γ m kl ¼ , kl À ,m kl g

The covariant vector components and their derivatives with respect to uk and ul are deﬁned as

∂φ ∂φ ∂2φ φ ¼ ¼ v ; φ ¼ ¼ v ; φ ¼ ¼ v , k ∂uk ÀÁk , m ∂um m ,kl ∂uk∂ul k,l ð5:38Þ Δφ Γ m kl kl ) ¼ vk,l À vm kl g vklj g

Obviously, Laplacian of a function is a scalar (cf. Table 5.1). 5.3 Laplacian Operator 191

5.3.2 Laplacian of a Contravariant Vector

Laplacian of a contravariant vector (ﬁrst-order tensor) is the divergence of grad v that can be computed as [3]

DivðÞ¼ Grad v Δv ∇ Á ∇v ¼ ∇2v k k Γ p p Γ k lm : ¼ v jl, m À v p lm þ v jl pm g gk ð5 39Þ k lm v jlmg gk

According to Eq. (5.39), Laplacian of a vector is a vector (cf. Table 5.1). The second covariant derivative of the contravariant vector component vk in Eq. (5.39) can be deﬁned as k k k Γ p p Γ k : v jlm v jl,m À v p lm þ v jl pm ð5 40Þ where ÀÁ k k k n Γ k nΓ k : v jl, m ¼ v jl v,lm þ v,m nl þ v nl,m ð5 41Þ , m k k nΓ k : v p v,p þ v np ð5 42Þ p p nΓ p : v jl v,l þ v nl ð5 43Þ

The vector triple product gives the relation of

a Â ðÞ¼b Â c baðÞÀÁ c ðÞa Á b c ð5:44Þ

Thus, Eq. (5.44) can be rewritten in the curl identity of the vector v is set into the position of the vector c, cf. Appendix C.

∇ Â ðÞ¼∇ Â v ∇∇ðÞÀÁ v ðÞ∇ Á ∇ v ð5:45Þ ¼ ∇∇ðÞÀÁ v ∇2v

The Laplacian of a vector v results from Eq. (5.44)as

Δv ∇2v ¼ ðÞ∇ Á ∇ v ¼ ∇ Á ðÞ∇v , ∇ Á ðÞ¼∇v ∇∇ðÞÀÁ v ∇ Â ðÞ∇ Â v ð5:46Þ , Laplacianv DivðÞ¼ Gradv GradðÞÀ Divv CurlðÞ Curlv 192 5 Applications of Tensors and Differential Geometry

5.4 Applying Nabla Operators in Spherical Coordinates

Spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 5.1 shows a point P in the spherical coordinates (ρ,φ,θ) embedded in orthonormal Cartesian coordinates (x1,x2,x3). However, the vector component changes as the spherical coordinates vary. The vector OP can be written in Cartesian coordinates (x1,x2,x3):

R ¼ ðÞρ sin φ cos θ e þ ðÞρ sin φ sin θ e þ ρ cos φ e 1 2 3 : 1 2 3 ð5 47Þ x e1 þ x e2 þ x e3 where e1, e2, and e3 are the orthonormal bases of Cartesian coordinates; φ is the equatorial angle; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u1, u2, and u3 can be used for ρ, φ, and θ, respectively. Therefore, the coordinates of the point P (u1,u2,u3) can be written in Cartesian coordinates: 8 9 ÀÁ< x1 ¼ ρ sin φ cos θ u1 sin u2 cos u3 = 1; 2; 3 2 ρ φ θ 1 2 3 : Pu u u ¼ : x ¼ sin sin u sin u cos u ; ð5 48Þ x3 ¼ ρ cos φ u1 cos u2

The covariant bases result from Chap. 2.

x3 (ρ,ϕ,θ) → (u1,u2,u3): ρ sinϕ u1 ≡ ρ ; u2 ≡ ϕ ; u3 ≡ θ

g P 3

e3 ϕ ρ e g2 0 2 x2 e1 θ ρ cosϕ

Fig. 5.1 Orthogonal spherical coordinates 5.4 Applying Nabla Operators in Spherical Coordinates 193 φ θ φ θ φ g1 ¼ ðÞsin cos e1 þ ðÞsin sin e2 þ cos e3 ) jjg1 ¼ gρ ¼1 ρ φ θ ρ φ θ ρ φ ρ g2 ¼ ðÞcos cos e1 þ ðÞcos sin e2 À ðÞsin e3 ) jjg2 ¼ gφ ¼ ρ φ θ ρ φ θ : ρ φ g3 ¼ÀðÞsin sin e1 þ ðÞsin cos e2 þ 0 e3 ) jj¼g3 jj¼gθ sin ð5:49aÞ

The covariant metric tensor M in the spherical coordinates can be computed from Eq. (5.49a). 2 3 2 3 g11 g12 g13 10 0 4 5 4 ρ2 5 : M ¼ g21 g22 g23 ¼ 0 0 ð5 49bÞ ρ φ 2 g31 g32 g33 00ðÞsin

Similarly, the contravariant bases result from Chap. 2. 1 φ θ φ θ φ 1 g ¼ ðÞsin cos e1 þ ðÞsinsin e2 þ cos e3 ) g ¼ 1 2 1 1 1 2 1 g ¼ cos φ cos θ e1 þ cos φ sin θ e2 À sin φ e3 ) g ¼ ρ ρ ρ ρ 1 sin θ 1 cos θ 1 g3 ¼À e þ e þ 0:e ) g3 ¼ ρ sin φ 1 ρ sin φ 2 3 ρ sin φ ð5:50aÞ

The contravariant metric coefﬁcients in the contravariant metric tensor MÀ1 can be calculated from Eq. (5.50a). 2 3 2 3 g11 g12 g13 10 0 MÀ1 ¼ 4 g21 g22 g23 5 ¼ 4 0 ρÀ2 0 5 ð5:50bÞ g31 g32 g33 00ðÞρ sin φ À2

5.4.1 Gradient of an Invariant

The gradient of an invariant Α 2 R can be written according to Eq. (5.9)in

∂A ∇A ¼ gi A gi ð5:51Þ ∂ui , i

Dividing the covariant basis by its vector length, the normalized covariant basis (covariant unitary basis) results in 194 5 Applications of Tensors and Differential Geometry

* gi gi gi : gi ¼ ¼ pffiffiffiffiffiffiffiffi ¼ ð5 52Þ jjgi gðÞii hi

The covariant basis in Eq. (5.52) in given in

* : gi ¼ higi ð5 53Þ where hi are the vector lengths, as given in Eq. (5.49a). pffiffiffiffiffiffi h1 ¼ g ¼ jjg gρ ¼ 1 pffiffiffiffiffiffi11 1 ρ 5:54 h2 ¼ ffiffiffiffiffiffig22 ¼ jjg2 gφ ¼ ð Þ p ρ φ h3 ¼ g33 ¼ jjg3 jjgθ ¼ sin

The contravariant bases can be transformed into the covariant bases in the orthogonal spherical contravariant basis, as given in Eq. (5.50b). 8 > g1 g11g g > ¼ 1 ¼ ρ <> 1 g2 ¼ g22g ¼ g gi ¼ gijg ) 2 ρ2 φ ð5:55aÞ j > > 3 33 1 : g ¼ g g3 ¼ gθ ðÞρ sin φ 2

Substituting Eqs. (5.53) and (5.54) into Eq. (5.55a), one obtains 8 > 1 * * > g ¼ gρ ¼ h1gρ ¼ gρ <> 2 1 1 * 1 * g ¼ gφ ¼ h2gφ ¼ gφ : > ρ2 ρ2 ρ ð5 55bÞ > ÀÁ > 3 1 1 * 1 * : g ¼ gθ ¼ h3gθ ¼ gθ ðÞρ sin φ 2 ðÞρ sin φ 2 ρ sin φ

Using Eqs. (5.51) and (5.55b), the gradient of A can be expressed in the physical vector components in the covariant unitary basis. ∂A A ∇A ¼ gi ¼ ,i g* ∂ui h i i ð5:56Þ ∂A 1 ∂A 1 ∂A ¼ g* þ g* þ g* ∂ρ ρ ρ ∂φ φ ρ sin φ ∂θ θ 5.4 Applying Nabla Operators in Spherical Coordinates 195

5.4.2 Divergence of a Vector

The divergence of v can be computed using the Christoffel symbols described in Eq. (5.15a). ∂ j ∂v v gj ∇ v gi gi Á ¼ i ¼ i : ∂u ∂u ð5 57Þ i jΓ i i ¼ v,i þ v ij v ji

At ﬁrst, the covariant derivatives of the contravariant vector components in Eq. (5.57) have to be computed.

1 1 Γ1 1 Γ1 2 Γ1 3 ; v j1 ¼ v,1 þ 11v þ 12v þ 13v for i ¼ 1 j ¼ 1, 2, 3 2 2 Γ2 1 Γ2 2 Γ2 3 ; : v j2 ¼ v,2 þ 21v þ 22v þ 23v for i ¼ 2 j ¼ 1, 2, 3 ð5 58Þ 3 3 Γ3 1 Γ3 2 Γ3 3 ; v j3 ¼ v,3 þ 31v þ 32v þ 33v for i ¼ 3 j ¼ 1, 2, 3

The second-kind Christoffel symbols in spherical coordinates can be calculated as [1]. 0 1 0 1 1 B 0 0 C 00 0 B ρ C Γ1 ¼ @ 0 Àρ 0 A; Γ2 ¼ B 1 C; ij ij @ 00A 00Àρ sin 2φ ρ 00 sin φ cos φ 0 1 À 5:59 1 ð Þ B 00 C B ρ C Γ3 B φ C ij ¼ @ 0 0 cot A 1 φ ρ cot 0

The physical vector components v*i result from the contravariant vector compo- * nents in the covariant unitary basis gi according to Eq. (B.11) in Appendix B. 8 1 > 1 *1 > v ¼ v vρ > h1 *i < i v 2 1 *2 1 v ¼ ) v ¼ v vφ ð5:60Þ h > h ρ i > 2 :> 3 1 *3 1 v ¼ v vθ h3 ρ sin φ

Using Eqs. (5.59) and (5.60), the covariant derivatives of the contravariant vector components can be computed as 196 5 Applications of Tensors and Differential Geometry

∂ 1 1 vρ v j1 ¼ v, 1 ¼ ∂ρ ∂ 2 2 2 1 2 1 1 1 vφ 1 Γ ρ 5:61 v j2 ¼ v, 2 þ 21v ¼ v,2 þ ρv ¼ ρ ∂φ þ ρv ð Þ ∂ 3 3 3 1 3 2 1 vθ 1 vφ v j ¼ v þ Γ v þ Γ v ¼ þ vρ þ cot φ 3 , 3 31 32 ρ sin φ ∂θ ρ ρ

Thus, the divergence of v results from Eq. (5.61)in

i 1 2 3 ∇ Á v ¼ v ji v j1 þ v j2 þ v j3 : ∂vρ 1 ∂vφ 1 ∂vθ 2vρ vφ ð5 62Þ ¼ þ þ þ þ cot φ ∂ρ ρ ∂φ ρ sin φ ∂θ ρ ρ

5.4.3 Curl of a Vector

The curl of a vector results from Eq. (5.33).

∇ ^ε ijk Â v ¼ vj, igk 1 ð5:63Þ ¼ ½ðÞv À v g þ ðÞv À v g þ ðÞv À v g J 3,2 2,3 1 1,3 3,1 2 2, 1 1, 2 3

The Jacobian of the spherical coordinates were calculated in Eq. (2.37) as

J ¼ ρ2 sin φ ð5:64Þ

Using Eqs. (B.19) and (5.49b), the covariant vector components can be computed in their physical vector components. 8 > *1 > v > v1 ¼ g11 ¼ vρ > h1 *j < *2 v v ρ : vi ¼ gij ) > v2 ¼ g22 ¼ vφ ð5 65Þ hj > h2 > *3 > v ρ φ : v3 ¼ g33 ¼ sin vθ h3

According to Eq. (5.53), the covariant bases can be written in the covariant unitary basis. 5.5 The Divergence Theorem 197 8 * * * < g1 ¼ h1g1 ¼ 1 Á g1 gρ * * ρ * ρ * : gi ¼ higi ) : g2 ¼ h2g2 ¼ g2 gφ ð5 66Þ * ρ φ * ρ φ * g3 ¼ h3g3 ¼ ðÞsin g3 ðÞsin gθ

Substituting Eqs. (5.64)–(5.66) into Eq. (5.63), the curl of v can be expressed in the * unitary covariant basis gi .

∇ 1 1 1 Â v ¼ ðÞv3, 2 À v2, 3 g1 þ ðÞv1,3 À v3,1 g2 þ ðÞv2,1 À v1,2 g3 J JÀÁ J ∂ðÞρ sin φ Á vθ ∂ ρ Á vφ 1 ¼ À g* ∂φ ∂θ ρ2 sin φ 1 ð5:67Þ ∂vρ ∂ðÞρ sin φ Á vθ 1 þ À g* ∂θ ∂ρ ρ sin φ 2 ÀÁ ∂ ρ ∂ Á vφ vρ 1 * þ ∂ρ À ∂φ ρg3

Computing the partial derivatives in Eq. (5.67), one obtains the curl of v in the unitary spherical coordinate bases. ∂ ∂ ∇ 1 vθ 1 vφ φvθ * Â v ¼ ρ ∂φ À ρ φ ∂θ þ cot ρ gρ sin ∂ ∂ 1 vρ vθ vθ * : þ ρ φ ∂θ À ∂ρ À ρ gφ ð5 68Þ sin ∂ ∂ vφ 1 vρ vφ * þ ∂ρ À ρ ∂φ þ ρ gθ

5.5 The Divergence Theorem

5.5.1 Gauss and Stokes Theorems

The divergence theorem, known as Gauss theorem deals with the relation between the flow of a vector or tensor field through the closed surface and the characteristics of the vector (tensor) in the volume closed by the surface. Gauss law states that the flux of a vector through any closed surface is proportional to the charge in the volume closed by the surface. This divergence theorem is a very useful tool that can be mostly applied to engineering and physics, such fluid dynamics and electrodynamics to derive the Navier-Stokes equations and Maxwell’s equations, respectively. 198 5 Applications of Tensors and Differential Geometry

Fig. 5.2 Fluid ﬂux through a closed surface S v n V ∈ R3

The Gauss theorem can be generally written in a three-dimensional space (see Fig. 5.2). þ ð v Á ndS ¼ ∇ Á v dV ð5:69Þ

S V where v is the fluid vector through the surface S; n is the normal vector on the surface; and ∇Áv is the divergence of the vector v. The outward fluid flux from the volume V causes the negative change rate of the volume mass with time. þ ð ∂ρ ρv Á ndS ¼À dV ð5:70Þ ∂t S V

Using Gauss divergence theorem, the balance of mass (also continuity equation) in the control volume V can be derived in þ ð ð ∂ρ ρv Á ndS ¼À dV ¼ ∇ Á ðÞρv dV ð5:71Þ ∂t S V V where ρ is the ﬂuid density. By rearranging the second and third terms in Eq. (5.71), the continuity equation can be written in the integral form for a control volume V: ð ∂ρ ∂ρ þ ∇ Á ðÞρv dV ¼ 0 , þ ∇ Á ðÞ¼ρv 0 ð5:72Þ ∂t ∂t V 5.5 The Divergence Theorem 199

Fig. 5.3 Fluid ﬂux through ∇× v an open surface So n v S dS o

dl (C)

Stokes theorem can be used for an open surface So, as shown in Fig. 5.3. The Stokes theorem indicates that the ﬂow velocity along the closed curve (C) is equal to the ﬂux of curl v going through the open surface So. þ ð v Á dl ¼ ðÞÁ∇ Â v ndS ð5:73Þ

ðÞC So where dl is the length differential on the curve closed (C) of the surface So; ∇Âv is the curl of the vector v.

5.5.2 Green’s Identities

The Green’s identities can be derived from the Gauss divergence theorem. Some- times, they can be usefully applied to the boundary element method (BEM) using the Green’s function [4–6]. Two Green’s identities are discussed in the following section.

5.5.3 First Green’s Identity

The vector v can be chosen as the product of two arbitrary scalars ψ and ϕ.

v ¼ ψ ∇ϕ ð5:74Þ

The divergence of v can be computed as follows: 200 5 Applications of Tensors and Differential Geometry

∇ Á v ¼ ∇ Á ðÞψ ∇ϕ ¼ ∇ψ Á ∇ϕ þ ψ ∇ Á ∇ϕ ð5:75Þ ¼ ∇ψ Á ∇ϕ þ ψ ∇2ϕ

Applying the Gauss divergence law of Eq. (5.69) to Eq. (5.75), one obtains ð þ ∇ Á v dV ¼ v Á ndS , ðV S þ þ : ÀÁ ∂ϕ ð5 76Þ ∇ψ Á ∇ϕ þ ψ ∇2ϕ dV ¼ ψðÞ∇ϕ Á n dS ¼ ψ dS ∂n V S S

5.5.4 Second Green’s Identity

The vector v can be chosen as the function of two arbitrary scalar products of ψ and ϕ.

v ¼ ψ ∇ϕ À ϕ ∇ψ ð5:77Þ

Thus, the divergence of v can be calculated as follows:

∇ Á v ¼ ∇ Á ðÞψ ∇ϕ À ϕ ∇ψ ¼ ∇ψ Á ∇ϕ þ ψ ∇2ϕ À ∇ϕ Á ∇ψ À ϕ ∇2ψ ð5:78Þ ¼ ψ ∇2ϕ À ϕ ∇2ψ

Applying the Gauss divergence law of Eq. (5.69) to Eq. (5.78), one obtains ð þ ∇ Á v dV ¼ v Á ndS , ðV S þ ÀÁ ψ ∇2ϕ ϕ ∇2ψ ψ ∇ϕ ϕ ∇ψ À dV ¼ ðÞÁÀ ndS ð5:79Þ V þ S ∂ϕ ∂ψ ¼ ψ À ϕ dS ∂n ∂n S where the gradients of the scalars in the normal direction can be deﬁned as

∂ϕ ∂ψ ∇ϕ Á n; ∇ψ Á n ð5:80Þ ∂n ∂n 5.5 The Divergence Theorem 201

5.5.5 Differentials of Area and Volume

In Gauss divergence theorem, the differentials of area dA and volume dV are changed from Cartesian coordinates to other general curvilinear coordinates by coordinate transformations.

5.5.6 Calculating the Differential of Area

Figure 5.4 shows the transformation of Cartesian coordinates {xi} into the curvilinear coordinate {ui}. The differential dr can be written in the covariant basis.

∂r dr ¼ dui ¼ g dui ð5:81Þ ∂ui i

Therefore, & 1 1 1 ðÞi ðÞi ðÞi dx ¼ e1dx ¼ g1du : dx ¼ eðÞi dx ¼ gðÞi du , 2 2 2 ð5 82Þ dx ¼ e2dx ¼ g2du

The area differential can be calculated in the curvilinear coordinates [7]. 1 2 1 2 1 2 dA ¼ dx Â dx ¼ jjðÞe1 Â e2 dx dx ¼ dx dx 5:83 1 2 ð Þ ¼ jjðÞg1 Â g2 du du

Using the Lagrange identity in Appendix E, one has the relation of

dA = gdu1du 2 u2 x2 g du 2 dA = dx1dx2 g2 1 u1 du1

1 0 e1 x

Fig. 5.4 Coordinate transformation in two-dimensional coordinates 202 5 Applications of Tensors and Differential Geometry qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 jjðÞg1 Â g2 ¼ g11g22 À ðÞg12 : pffiffiffi ð5 84Þ g where gij are the covariant metric coefficients, as defined in Eq. (2.43). Substituting Eq. (5.84) into Eq. (5.83), the area differential becomes

dA ¼ dx1dx2 ¼ jjðÞg Â g du1du2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 ð5:85Þ 2 1 2 ¼ g11g22 À ðÞg12 du du pffiffiffi gdu1du2

5.5.7 Calculating the Differential of Volume

The volume differential can be calculated in the curvilinear coordinates (Fig. 5.5). ÀÁ 1 2 3 1 2 3 dV ¼ dx Â dx Á dx ¼ jjðÞÁe1 Â e2 e3 dx dx dx 1 2 3 1 2 3 ¼ ½e1; e2; e3 dx dx dx ¼ dx dx dx ð5:86Þ 1 2 3 ; ; 1 2 3 ¼ jjðÞÁg1 Â g2 g3 du du du ¼ ½g1 g2 g3 du du du ¼ Jdu1du2du3

The scalar triple product of the covariant bases of the curvilinear coordinates can be deﬁned as

; ; : ðÞÁg1 Â g2 g3 ¼ ½g1 g2 g3 ð5 87Þ

Fig. 5.5 Coordinate u3 transformation in three- dimensional coordinates g 3 dV = J du1du 2du3 dV = dx1dx2dx3 x3 g2 u2 g1 e3 u1 e1 0 e2 x2 x1 5.5 The Divergence Theorem 203

The determinant of the covariant basis tensor equals the scalar triple product of the covariant bases, as given in Eq. (1.10). 1 1 1 ∂x ∂x ∂x ∂u1 ∂u2 ∂u3 ∂ 2 ∂ 2 ∂ 2 ; ; x x x > : ½¼g1 g2 g3 ¼ J 0 ð5 88Þ ∂u1 ∂u2 ∂u3 ∂ 3 ∂ 3 ∂ 3 x x x ∂u1 ∂u2 ∂u3 where J (¼Jacobian) is the determinant of the covariant basis tensor. Using Eqs. (5.15a), (5.85) and (5.86), the Gauss divergence theorem in general curvilinear coordinates {ui} 2 R3 can be written in the tensor integral equation: þ ð þ ffiffiffi i p 1 2 v Á ndS ¼ ∇ Á v dV , v ni gdu du ¼ ðS V S ð ð5:89Þ i jΓ i 1 2 3 i 1 2 3 v, i þ v ij Jdu du du v iJdu du du V V

The physical vector components v*i result from the contravariant vector compo- * nents in the normalized covariant basis (unitary basis) gi according to Eq. (B.11) in Appendix B.

v*i vi ¼ for i ¼ 1, 2, 3 ð5:90Þ hi

The scale factor hi can be defined as the covariant basis norm |gi| of the curvilinear coordinates {ui}. pffiffiffiffiffiffiffiffi : hi ¼ jjgi ¼ gðÞii ðÞðno summation over i 5 91Þ

Therefore, the divergence theorem in Eq. (5.89) can be rewritten in the curvilinear coordinate {ui} with the physical vector components as follows: þ ð ! *i ffiffiffi *i v p 1 2 v v*j ni gdu du , i Γ i 1 2 3 h ¼ þ ij Jdu du du i hi hj S ðV : ð5 92Þ 1 *i 1 2 3 v iJdu du du hi V

In the following sections, some applications of tensor analysis and differential geometry are applied to computational fluid dynamics (CFD), continuum mechanics, classical electrodynamics, electrodynamics in relativity fields, and the Einstein field theory as well. 204 5 Applications of Tensors and Differential Geometry

5.6 Governing Equations of Computational Fluid Dynamics

Navier-Stokes equations describe fluid flows in Computational Fluid Dynamics (CFD). In this book, Navier-Stokes equations are derived for compressible flows in a general rotating frame of the turbomachinery. The rotating frame rotates at an angular velocity ω(t) with respect to the inertial coordinate system [8, 9]. At ω ¼ 0, the Navier-Stokes equations can be used for a non-rotating frame (a special case). In this case, the velocity w in Eq. (5.94) is used in the equations instead of the absolute fluid velocity v.

5.6.1 Continuity Equation

The continuity equation satisﬁes the mass balance of the ﬂuid in the given control volume. According to Eq. (5.72), the continuity equation can be written as

∂ρ þ ∇ Á ðÞ¼ρv 0 ð5:93Þ ∂t

The absolute velocity v is the sum of the relative velocity w and the circumferential velocity u.

v ¼ w þ u ¼ w þ ðÞω Â r ð5:94Þ

Substituting the absolute velocity v in Eq. (5.94) into Eq. (5.93), one obtains the continuity equation of compressible ﬂuids in a rotating frame.

∂ρ þ ∇ Á ρðÞ¼w þ ðÞω Â r 0 , ∂t ∂ρ þ ∇ Á ðÞþρw ∇ Á ρðÞ¼ω Â r 0 , ð5:95Þ ∂t ∂ρ þ ∇ Á ðÞþρw ρ∇ Á ðÞþω Â r ðÞÁω Â r ∇ρ ¼ 0 ∂t

Using the Gauss divergence theorem, the volumetric ﬂow rate of the circumferential velocity u perpendicular to n over the control surface S equals zero (see Fig. 5.2). þ ð u Á ndA ¼ ∇ Á udV ¼ 0

S V

Thus, the third term in Eq. (5.95) is equal to zero for any control volume V 6¼ 0. 5.6 Governing Equations of Computational Fluid Dynamics 205

∇ Á u ¼ ∇ Á ðÞ¼ω Â r 0 ðÞq:e:d: ð5:96Þ

Using Eq. (5.96), the continuity equation (5.95) is written as ∂ρ ∂ ρ þ ðÞÁω Â r ∇ρ þ ∇ Á ðÞρw R þ ∇ Á ðÞ¼ρw 0 ð5:97Þ ∂t ∂t

The partial derivative with respect to the rotating frame can be deﬁned by [9].

∂ ρ ∂ρ R þ ðÞÁω Â r ∇ρ ð5:98Þ ∂t ∂t

The continuity equation (5.97) can be written in the tensor equation of

∂ ρ R þ ∇ Á ðÞ¼ρw 0 ∂t ∂ ρ ÀÁ , R þ ρwi j ¼ 0 ð5:99aÞ ∂t i ∂ ρ ÀÁ ÀÁ , R þ ρwi þ ρwj Γ i ¼ 0 ∂t , i ij

According to Eq. (5.18a), the continuity equation can be written in

∂ ρ ÀÁ R i þ ρw j¼i 0 ∂t ÀÁffiffiffi ∂ p ρ i ∂Rρ 1 g w , þ pffiffiffi ¼ 0 ð5:99bÞ ∂t g ∂ui ÀÁ ∂Rρ 1 pffiffiffi , þ pffiffiffi gρwi ¼ 0 ∂t g ,i

* The physical vector component in the normalized covariant basis gi (unitary basis) can be obtained from Eq. (B.11) in Appendix B.

*i *i i i w w ¼ hiw ) w ¼ ð5:100Þ hi where hi is the norm of the covariant basis gi. Therefore, the continuity tensor equation (5.99a) can be written in the physical velocity components of w* using Eqs. (2.244) and (5.100). 206 5 Applications of Tensors and Differential Geometry ∂ ρ w*i w*j R þ ρ þ ρ Γ i ¼ 0 ∂t h h ij i ,i j pffiffiffi ð5:101aÞ ∂ ρ w*i 1 w*j ∂ g , R þ ρ þ ffiffiffi ρ ¼ 0 ∂ p ∂ j t hi ,i g hj u and the continuity tensor equation (5.99b) becomes using Eq. (5.100) ÀÁpffiffiffi ∂ ρ 1 ∂ gρwi R þ ffiffiffi ¼ 0 ∂ p ∂ i t g u *i ∂Rρ 1 pffiffiffi w , þ pffiffiffi gρ ¼ 0 ð5:101bÞ ∂t g h i , ipffiffiffi ∂ ρ w*i 1 w*i ∂ g , R þ ρ þ ffiffiffi ρ ¼ 0 ∂ p ∂ i t hi ,i g hi u

5.6.2 Navier-Stokes Equations

The Navier-Stokes equations describe the balance of forces in a given control volume V. According to Newton’s second law, the balance of speciﬁc forces 3 fi (N/m ) for an arbitrary unit volume is written as

Dv Xn ρ ¼ fi ð5:102Þ Dt i¼1

The substantial derivative in the inertial coordinate system is deﬁned as

D ∂ þ v Á ∇ ð5:103Þ Dt ∂t where the first term is the time derivative; v is the absolute fluid velocity. Substituting the absolute velocity v in Eq. (5.94) into Eq. (5.103), one obtains the acceleration force acting upon a volumetric unit of the fluid in the inertial coordinate system.

∂ ∂ ω ρ Dv ρ v ρ ∇ ρ ðÞw þ Â r ρ ∇ ω ¼ ∂ þ v Á v ¼ ∂ þ v Á ðÞw þ Â r Dt t t ð5:104Þ ∂w ∂ðÞω Â r ¼ ρ þ ρ þ ρðÞÁw þ ω Â r ∇ðÞw þ ω Â r ∂t ∂t

The second term on the RHS of Eq. (5.104) can be calculated as 5.6 Governing Equations of Computational Fluid Dynamics 207 ∂ðÞω Â r ∂r ∂ω ρ ¼ ρ ω Â þ ρ Â r ð5:105Þ ∂t ∂t ∂t

The ﬁrst term on the RHS of Eq. (5.105) equals zero since the radius vector r does not vary with time. Thus, Eq. (5.105) is simply written as ∂ðÞω Â r ∂ω ρ ¼ ρ Â r ð5:106Þ ∂t ∂t

The third term on the RHS of Eq. (5.104) can be written as

ρv Á ∇v ¼ ρðÞÁw þ ω Â r ∇ðÞw þ ω Â r ¼ ρw Á ∇w þ ρw Á ∇ðÞþω Â r ρðÞÁω Â r ∇w ð5:107aÞ þ ρðÞÁω Â r ∇ðÞω Â r where the last three terms on the RHS of Eq. (5.107a) result from [8, 9]:

ρw Á ∇ðÞ¼ω Â r ρω Â w; ρðÞÁω Â r ∇w ¼ ω Â ρw ¼ ρω Â w; ð5:107bÞ ρðÞÁω Â r ∇ðÞ¼ω Â r ρω Â ðÞω Â r

Substituting Eqs. (5.106), (5.107a) and (5.107b) into Eq. (5.104), the acceleration force acting upon a volumetric unit of the ﬂuid results as Dv ∂w ∂ω ρ ¼ ρ þ ρw Á ∇w þ ρ Â r þ ½þρω Â ðÞω Â r 2ρðÞðω Â w 5:108Þ Dt ∂t ∂t

The acceleration force on the LHS of Eq. (5.108) consists of the following forces acting upon the volumetric unit of the fluid: the first term is the unsteady state term; the second term denotes the convective term; the third term is the circumferential force; the fourth term displays the centripetal force; and the last term is the Coriolis force. The external forces acting upon the fluid control volume comprise the pressure, fluid viscous, and gravity forces. As a result, the unsteady-state Navier-Stokes equations for a compressible viscous fluid in a rotating frame with an angular velocity ω(t) are written as ∂w ∂ω ρ þ ρw Á ∇w þ ρ Â r þ ½þρω Â ðÞω Â r 2ρðÞω Â w ∂t ∂t ð5:109Þ ¼À∇p þ ∇ Á Π À ∇ðÞρgz where Π is the viscous stress tensor of fluid that is written using the Stokes relation [8]. 208 5 Applications of Tensors and Differential Geometry

2 Π ¼ 2μ E À μ ∇ Á ðÞwI 3 ð5:110aÞ πij π i j ¼ gigj ¼ ijg g in which μ is the dynamic viscosity of ﬂuid; I is the unit tensor; Ε is the strain tensor of ﬂuid that is written in the strain components, cf. Eq. (5.176):

εij ε i j : E ¼ gigj ¼ ijg g ð5 110bÞ

The ﬁrst two terms on the LHS of Eq. (5.109) can be written in the tensor equation with the physical components as ÀÁ ∂w ∂ wkg ρ þ ρw Á ∇w ¼ ρ k þ ρ wjg Á ∇w ∂t ∂t j ∂ k ÀÁ ∂ k ρ w ρ j k i ρ w ρ j k δ i ¼ ∂ gk þ w gj Á w igkg ¼ ∂ gk þ w Á w i j gk t t ∂wk ∂wk ¼ ρ g þ ρwi Á wkj g ¼ ρ þ wi Á wkj g ð5:111aÞ ∂t k i k ∂t i k

Equation (5.111a) can be written in the unitary covariant basis with the physical velocity components of w* using Eq. (5.100). ∂ ∂ *k *i ρ w ρ ∇ ρ 1 w w *k * þ w Á w ¼ hk þ Áw i gk ∂t hk ∂t hihk ð5:111bÞ ∂ *k ρ w 1 *i *k * ¼ þ w Áw i gk ∂t hi where hi and hk are the norms of the covariant bases gi and gk, respectively. Assumed that the ﬂuid be a perfect gas. Using the state equation of gas, its density ρ is calculated from the pressure p and temperature T at any point P and time t as

pPðÞ; t ρ ¼ ð5:112Þ R Á TPðÞ; t where R is the gas constant. The third term on the LHS of Eq. (5.109) can be written in the tensor equation of the unitary covariant basis with the physical vector components and the contravariant permutation symbols (cf. Appendix A). 5.6 Governing Equations of Computational Fluid Dynamics 209 ∂ω ∂ω ÀÁ ∂ω ρ ρ i i j ^ε ijkρ i Â r ¼ rj g Â g ¼ rjgk ∂t ∂t ∂t : ∂ω ð5 113Þ ¼ ^ε ijkρh i r g* k ∂t j k

The fourth term on the LHS of Eq. (5.109) can be written in the tensor equation and the covariant permutation symbols (cf. Appendix A). ÀÁÀÁ ρω Â ðÞ¼ω Â r ρω gl Â ωir j^ε gk l ÀÁijk i j l k ¼ ρωlω r ^ε ijk g Â g ρω ωi j^ε ^ε lkm : ¼ l r ijk gm ð5 114Þ ρω ωi jδlkm ¼ l r ijk gm ρω ωi jδ lm ¼ l r ij gm

It can be written in the tensor equation of the unitary covariant basis with the physical vector components.

ρω ω ρω ωi jδ lm * Â ðÞ¼Â r hm l r ij gm ð5:115Þ ρω ωi jδlk * ¼ hk l r ij gk

The Coriolis term on the LHS of Eq. (5.109) can be written in the tensor equation of the unitary covariant basis with the physical vector components and the contravariant permutation symbols (cf. Appendix A). ÀÁ 2ρðÞ¼ω Â w 2ρω wm gi Â g i m ρω m i j ¼ 2 iw g Â gmjg : ^ε ijkρω m ð5 116Þ ¼ 2 igmjw gk ^ε ijkρhk ω *m * ¼ 2 igmjw gk hm

Using Eqs. (B.2) and (B.3), the pressure tensor on the RHS of Eq. (5.109) can be written in the unitary covariant basis with the physical vector components. ÀÁ ∇ j j * *j * *k *; À p ¼Àp, gj ¼À p, hÀÁj gj Àp, gj ¼Àp, gk : ∇ i ik * *k * ð5 117Þ À p ¼Àp, ig ¼À p, ig hk gk Àp, gk

* Using Eq. (B.3), the physical covariant stress tensor components πij in the unitary contravariant basis can be computed as 210 5 Applications of Tensors and Differential Geometry ÀÁÀÁ Π ¼ π gigj ¼ π gikg gjlg ÀÁij ij k l ¼ π gikgjlh h g*g* π* g*g* ð5:118Þ ÀÁij k l k l ij k l π* ik jl π ) ij ¼ g g hkhl ij

Using Eqs. (5.25), (5.118) and (B.2) and interchanging the indexlwith k, the friction covariant stress tensor on the RHS of Eq. (5.109) can be rewritten in the unitary contravariant basis with the physical tensor components.

∇ Á Π ¼ π j gjkgi ij k ÀÁ ÀÁ ¼ π j gjk glig ¼ π j gjkgli h g* ij k l ij k l l ð5:119Þ jk li jl ki g g π* * g g π* * ¼ ik jl ijjkgl ¼ il jk ijjlgk g g hk g g hl

The physical contravariant stress tensor components π*ij in the unitary covariant basis can be computed using Eq. (B.2). ÀÁ Π π ij π ij * * ¼ gigj ¼ higi hjgj πij * * π*ij * * ð5:120Þ ¼ hihjgi gj gi gj *ij ij ) π ¼ hihjπ

Using Eqs. (5.24a), (5.120) and (B.2), the friction contravariant stress tensor on the RHS of Eq. (5.109) can be written in the unitary covariant basis with the physical tensor components.

∇ Π πij π ij * Á ¼ jigj ¼ ji hjgj ð5:121Þ 1π*ij * 1π*ik * ¼ ji gj ¼ ji gk hi hi

Using Eqs. (B.2) and (B.3), the gravity tensor on the RHS of Eq. (5.109) can be written in the unitary covariant basis with the physical vector components. ÀÁ ∇ ρ ρ j ρ j * ρ *k *; À ðÞ¼Àgz gz, gj ¼À gzÀÁ, hj gj À gz, gk : ∇ ρ ρ i ρ ik * ρ *k * ð5 122Þ À ðÞ¼Àgz gz, ig ¼À gz, ig hk gk À gz, gk where g (¼9.81 m/s2) is the earth gravity.

5.6.3 Energy (Rothalpy) Equation

The energy equation describes the balance of energies in a control volume. They are based on the first law of thermodynamics for an open system. The specific rothalpy I for a unit volume is defined as 5.6 Governing Equations of Computational Fluid Dynamics 211

1 2 I h þ v À uvu 2 ð5:123Þ 1 ¼ c T þ v2 À uv p 2 u where h is the speciﬁc enthalpy of ﬂuid. Using the trigonometric calculation with the velocity triangle, the circumferential absolute velocity results as (Fig. 5.6)

vu ¼ u þ wu ð5:124Þ Similarly, the absolute velocity can be written using Eq. (5.124).

v ¼ vu þ vm ¼ ðÞþu þ w v u m : 2 2 2 ð5 125Þ ) v ¼ ðÞu þ wu þ vm 2 2 2 ¼ u þ wu þ 2uwu þ vm

Substituting Eqs. (5.124) and (5.125) into Eq. (5.123), one obtains the speciﬁc rothalpy of the turbomachinery.

1 I h þ v2 À uv 2 u 1ÀÁ ¼ h þ u2 þ w2 þ 2uw þ v2 À uuðÞþ w 2 u u m u 1ÀÁ1 ¼ h þ w2 þ v2 À u2 ð5:126Þ 2 u m 2 1 1 ¼ h þ w2 À u2 2 2 1 1 ¼ h þ w Á w À ðÞω Â r 2 2 2

Fig. 5.6 Triangle of velocities in an axial turbomachinery 212 5 Applications of Tensors and Differential Geometry

Therefore, the speciﬁc rothalpy I is written as

1 1 I ¼ c T þ w Á w À ðÞω Â r 2 ð5:127Þ p 2 2

Using the first law of thermodynamics for an open system in a rotating coordinate system, the energy equation can be written as [8]. DI ∂I ρ ¼ ρ þ ρw Á ∇I Dt ∂t rot ð5:128Þ ∂p ¼ þ ∇ Á q_ þ ½w Á ðÞ∇ Á Π þ ðÞΠ Á E ∂t in which the first term on the RHS of Eq. (5.128) denotes the specific pressure power, the second term is the specific heat transfer power, and both last terms are the induced specific viscous power of the stress and strain tensors Π and E of fluid acting upon the control volume. In the following section, the relating terms in the energy equation (5.128) are calculated. At first, the substantial time derivative of the specific rothalpy I with respect to the rotating frame can be written in the tensor equation. ∂ ρ DI ρ I ρ ∇ ∂ þ w Á I Dt rot t ∂I ∂I : ¼ ρ þ ρwjI g Á gi ¼ ρ þ ρwjI δ i ð5 129aÞ ∂t ,i j ∂t ,i j ∂I ¼ ρ þ ρwiI ∂t ,i

Using Eq. (5.100), the tensor equation (5.129a) can be written in the velocity physical component w*i.

∂I ∂I w*i ρ þ ρw Á ∇I ¼ ρ þ ρ I, i ð5:129bÞ ∂t ∂t hi in which the gas density ρ is calculated from Eq. (5.112). The time change rate of the rothalpy results from Eq. (5.127)as

∂I ∂T ∂w ∂ðÞω Â r ¼ c þ w Á À ðÞÁω Â r ð5:130Þ ∂t p ∂t ∂t ∂t

The heat transfer speciﬁc power due to heat conduction through the control volume can be written using the Laplacian of ﬂuid temperature T, as given in Eq. (5.37). 5.7 Basic Equations of Continuum Mechanics 213

q_ ¼ λ∇T ∇ _ ∇ λ∇ λ∇2 ) Á q ¼ ÁðÞ¼T T ð5:131Þ λ ij Γ k ¼ g T, ij À T,k ij where λ is the constant thermal conductivity. The ﬂuid viscous power on the control volume surface can be written in the ij tensor equation with the stress and strain tensor components of π and εij. ÀÁ ∇ Π Π πij πijε kπij πijε : ½¼w Á ðÞþÁ ðÞÁ E wj ji þ ij ¼ gjkw ji þ ij ð5 132aÞ

Using Eqs. (5.100) and (5.118), Eq. (5.132a) can be expressed in the physical vector components w*. ! π*ijε* ∇ Π Π 1 *kπ*ij ij : : ½¼w Á ðÞþÁ ðÞÁ E gjkw i þ ik jl ð5 132bÞ hihjhk g g hl

5.7 Basic Equations of Continuum Mechanics

Continuum mechanics deals with the mechanical behavior of continuous materials on the macroscopic scale. The basic equations of continuum mechanics consist of two kinds of equations. First, the equations of conservation of mass, force, and energy can be applied to all materials in any coordinate system. Such equations are the Cauchy’s laws of motion, which concern with the kinematics of a continuum medium (solids and ﬂuids). Second, the constitutive equations describe the macroscopic responses resulting from the internal characteristics of an individual material. The additional constitutive equations help the Cauchy’s equations of continuum mechanics in order to predict the responses of the individual material to the applied loads. Physical laws must always be invariant and independent of the observers from different coordinate systems. Therefore, such equations describing physical laws are generally written in the tensor equations that are always valid for any coordinate system.

5.7.1 Cauchy’s Law of Motion

The Cauchy’s law of motion is based on the conservation of forces in a continuum medium. Figure 5.7 displays the balance of forces acting on the moving body in a general curvilinear coordinate system {u1,u2,u3}. The Cauchy’s stress tensor T can be written in the contravariant second-order tensor as 214 5 Applications of Tensors and Differential Geometry

Fig. 5.7 Acting forces on a : stress tensor u3 T moving body n V g3 dS : inertial finert 0 g2 dV force S

g1 1 u f : body force u2

ij : T ¼ T gigj ð5 133Þ

The body force f per volume unit can be expressed in the contravariant ﬁrst-order tensor (vector).

j : f ¼ f gj ð5 134Þ

The inertial force acting upon the body can be written in the contravariant ﬁrst- order tensor (vector).

ρ j ρ€ j : finert ¼À a gj ¼À u gj ð5 135Þ where ρ is the body density. Using the D’Alembert principle, one obtains the Cauchy’s law of motion in the integral form. X þ ð ð F ¼ T Á ndS þ fdV þ finertdV ¼ 0 ð5:136Þ S V V

Applying the Gauss divergence theorem to Eq. (5.136), the ﬁrst integral over surface S can be transformed into the integral over volume V. þ ð ð ∇ ij : T Á ndS ¼ Á TdV ¼ T igjdV ð5 137Þ S V V where the divergence ∇ÁT of a second-order tensor T results from Eq. (5.24a)

∇ Á T ¼ Tijj g i j ð5:138Þ ij Γ i mj Γ j im ¼ T,i þ imT þ imT gj

Applying the coordinate transformation in Eq. (5.89), Eq. (5.137) can be written in the curvilinear coordinates using the Jacobian J. 5.7 Basic Equations of Continuum Mechanics 215 ð ð ∇ ij 1 2 3 : Á TdV ¼ T jigj Jdu du du ð5 139Þ V V

Thus, the conservation of forces can be rewritten in the curvilinear coordinates. ð À Á ij j ρ€ j 1 2 3 : T jþi f À u gj Jdu du du ¼ 0 ð5 140Þ V

The Cauchy’s tensor equation for an arbitrary volume V becomes

ij j j T ji þ f ¼ ρu€ ð5:141Þ

In case of equilibrium, the Cauchy’s tensor equation becomes

ij j T ji þ f ¼ 0 ð5:142Þ

The Cauchy stress tensor T contains nine tensor components:

Tij ¼ τij for i, j ¼ 1, 2, 3 ð5:143Þ

Using Eq. (5.143), one obtains the Cauchy’s tensor equations for both cases:

τij j ρ€ j; ji þ f ¼ u : ij j ð5 144Þ τ ji þ f ¼ 0 within using the Christoffel symbols of second kind, the covariant derivative of the stress tensor components with respect to ui is deﬁned as

τij τij Γ i τmj Γ j τim : ji ¼ ,i þ im þ im ð5 145Þ

The Cauchy’s stress tensor T can be written in a three-dimensional space as 2 3 0 1 T1 σ11 τ12 τ13 ÀÁ 4 5 @ A ij T ¼ T2 ¼ τ21 σ22 τ23 τ ð5:146Þ T3 τ31 τ32 σ33 where σ is the normal stress; τ is the shear stress. The stress tensor Ti on the surface S is deﬁned as the acting force F per unit of surface area.

ΔF ∂F Ti ¼ lim ¼ ð5:147Þ ΔS!0 ΔS ∂S 216 5 Applications of Tensors and Differential Geometry

33 Fig. 5.8 Stress tensor σ 3 components Ti acting upon T dV a volume element 32 31 τ τ τ 23 T2 13 3 τ σ 22 u T1 21 τ 12 τ σ 11

0 u 2

At each point in the surface S, there are a set of three stress tensor components, one normal stress component σ (pressure) perpendicular to the surface and two shear stress components τ parallel to the surface, as shown in Fig. 5.8. Note that the tensile normal stress denotes a positive normal stress; the compressive stress, a negative normal stress. In the following section, it is to prove that the second-order Cauchy’s stress tensor is symmetric in a free couple-stress body of non-polar materials. Note that polar materials contain couple stresses. The conservation of angular momentum of the body in equilibrium (cf. Fig. 5.7) can be written as X þ ð M ¼ ðÞr Â T dS þ ðÞr Â f dV ¼ 0 ð5:148Þ S V

The vector r can be written in the covariant basis of curvilinear coordinates:

j : r ¼ x gj ð5 149aÞ

The Cauchy’s stress tensor T can be written in the covariant basis of curvilinear coordinates:

mk k : T ¼ T gmgk T gk ð5 149bÞ

Using Eqs. (5.149a) and (5.149b) and the Levi-Civita symbols in Eq. (A.6) in Appendix A, the angular momentum equation can be expressed as 5.7 Basic Equations of Continuum Mechanics 217 þ ð j k i j k i ^ε ijkx T g dS þ ^ε ijkx f g dV ¼ 0 , þS V ð ÀÁ ð5:150Þ ^ε j mk i ^ε j k i ijkx T gm g dS þ ijkx f g dV ¼ 0 S V

Applying the Gauss divergence theorem and using the transformation of curvilinear coordinates, Eq. (5.150) results in ð ÀÁ ^ε j mk j k i ijk x T m þ x f g dV ¼ 0 ) ðV ÀÁ ð5:151Þ ^ε j mk j k i 1 2 3 ijk x T m þ x f g Jdu du du ¼ 0 V

Calculating the covariant derivative of the ﬁrst term in Eq. (5.151), one obtains ð j mk j mk j k i 1 2 3 ^ε ijk x T þ x T þ x f g Jdu du du ¼ 0 ð5:152Þ m m V

Rearranging the second and third terms in Eq. (5.152), one obtains ð ð ^ε j mk i 1 2 3 ^ε j mk k i 1 2 3 : ijkx mT g Jdu du du þ ijkx T m þ f g Jdu du du ¼ 0 ð5 153Þ V V

Due to the balance of forces in Eq. (5.142), the second integral in Eq. (5.153) equals zero. Therefore, the tensor equation of angular momentum Eq. (5.153) results in ð ^ε j mk i 1 2 3 ijkx jmT g Jdu du du V ð ð5:154Þ ^ε j mk i ¼ ijkx jmT g dV ¼ 0 V

j δ j Using the relation of x jm ¼ m, one obtains for an arbitrary volume dV

^ε j mk ^ε δ j mk ^ε jk ijkx jmT ¼ ijk mT ¼ ijkT ¼ 0 : jk ð5 155Þ ) εijkτ ¼ 0

Note that εikj ¼Àεijk, one obtains from Eq. (5.155) 218 5 Applications of Tensors and Differential Geometry

jk kj εijkτÀÁ¼ εikjτ ¼ 0 ÀÁ jk kj jk kj ) εijkτ þ εikjτ ¼ εijk τ À τ ¼ 0 ð5:156Þ ) τjk ¼ τkj for i 6¼ j 6¼ k; j, k ¼ 1, 2, 3

This result proves that the Cauchy’s stress tensor is symmetric in the free couple- stress body. In a three-dimensional space, there are six symmetric tensor components of shear stress τ ij and three diagonal tensor components of normal stress σ ii, as shown in Eq. (5.146). It is obvious, the Cauchy’s stress tensor becomes non- symmetric if the couple stresses act on the body.

5.7.2 Principal Stresses of Cauchy’s Stress Tensor

The Cauchy’s stress tensor T in a three-dimensional space generally has three invariants that are independent of any chosen coordinate system. These invariants are the normal stresses in the principal directions perpendicular to the principal planes. In fact, the principal normal stresses are the eigenvalues of the stress tensor where only the normal stresses act on the principal planes in which the remaining shear stresses equal zero. The eigenvectors related to their eigenvalues have the same directions of the principal directions, as shown in Fig. 5.9. i The principal stress vector T of the stress tensor on the normal unit vector ni (parallel to the principal direction) can be expressed as

i T ¼ λ ni ð5:157Þ

Furthermore, the principal stress vector can be written in a linear form of the stress tensor components and their relating normal unit vectors.

Fig. 5.9 Acting principal i ni stress T on a principal stress tensor T principal direction plane i = λ T ni

principal plane

5.7 Basic Equations of Continuum Mechanics 219

i ij T ¼ σ nj for j ¼ 1, 2, 3 ð5:158Þ where using the Kronecker delta, the normal unit vector nj is deﬁned as

δ j : ni ¼ nj i for i, j ¼ 1, 2, 3 ð5 159Þ

The relation between the unit vectors in Eq. (5.159) denotes that all shear stresses vanish in the principal planes perpendicular to the unit vectors nj with all indices j 6¼ i. In case of j i, only the principal stresses act on the principal planes. Substituting Eqs. (5.158) and (5.159) into Eq. (5.157), one obtains σij λδ j : À i nj ¼ 0 for i, j ¼ 1, 2, 3 ð5 160Þ

For nontrivial solutions of Eq. (5.160), its determinant must equal zero. Therefore, ðÞσ11 À λ σ12 σ13 j det σij À λδ ¼ σ21 ðÞσ22 À λ σ23 ¼ 0 ð5:161Þ i σ31 σ32 ðÞσ33 À λ

This equation is called the characteristic equation of the eigenvalues of the second- order stress tensor T. Calculating the determinant of Eq. (5.161), the third-order characteristic equation of λ results in

3 2 Àλ þ I1λ À I2λ þ I3 ¼ 0 ð5:162Þ within 8 σii ; <> I1 ¼ TrðÞ 1ÀÁ I ¼ σiiσ jj À σijσ ji ; ð5:163Þ :> 2 2 ij I3 ¼ detðÞσ

Generally, there are three real roots of Eq. (5.162) for the eigenvalues in the principal directions. 8 1 max < λ1 ¼ σ σ λ σ1 σ3 σ2 : : 2 ¼ I1 À À ¼ ð5 164Þ 3 min λ3 ¼ σ σ

The eigenvalues of Eq. (5.164) are the roots of the Cayley-Hamilton theorem. The stress tensor T can be written in the principal directions as follows: 220 5 Applications of Tensors and Differential Geometry 0 1 ÀÁ σ1 00 T σij ¼ @ 0 σ2 0 A ð5:165Þ 00σ3

The maximum shear stress occurs at the angle of 45 between the smallest and largest principal stress planes with the value σmax À σmin σ1 À σ3 σ1 þ σ3 τmax ¼ ¼ at σmiddle ¼ : ð5:166Þ 2 2 2

5.7.3 Cauchy’s Strain Tensor

The Cauchy’s strain tensor describes the inﬁnitesimal deformation of a solid body in which the displacement between two arbitrary points in the material is much less than any relevant dimension of the body. In this case, the Cauchy’s strain tensor is based on the small deformation theory or linear deformation theory [18]. The vector R(u1,u2,u3,t) of an arbitrary point P of the body at the time t after 1 2 3 deformation results from the vector r(u ,u ,u ) of the same point at the time t0 before deformation and the small deformation vector v(u1,u2,u3,t), as shown in Fig. 5.10. ÀÁÀÁÀÁ R u1; u2; u3; t ¼ r u1; u2; u3 þ v u1; u2; u3; t ð5:167Þ

The covariant basis vectors of both curvilinear coordinates at the times t0 and t result from

∂R ∂r ∂v ¼ þ , G ¼ g þ v ð5:168Þ ∂ui ∂ui ∂ui i i ,i

The difference of two segments can be calculated by

Fig. 5.10 Inﬁnitesimal at time t deformation of a solid body at time t 3 0 G3 u

g3 P v G2 P 1 0 u 2 G1 u g2 g1 R r 0 5.7 Basic Equations of Continuum Mechanics 221

2 2 i j i j dR À dr ¼ GiGjdu du À gigjdu du 5:169 i j γ i j ð Þ ¼ Gij À gij du du 2 ijdu du

where γij are the components of the second-order strain tensor. Calculating the covariant metric coefﬁcients, one obtains G Á G ¼ ðÞÁg þ v g þ v ¼ g Á g þ g Á v þ g Á v þ v Á v i j i ,i j , j i j i ,j j ,i , i ,j ð5:170Þ ) Gij À gij ¼ gi Á v,j þ gj Á v,i þ v, i Á v,j

Thus, the stress tensor components can be calculated as γ 1 1 ij ¼ Gij À gij ¼ gi Á v, j þ gj Á v, i þ v,i Á v,j 2 2 1 ∂v ∂v ∂v ∂v ¼ g Á þ g Á þ Á ð5:171Þ 2 i ∂uj j ∂ui ∂ui ∂uj

The deformation vector v can be written in the contravariant and covariant bases.

k k : v ¼ vkg ¼ v gk ð5 172Þ

Using Eqs. (2.200) and (2.207), one obtains the covariant partial derivatives

k; v, i ¼ vkji g k ð5:173Þ v, j ¼ v j gk within the covariant derivatives result from Eqs. (2.201) and (2.208) in

Γ j ; vkji ¼ vk,i À kivj : k k Γ k i ð5 174Þ v j ¼ v,j þ jiv

Substituting Eq. (5.173) into Eq. (5.171), the strain tensor components of the Cauchy’s strain tensor γ can be rewritten as 1 γ ¼ g Á v þ g Á v þ v Á v ij 2 i , j j , i ,i , j 1 ¼ v j gk Á g þ v j gk Á g þ v j vk g Á gl 2 k j i k i j l i j k ð5:175Þ 1 ¼ v j δ k þ v j δ k þ v j vk δ l 2 k j i k i j l i j k 1 ¼ v j þ v þ v j Á vk 2 i j j i k i j 222 5 Applications of Tensors and Differential Geometry

The third term on the RHS of Eq. (5.175) is very small in the inﬁnitesimal deformation. Therefore, the components of the Cauchy’s strain tensor γ become 1 γ v j þ v ð5:176Þ ij 2 i j j i

Substituting Eq. (5.174) into Eq. (5.176) and using the symmetry of the Christoffel symbols, the Cauchy’s tensor component can be written in the linear elasticity.

1ÀÁ γ ¼ v þ v À Γ kv ij 2 i,j j, i ij k ÀÁ 1 5:177 ¼ v þ v À Γ kv ð Þ 2 j,i i, j ji k γ : : : ¼ ji ðÞq e d

This result proves that the Cauchy’s strain tensor γ is also symmetric. As an example, the Cauchy’s strain tensor γ can be written in a three- dimensional space: 0 1 ε γ γ 11 12 13 ÀÁ γ @ γ ε γ A γ : ¼ 21 22 21 ij ð5 178Þ γ γ ε 31 32 33 where εii are the normal strains; γij the shear strains of the second-order strain tensor (matrix). Note that the Christoffel symbols in Eq. (5.177) vanish in Cartesian coordinates (x,y,z) in a three-dimensional space E3. Therefore, the normal and shear strains of Eq. (5.178) can be computed as follows: 9 ∂ ∂ ∂ ε u ; ε v ; ε w > xx ¼ yy ¼ zz ¼ > ∂x ∂y ∂z > ∂ ∂ > 0 1 γ 1 u v γ > ε γ γ xy ¼ þ ¼ yx = xx xy xz 2∂y ∂x @ A ) γ ¼ γ εyy γ ð5:179Þ 1 ∂v ∂w > yx yz γ ¼ þ ¼ γ > γ γ εzz yz 2 ∂z ∂y zy > zx zy > 1 ∂u ∂w > γ ¼ þ ¼ γ ; xz 2 ∂z ∂x zx where u, v, and w are the vector components of the deformation vector v. Similar to the principal stresses, the principal strains are invariants that are independent of any chosen coordinate system. In this case, only the principal strains (eigenvalues) exist in the principal directions (eigenvectors) of the principal planes in which the shear strains equal zero. The characteristic equation of the eigenvalues λ of the strain tensor γ results from 5.7 Basic Equations of Continuum Mechanics 223 γ λδ j ij À i nj ¼ 0 for i, j ¼ 1, 2, 3 ð5:180Þ γ λδ j ) det ij À i ¼ 0

There are three real roots of Eq. (5.180) for the principal strains (eigenvalues) in the principal directions (eigenvectors).

λ1 ¼ ε1; λ2 ¼ ε2; λ3 ¼ ε3 ð5:181Þ

The Cauchy’s strain tensor γ can be written in the principal directions: 0 1 ε1 00 ÀÁ @ A γ ¼ 0 ε2 0 εij ð5:182Þ 00ε3

5.7.4 Constitutive Equations of Elasticity Laws

The constitutive equations describe the macroscopic responses resulting from the internal characteristics of an individual material. The linear elasticity law of materials shows the relation between the stress tensor T and strain tensor γ in general curvilinear coordinates of an N-dimensional space. In general, the Hooke’s law is valid in the linear elasticity of material. As a result, the stress tensor component is proportional to the strain tensor component in the elasticity range of an individual material compared to the spring force that is proportional to its displacement by a spring constant. The Hooke’s law of the linear elasticity law is written as

τij ijklγ ... : ¼ E kl for i, j, k, l ¼ 1, 2, , N ð5 183Þ where Eijkl is the fourth-order elasticity tensor that only depends on the material characteristics. The elasticity tensor has 81 components (¼34) in a three- dimensional space; N4 components in an N-dimensional space. The elasticity tensor is a function of the elasticity modulus E (Young’s modulus) and the Poisson’s ratio v [10]. 2v Eijkl ¼ μ gikgjl þ gilgjk þ gijgkl ð5:184Þ ðÞ1 À 2v 224 5 Applications of Tensors and Differential Geometry where μ is the shear modulus (modulus of rigidity) that can be deﬁned as

E μ ¼ ð5:185Þ 21ðÞþ v

Some moduli of steels are mostly used in engineering applications: • E 212 GPa (low-alloy steels); 230 GPa (high-alloy steels); • μ 0.385E ...0.400E; • ν 0.25...0.30 (most metals). According to the Hooke’s law, the elasticity equation results from Eqs. (5.183) to (5.185) for i, j, k, l ¼ 1,2,...,N. 2v τij ¼ μ gikgjl þ gilgjk þ gijgkl γ ð5:186Þ ðÞ1 À 2v kl

The basic tensor equations of the linear elasticity theory comprise Eqs. (5.144), (5.177) and (5.186):

τij Γ i τmj Γ j τim ρ€j j ,i þ im þ im ¼ u À f 1ÀÁ γ ¼ v þ v À Γ kv ij 2 i, j j, i ij k ð5:187Þ 2v τij ¼ μ gikgjl þ gilgjk þ gijgkl γ : ðÞ1 À 2v kl

5.8 Maxwell’s Equations of Electrodynamics

Maxwell’s equations are the fundamental equations for electrodynamics, telecommunication technologies, quantum electrodynamics, special and general relativity theories. They describe mutual interactions between the charges, currents, electric, and magnetic fields in a matter. The Maxwell’s equations of electromagnetism are a system of four inhomogeneous partial differential equations of the electric field strength E, magnetic field density B, electric displacement D, and magnetic field strength H in N-dimensional spaces and the four-dimensional spacetime (x,y,z,t)[11, 12]. 5.8 Maxwell’s Equations of Electrodynamics 225

5.8.1 Maxwell’s Equations in Curvilinear Coordinate Systems

The Maxwell’s equations in a matter can be written in integral, differential, and tensor equations with physical components using Eqs. (5.31) and (5.33) and SI units as – Gauss’s law for electric fields: þ : D Á ndA ¼ qenc ðÞðIntegral equation 5 188aÞ S ∇ Á D ¼ ρ ðÞðDifferential equation 5:188bÞ 1 ∂ ÀÁpffiffiffi pffiffiffi gDi ¼ ρ ðÞðTensor equation 5:188cÞ g ∂ui 1 ∂ ffiffiffi D*i ffiffiffi p ρ : p i g ¼ ðÞð* Physical components 5 188dÞ g ∂u hi

– Gauss’s law for magnetic fields: þ B Á ndA ¼ 0ðÞð Integral equation 5:189aÞ S ∇ Á B ¼ 0ðÞð Differential equation 5:189bÞ 1 ∂ ÀÁpffiffiffi pffiffiffi gBi ¼ 0ðÞð Tensor equation 5:189cÞ g ∂ui 1 ∂ ffiffiffi B*i ffiffiffi p : p i g ¼ 0 ðÞð* Physical components 5 189dÞ g ∂u hi

– Faraday’s law: þ ð d E Á dl ¼À B Á ndA ðÞðIntegral equation 5:190aÞ dt C S ∂B ∇ Â E ¼À ðÞðDifferential equation 5:190bÞ ∂t ∂Bi ^ε ijkE ¼À ðÞðTensor equation 5:190cÞ k,j ∂t 226 5 Applications of Tensors and Differential Geometry

1 1 ∂B*i ffiffiffi * * : p hkEk, j À hjEj, k ¼À ðÞð* Physical components 5 190dÞ g hi ∂t

– Ampere-Maxwell law: þ ð d H Á dl ¼ I þ D Á ndA ðÞðIntegral equation 5:191aÞ enc dt C S ∂D ∇ Â H ¼ J þ ðÞðDifferential equation 5:191bÞ ∂t ∂Di ^ε ijkH ¼ Ji þ ðÞðTensor equation 5:191cÞ k,j ∂t 1 J*i 1 ∂D*i ffiffiffi * * : p hkHk,j À hjHj, k ¼ þ ðÞð* Physical components 5 191dÞ g hi hi ∂t where n is the normal unit vector to the surface S; qenc (r, t) is the four-dimensional spacetime enclosed electric charge; ρ (r, t) is the four-dimensional spacetime electric charge volumetric density; Ienc (r, t) is the four-dimensional spacetime enclosed electric current: J (r, t) is the four-dimensional spacetime electric current volumetric density; J* is the physical component of J; hi is the norm of the covariant basis gi. The electric displacement D is related to the electric field strength E by the matter permittivity ε.

D ¼ εE ð5:192Þ in which ε ¼ 1/(μc2) is the matter permittivity; c 3 Â 108 m/s is the light speed in À7 2 vacuum; μ is the matter permeability (μ0 ¼ 4π Â 10 N/A for vacuum). Analogously, the relation between the magnetic ﬁeld strength H and magnetic ﬁeld density B can be written as

B ¼ μH ð5:193Þ where μ is the matter permeability. Taking curl (∇Â) of the curl Eqs. (5.190b) and (5.191b) and using the curl identity, as given in Eq. (C.23), the homogenous wave equations for the electric and magnetic ﬁeld strengths of E and H in vacuum (i.e. without any source exists, J ¼ ρ ¼ 0) can be derived in 5.8 Maxwell’s Equations of Electrodynamics 227

∂2E À c2∇2E ¼ 0 ð5:194Þ ∂t2 ∂2H À c2∇2H ¼ 0 ð5:195Þ ∂t2 where c is the propagating speed of the electromagnetic waves in vacuum, which equals the light speed.

1 ffiffiffiffiffiffiffiffiffi : 8 À1 : c ¼ pμ ε ¼ 2 998 Â 10 ms ð5 196Þ 0 0

À7 2 À12 2 2 with μ0 ¼ 4π Â 10 N/A (H/m); ε0 ¼ 8.85 Â 10 C /(Nm ). Therefore, Maxwell postulated that light is induced by the electromagnetic disturbance propagated through the ﬁeld according to the electromagnetic laws. Furthermore, the law of conservation of the four-current density vector J is similar to the continuity equation of ﬂuid dynamics, as given in Eq. (5.93).

∂ρðÞr; t þ ∇ Á JrðÞ¼; t 0 ð5:197Þ ∂t

Note that the Maxwell’s equations are invariant at the Lorentz transformation [13]. However, they will be studied on the four-dimensional spacetime manifold by the Poincare´ transformation.

5.8.2 Maxwell’s Equations in the Four-Dimensional Spacetime

In the following section, the Maxwell’s equations can be expressed on the spacetime manifold in which the Einstein special relativity theory has been usually formulated. The spacetime coordinates can be deﬁned by Poincare´ in the Minkowski space where three real space dimensions in Euclidean space of x, y, and z are combined with a single time dimension t to generate a four-dimensional spacetime manifold with a fourth imaginary dimension of x4 ¼ jct in the Poincare´ group [12, 13]. The coordinates of this four-dimensional ﬂat spacetime are called the pseudo-Euclidean coordinates that can be written as (cf. Appendix G)

x1 ¼ x; 2 ; x ¼ y : 3 ; ð5 198aÞ x ¼ zpffiffiffiffiffiffiffi x4 ¼ À1 ct jct 228 5 Applications of Tensors and Differential Geometry

Fig. 5.11 Minkowski four- timelike ct dimensional spacetime diagram future lightcone t → ∞ t > 0 spacelike y observer ds

= = now hyperspace t 0 t 0 spacelike x O

t < 0

past lightcone

The spacetime interval ds in the Minkowski spacetime for the special relativity is written as

2 μ v ds ¼ g ¼ gμvdx dx ¼ dx2 þ dy2 þ dz2 þ j2c2dt2 ð5:198bÞ ¼ dx2 þ dy2 þ dz2 À c2dt2

The spacetime interval is the shortest path length in the four-dimensional spacetime. If the spacetime interval ds is zero, it is called the lightcone (lightlike); ds is negative, it is called the timelike; and ds is positive, it is called the spacelike, cf. Fig. 5.11. Thus, the Minkowski metric with four spacetime coordinates (x, y, z, jct) results from Eq. (5.198b). 0 1 1000 B 0100C g ¼ g ¼ B C ð5:198cÞ μv @ 0010A 0001

Obviously, the Minkowski metric with four spacetime coordinates (x, y, z, ct) can be written as 0 1 100 0 B 010 0C g ¼ g ¼ B C ð5:198dÞ μv @ 001 0A 000À1 5.8 Maxwell’s Equations of Electrodynamics 229

The skew-symmetric tensor components Fkl and Gkl of the electromagnetic ﬁelds B and H are expressed in the pseudo-Euclidean coordinates for k, l ¼ 1,2,3,4. & F ¼ B ; F ¼ B ; F ¼ B ; F ¼ jE =c 12 z 23 x 31 y 4k k ð5:199Þ G12 ¼ Hz; G23 ¼ Hx; G31 ¼ Hy; G4k ¼ jcDk

Similarly, the four-current density vector J can be defined as 8 > <> J1 ¼ Jx J2 ¼ Jy : J > ð5 200Þ :> J3 ¼ Jz pffiffiffiffiffiffiffi J4 ¼ jcρ ¼ À1 cρ

According to Eqs. (5.192), (5.193) and (5.196), the relation between Gkl and Fkl in Eq. (5.199) for vacuum results in

Fkl : Gkl ¼ μ ð5 201Þ 0

Therefore, the ﬁrst and fourth inhomogeneous Maxwell’s equations (5.188c) and (5.191c) can be expressed in the tensor equation of Gkl.

∂Gkl Gkl, l ¼ Jk : ∂xl ð5 202Þ μ ) Fkl, l ¼ 0Jk for k, l ¼ 1, 2, 3, 4

Thus, Eq. (5.202) can be written in the four-dimensional spacetime coordinates: 8 ∂ ∂ ∂ > F12 F13 F14 > þ þ ¼ μ J1 > ∂x2 ∂x3 ∂x4 0 > ∂ ∂ ∂ <> F21 F23 F24 þ þ ¼ μ J2 ∂x1 ∂x3 ∂x4 0 : > ∂ ∂ ∂ ð5 203Þ > F31 F32 F34 μ > 1 þ 2 þ 4 ¼ 0J3 > ∂x ∂x ∂x > ∂F ∂F ∂F : 41 þ 42 þ 43 ¼ μ J ∂x1 ∂x2 ∂x3 0 4

Similarly, the second and third homogenous Maxwell’s equations (5.189c) and (5.190c) can be written in the tensor equation of Fkl.

Fkl,m þ Flm,k þ Fmk, l ¼ 0fork, l, m ¼ 1, 2, 3, 4 ð5:204Þ where the covariant electromagnetic ﬁeld tensors can be deﬁned by [12]. 230 5 Applications of Tensors and Differential Geometry 0 1 0 Bz ÀBy ÀjEx=c ÀÁ B C B ÀBz 0 Bx ÀjEy=c C Fij @ A ð5:205Þ By ÀBx 0 ÀjEz=c jEx=cjEy=cjEz=c 0

Equation (5.204) can be written in the four-dimensional spacetime coordinates: 8 > ∂F ∂F ∂F > 23 þ 31 þ 12 ¼ 0 > ∂ 1 ∂ 2 ∂ 3 > x x x > ∂F ∂F ∂F < 34 þ 42 þ 23 ¼ 0 ∂x2 ∂x3 ∂x4 : > ∂ ∂ ∂ ð5 206Þ > F41 F13 F34 > 3 þ 4 þ 1 ¼ 0 > ∂x ∂x ∂x > ∂F ∂F ∂F : 12 þ 24 þ 41 ¼ 0 ∂x4 ∂x1 ∂x2

The Maxwell’s equations written in the tensor components Fij result from Eqs. (5.202) and (5.204) in the four-dimensional Minkowski spacetime (special relativity):

μ Fkl,l ¼ 0Jk for k, l ¼ 1, 2, 3, 4 Fkl,m þ Flm,k þ Fmk, l ¼ 0fork, l, m ¼ 1, 2, 3, 4

The potential vector A (r, t) can be deﬁned as a potential scalar φ. 8 > A1 ¼ Ax <> A2 ¼ Ay ArðÞ¼; t A ¼ A ð5:207Þ > 3 z :> jφðÞr; t A ¼ 4 c

The skew-symmetric electromagnetic ﬁeld tensor component Fkl can be written in the potential vector components.

∂A ∂A F ¼ l À k kl ∂xk ∂xl ð5:208Þ ¼ Al,k À Ak, l for k, l ¼ 1, 2, 3, 4

Substituting Eqs. (5.202), (5.204) and (5.208), the Maxwell’s equations can be expressed in the potential vector A. 5.8 Maxwell’s Equations of Electrodynamics 231

∂ F ¼ ðÞAl,k À Ak, l kl,l ∂xl ∂2A ∂2A ¼ l À k ¼ μ J ð5:209aÞ ∂xk∂xl ∂xl∂xl 0 k

μ , Fkl,l ¼ Al,kl À Ak, ll ¼ 0Jk for k, l ¼ 1, 2, 3, 4

According to Eqs. (5.204) and (5.209a), one obtains

Fkl,m þFlm,k þFmk,l ¼ ∂ ∂ ∂ ðÞþA ÀA ðÞþA ÀA ðÞ¼A ÀA 0 ∂xm l,k k,l ∂xk m,l l,m ∂xl k,m m,k ð5:209bÞ

,ðÞþAl,km ÀAk,lm ðÞþAm,lk ÀAl,mk ðÞ¼Ak,ml ÀAm,kl 0fork,l,m¼1,2,3,4

The Maxwell’s equations written in the potential vector components Al result from Eqs. (5.209a) and (5.209b) in the four-dimensional Minkowski spacetime (special relativity):

μ Al, kl À Ak,ll ¼ 0Jk for k, l ¼ 1, 2, 3, 4

ðÞþAl, km À Ak,lm ðÞþAm,lk À Al,mk ðÞ¼Ak,ml À Am,kl 0fork, l, m ¼ 1, 2, 3, 4

Using Eqs. (5.199) and (5.208), the magnetic ﬁeld strength H can be rewritten in the potential vector A.

B ¼ μ H ¼ ∇ Â A ð5:210Þ

Using Faraday’s law, the electric ﬁeld strength E can be expressed in the potential vector A and the potential scalar φ.

∂A E ¼À∇φ À ð5:211Þ ∂t

In the Lorenz gauge, the relation between the potential scalar and vector can be written as [12].

1 ∂φ þ ∇ Á A ¼ 0 ð5:212Þ c2 ∂t

Using the Maxwell’s equations and product rules of vector calculus, the wave equations of the potential scalar and vector result in

∂2φ ρ ∇2φ 1 : À 2 2 ¼À ð5 213Þ c ∂t ε0 232 5 Applications of Tensors and Differential Geometry

1 ∂2A ∇2A À ¼Àμ J: ð5:214Þ c2 ∂t2 0

5.8.3 The Maxwell’s Stress Tensor

Newton’s third law (force and reaction force) is only valid for electrostatics and magnetostatics but not valid for electrodynamics. However, the momentum conservation law is still valid in electrodynamics. In the following section, the speciﬁc electromagnetic force that consists of the electrostatic and magnetostatic forces is derived from the Maxwell’s stress tensor. First, the electrostatic force is written as ð þ

Fe ¼ Edq ¼ EρdV V where ρ is the electric charge volumetric density. Second, the magnetostatic force is written as ð þ

Fm ¼ ðÞv Â B dq ¼ ðÞv Â B ρdV V where v is the charge velocity of the electric charge q. Thus, the electromagnetic force results from both electro- and magnetostatic forces as þ

F ¼ Fe þ Fm ¼ ðÞE þ v Â B ρdV þ V ρ ρ ¼ ðÞE þ v Â B dV ð5:215Þ þV ¼ ðÞρE þ J Â B dV V in which the current density vector J is deﬁned as product of the electric charge density and its charge velocity.

J ρv 5.8 Maxwell’s Equations of Electrodynamics 233

The speciﬁc electromagnetic force f is deﬁned as the electromagnetic force per unit volume.

f ¼ ρE þ ðÞJ Â B ð5:216Þ

Using the Gauss’s law for electric ﬁelds, one obtains ρ ∇ Á E ¼ ) ρ ¼ ε0∇ Á E ε0

Thus, the ﬁrst term on the RHS of Eq. (5.216) is written as

ρE ¼ ε0ðÞ∇ Á E E ð5:217Þ

The Ampere-Maxwell’s law gives the electric current density

∂E ∇ Â B ¼ μ J þ μ ε 0 0 0 ∂t ∂ 1 ∇ ε E ) J ¼ μ ðÞÀÂ B 0 ∂ 0 t

Therefore, the second term on the RHS of Eq. (5.216) is written as ! 1 ∂E ðÞ¼J Â B ðÞÀ∇ Â B ε Â B μ 0 ∂t 0 5:218 ∂ ð Þ B ∇ ε E ¼Àμ Â ðÞÀÂ B 0 ∂ Â B 0 t

Furthermore, the last term on the RHS of Eq. (5.218) is calculated using the product rule of vector calculus as ∂E ∂ðÞE Â B ∂B Â B ¼ À E Â ∂t ∂t ∂t

Using the Faraday’s law, one obtains

∂B ∇ Â E ¼À ∂t

Thus, ∂E ∂ðÞE Â B Â B ¼ þ E Â ðÞ∇ Â E ð5:219Þ ∂t ∂t 234 5 Applications of Tensors and Differential Geometry

Substituting Eqs. (5.217)–(5.219) into Eq. (5.216), the speciﬁc electromagnetic force f is written as

∂ ε ∇ ∇ B ∇ ε ðÞE Â B : f ¼ 0½ÀðÞÁ E E À E Â ðÞÂ E μ Â ðÞÀÂ B 0 ∂ ð5 220Þ 0 t

Using product rule of vector calculus, gradient of a vector product is calculated as (cf. Appendix C)

∇ðÞ¼A Á B ðÞA Á ∇ B þ ðÞB Á ∇ A þ A Â ðÞþ∇ Â B B Â ðÞ∇ Â A where A and B are two arbitrary vectors in RN. In the case of A ¼ B E, one obtains

∇ðÞE Á E ¼ ∇E2 ¼ 2ðÞE Á ∇ E þ 2E Â ðÞ∇ Â E

Thus, the relation of the electric ﬁeld strength E

1 E Â ðÞ¼∇ Â E ∇E2 À ðÞE Á ∇ E ð5:221Þ 2

Similarly, one obtains the relation of the magnetic ﬁeld density B

1 B Â ðÞ¼∇ Â B ∇B2 À ðÞB Á ∇ B ð5:222Þ 2

Using Gauss’s law for magnetic ﬁelds (∇ÁB ¼ 0), Eq. (5.222) can be written as

1 B Â ðÞ¼∇ Â B ∇B2 À ðÞB Á ∇ B 2 ð5:223Þ 1 ¼ ∇B2 À ðÞB Á ∇ B À BðÞ∇ Á B 2

Substituting Eqs. (5.221) and (5.223) into Eq. (5.220), the speciﬁc electromagnetic force is expressed as

ε ∇ ∇ 1 ∇ ∇ f ¼ 0½þðÞÁ E E þ ðÞE Á E μ ½ðÞÁ B B þ ðÞB Á B 0 : ∂ ð5 224Þ 1∇ ε 2 1 2 ε À 0E þ μ B À 0 ∂ ðÞE Â B 2 0 t

The Maxwell’s stress tensor M is defined as a second-order tensor with the covariant components of ε 1δ 2 1 1δ 2 Mij ¼ 0 EiEj À ijE þ μ BiBj À ijB for i, j ¼ x, y, z 2 0 2 5.8 Maxwell’s Equations of Electrodynamics 235 where δij is the Kronecker delta; th Ei is the i component of the electric field strength E; th Bi is the i component of the magnetic field density B; E and B are the magnitudes of the electric field strength and the magnetic field density, respectively. The magnitudes of E and B in a 3-D space R3 are calculated as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E jjE ¼ E2 þ E2 þ E2; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix y z 2 2 2: B jjB ¼ Bx þ By þ Bz

3 There are nine components of the Maxwell’s stress tensor Mij in a 3-D space R that are written as 1 1 M ¼ ε E2 À E2 À E2 þ B2 À B2 À B2 ; xx 2 0 x y z 2μ x y z 0 ð5:225aÞ ε 1 Mxy ¼ Myx ¼ 0ExEy þ μ BxBy; 0 1 1 M ¼ ε E2 À E2 À E2 þ B2 À B2 À B2 ; yy 2 0 y x z 2μ y x z 0 ð5:225bÞ ε 1 Myz ¼ Mzy ¼ 0EyEz þ μ ByBz; 0 1 1 M ¼ ε E2 À E2 À E2 þ B2 À B2 À B2 ; zz 2 0 z x y 2μ z x y 0 ð5:225cÞ ε 1 : Mzx ¼ Mxz ¼ 0EzEx þ μ BzBx 0

The second-order Maxwell’s stress tensor M is written as in the contravariant bases

M ¼ M gigj for i, j ¼ x, y, z 0ij 1 Mxx Mxy Mxz B C ¼ @ Myx Myy Myz A

Mzx Mzy Mzz

Using the Christoffel symbols, the ith component of the divergence ∇ÁM is generally written in an N-dimensional space RN as ∇ Γ m Γ m jk ðÞÁ M i ¼ Mij,k À ik Mmj À jk Mim g for i, j, k, m ¼ 1, 2, ..., N 236 5 Applications of Tensors and Differential Geometry

For an orthonormal coordinate system in R3, the ith component of the divergence ∇ÁM results as [11] ÀÁ ∇ ε ∇ ∇ 1 ∇ 2 ðÞÁ M i ¼ 0 ðÞÁ E Ei þ ðÞE Á Ei À Á E i ÀÁ2 : 1 1 ð5 226Þ þ ðÞ∇ Á B B þ ðÞB Á ∇ B À ∇ Á B2 for i ¼ x, y, z μ i i 2 i 0

The divergence of the Maxwell’s stress tensor is calculated as jk i ∇ Á M ¼ Mij g g k ð5:227Þ ∇ i ¼ ðÞÁ M ig for i ¼ x, y, z

Substituting Eqs. (5.224), (5.226) and (5.227), one obtains the speciﬁc electromagnetic force in a simple form

∂ f ¼ ∇ Á M À ε ðÞE Â B 0 ∂t ∂S ¼ ∇ Á M À ε μ ð5:228Þ 0 0 ∂t 1 ∂S ¼ ∇ Á M À c2 ∂t

The Poynting’s vector S in Eq. (5.228) is defined as the energy flux density of the electromagnetic field.

1 : S ¼ μ ðÞE Â B ð5 229Þ 0

Using Gauss’s divergence theorem, the total electromagnetic force is calculated from Eq. (5.229)as þ þ 1 ∂S F ¼ fdV ¼ ∇ Á M À dV c2 ∂t V V þ þ ð5:230Þ 1 ∂S ¼ MÁndA À dV c2 ∂t A V where n is the normal unit vector on the surface A. The momentum density in the electromagnetic ﬁeld is deﬁned as

ε μ ε g ¼ 0 0S ¼ 0ðÞE Â B 5.8 Maxwell’s Equations of Electrodynamics 237

5.8.4 The Poynting’s Theorem

The Poynting’s theorem is used to calculate the generated power of the electromagnetic force, as shown in Eqs. (5.215) and (5.216). The generated work of the electromagnetic force is calculated as

dWt ¼ F Á dl ¼ F Á vdt ¼ qðÞÁE þ v Â B vdt ¼ qE Á vdt þ qðÞÁv Â B vdt

Due to the orthogonality, the dot product of two vectors (v Â B) and v equals zero.

ðÞÁv Â B v ¼ 0

As a result, the electromagnetic power Pt becomes

dWt ¼ qE Á vdt þ dW ) P t ¼ qE Á v ¼ ρE Á vdV t dt þ V ð5:231Þ À Á ¼ E Á J dV V where q is the electric charge; J is the electric current density. Using the Ampere and Maxwell’s equation, the dot product between E and J is calculated as ! 1 ∂E E Á J ¼ E Á ðÞÀ∇ Â B ε μ μ 0 0 ∂t 0 5:232 ∂ ð Þ 1 ∇ ε E ¼ μ E Á ðÞÀÂ B 0E Á ∂ 0 t

Applying the product rule of vector calculus, one obtains using the Faraday’s law

∇ Á ðÞ¼E Â B ðÞÁ∇ Â E B À E Á ðÞ∇ Â B ∂B ¼À Á B À E Á ðÞ∇ Â B ∂t

Thus, the ﬁrst term on the RHS of Eq. (5.232) results as

∂B E Á ðÞ¼À∇ Â B B Á À ∇ Á ðÞE Â B ∂t ÀÁ ð5:233Þ 1 ∂ B2 ¼À À ∇ Á ðÞE Â B 2 ∂t

Substituting Eq. (5.233) into Eq. (5.232), one obtains 238 5 Applications of Tensors and Differential Geometry

"#ÀÁ ∂ B2 ∂ 1 1 ∇ ε E E Á J ¼Àμ ∂ þ Á ðÞE Â B À 0E Á ∂ 0 2 t t "#ÀÁ ÀÁ ∂ B2 ∂ E2 1 1 ∇ 1ε ð5:234Þ ¼Àμ ∂ þ Á ðÞE Â B À 0 ∂ 0 2 t 2 t ∂ 2 1 ε 2 B 1 ∇ ¼À ∂ 0E þ μ À μ Á ðÞE Â B 2 t 0 0

Inserting Eq. (5.234) into Eq. (5.231), one obtains the electromagnetic power þ þ ∂ 2 1 ε 2 B ∇ 1 Pt ¼À 0E þ μ dV À Á μ ðÞE Â B dV ∂t 2 0 0 2 V 3 V þ þ ∂ ¼À4 udV þ ∇ Á SdV5 ð5:235Þ ∂t þ V V ∂u ¼À þ ∇ Á S dV ∂t V

In Eq. (5.235), the speciﬁc work u is deﬁned as 2 1 ε 2 B : u 0E þ μ ð5 236Þ 2 0

The speciﬁc work results from the generated electro-magnetostatic power of the electro and magnetostatic forces. The electrostatic work results from the electric ﬁeld strength as þ ε W ¼ 0 E2dV e 2 V

The magnetostatic work results from the magnet ﬁeld density as þ 1 2 Wm ¼ μ B dV 2 0 V

Therefore, the generated electro-magnetostatic power results as þ þ 2 1 ε 2 B Wems ¼ 0E þ μ dV udV 2 0 V V

In an empty space, no electromagnetic power is generated. According to Eq. (5.235), the energy continuity equation is given as 5.9 Einstein Field Equations 239 þ ∂u P ¼À þ ∇ Á S dV ¼ 0 t ∂t V ð5:237Þ ∂ ∂ u ∇ u 1 ∇ ) ∂ þ Á S ¼ ∂ À μ Á ðÞ¼B Â E 0 t t 0

5.9 Einstein Field Equations

Einstein field equations (EFE) are the fundamental equations in the general relativity theory. They describe the interactions between the gravitational field, physical characteristics of matters, and energy momentum tensor [13]. According to Einstein, due to gravity the matter curves and distorts the spacetime. In return, the curved spacetime shows the matter how to move by means of energy momentum and its curvature. Obviously, the larger the mass is, the more the spacetime is wrapped. Hence, the mass is accelerated in the direction perpendicular to its moving direction due to centrifugal force. Gravitational waves that are caused by merging two rotating binary black holes could vibrate and wrap the universe spacetime. Recently, the wrapped spacetime with very tiny vibrations in different directions generating by gravitational waves has been confirmed on February 11th 2016 by LIGO Scientific Collaboration (Laser Interferometer Gravitational wave Observatory). All tensors using in the general relativity theory have been mostly written in the abstract index notation defined by Penrose [14]. This index notation uses the indices to express the tensor types rather than their covariant components in the basis {gi}. According to Eqs. (2.250a) and (2.251) and using the tensor contraction laws, the Einstein tensor can be written in the covariant tensor components.

k Gij ¼ gikGj 1 ¼ g R k À δ kR ik j 2 j 1 ¼ R À g R ij 2 ij

Therefore, the Einstein ﬁeld equations can be expressed as 1 8πG G R À g R ¼À T ð5:238Þ ij ij 2 ij c4 ij where Gij is the covariant Einstein tensor in Eq. (2.251); Rij is the Ricci tensor in Eq. (2.242); gij is the covariant metric tensor components; R is the Ricci curvature tensor in Eq. (2.248); 240 5 Applications of Tensors and Differential Geometry

G is the universal gravitational constant (¼6.673 Â 10À11 Nm2/kg2); c is the light speed (3 Â 108 m/s); Tij is the kinetic energy-momentum tensor. The kinetic energy-momentum tensor in vacuum can be calculated from the cosmological constant Λ and the covariant metric tensor components gij.

ÀΛc4 T ¼ g ð5:239Þ ij 8πG ij in which the cosmological constant is equivalent to an energy density in a vacuum space. According to a recent measurement, the cosmological constant Λ is on the order of 10À52 mÀ2 and proportional to the dark-energy density ρ with a factor of 8πG used in the general relativity.

Λ ¼ 8πGρ ð5:240Þ

In case of a positive energy density of the vacuum space (Λ > 0), the related negative pressure will cause an accelerating expansion of the universe. In relativity electromagnetism, the kinetic energy-momentum tensor Tij can be calculated from the energy-momentum tensor of the electromagnetic ﬁeld Sij (electromagnetic stress-energy tensor) according to [12]. 1 1δ : ÀTij ¼ Sij ¼ μ FikFjk À ijFklFkl ð5 241Þ 0 4 in which δij is the Kronecker delta. Substituting Eq. (5.241) into Eq. (5.238), the Einstein-Maxwell equations can be generally written with the cosmological constant Λ. 1 1 G À g Λ R À g R À g Λ ¼ R À R þ Λ g ij ij ij 2 ij ij ij 2 ij 8πG : ¼À 4 Tij ð5 242Þ c 8πG 1 ¼ F F À δ F F 4μ ik jk ij kl kl c 0 4 where R is the Ricci curvature tensor can be obtained according to Eqs. (2.248) and (2.249).

R ¼ R gij 0ij 1 ÀÁffiffiffi ∂2 p ∂ JΓ m ð5:243Þ @ ln g 1 ij m n A ij ¼ À pffiffiffi þ Γ Γ g ∂ui∂uj g ∂um in jm 5.10 Schwarzschild’s Solution of the Einstein Field Equations 241

The Ricci curvature tensor can be expressed in the kinetic energy-momentum tensor T [15]:

8πG 8πG R ¼ T i ¼ T ð5:244Þ c4 i c4

The covariant electromagnetic field tensors Fij in Eq. (5.242) are given according to Eq. (5.205). 0 1 0 Bz ÀBy ÀjEx=c ÀÁ B C B ÀBz 0 Bx ÀjEy=c C Fij ¼ @ A ð5:245Þ By ÀBx 0 ÀjEz=c jEx=c jEy=c jEz=c 0 where E is the electric field strength; B is the magnetic field density; c is the light speed in vacuum.

5.10 Schwarzschild’s Solution of the Einstein Field Equations

In case of ignoring the cosmological constant Λ and the negligibly small energy- momentum tensor (Tij ¼ 0) in an empty space of small scales (R ¼ 0), the Einstein ﬁeld equation in Eq. (5.242) can be written in a simple tensor equation as

Rij ¼ 0 ð5:246Þ

The Schwarzschild’s solution of Eq. (5.246) can be derived for a spherically symmetrical empty space with four spacetime coordinates ( jct, r, φ, θ) around an object with a mass M [15, 16]. The purely imaginary spacetime distance ds along a curve C in the spherical space is written as (cf. Appendix G)

2 2 τ2 μ v ds ¼Àc d ¼ gμvdx dx ¼Àg c2dt2 þ g dr2 þ r2dΩ2 00 rr ð5:247aÞ 2MG 2MG À1 ¼À 1 À c2dt2 þ 1 À dr2 þ r2dφ2þr2 sin 2φ dθ2 c2r c2r

According to Eq. (5.247a), the Schwarzschild metric describes the spacetime around the spherical black hole with four spacetime coordinates ( jct, r, φ, θ) 242 5 Applications of Tensors and Differential Geometry 2 3 2MG 6 1 À 2 0007 6 c r ÀÁ 7 6 2MG À1 7 : g ¼ gμv ¼ 6 01À 007 ð5 247bÞ 4 c2r 5 00r2 0 000r2 sin 2φ

The proper time in the Schwarzschild space results from Eq. (5.247a)in 2MG 1 2MG À1 r2 ÀÁ dτ2 ¼ 1 À dt2 À 1 À dr2 À dφ2þ sin 2φ dθ2 ð5:247cÞ c2r c2 c2r c2 in which dτ is the proper time (intrinsic time) unaffected by any gravitational ﬁeld; dt is the apparent time (laboratory time) in a rest frame of the clock affected by a gravitational ﬁeld. Generally, dt > dτ due to gravitational time dilation;

dΩ2 dφ2 þ sin 2φdθ2; r is the radius of spherical coordinates (r,φ,θ), as shown in Fig. 5.1; G is the gravitational constant (G ¼ 6.673 Â 10À11 m3 kgÀ1sÀ2); M is the object mass; c is the light speed. Two singularities of the Schwarzschild’s solution in Eq. (5.247a) exist as the space distance ds increases inﬁnitely. Besides r ¼ 0 (an unavoidable case), the space metric coefﬁcient must be

1 : grr ¼ 2MG !1 ð5 248Þ 1 À c2r

Thus,

2MG 1 À ¼ 0 ð5:249Þ c2r

The Schwarzschild’s radius can be deﬁned at the singularity by 2MG r r ¼ ð5:250Þ S grr!1 c2

The proper time dτ without effect of any gravitational ﬁeld results at r rS, i.e. the clock runs slower by the factor given in Eq. (5.251) when it comes near the gravitation ﬁeld (dt > dτ). This effect is called the gravitational time dilation [15]. 5.11 Schwarzschild Black Hole 243

Fig. 5.12 Proper time time-like ct → ∞ vs. apparent time in the ct(rS ) Schwarzschild spacetime black hole (r ≤ rS)

cτ (r)

t = 0 space-like r r 0 rS 0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 2MG dτ < g dt ¼ 1 À dt < dt ð5:251Þ 00 c2r

According to Eq. (5.251), the laboratory time dt becomes inﬁnite as r reaches the Schwarzschild’s radius rS. The clock would stop running near the black hole because dt goes to inﬁnity (cf. Fig. 5.12).

1 1 dt > pffiffiffiffiffiffidτ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidτ g00 2MG ð5:252Þ 1 À 2 ffiffiffiffiffiffi c r p τ : ¼ grrd !1 as r ! rS

5.11 Schwarzschild Black Hole

In case of r < rS, the uncharged spherically symmetric dwarf star (neutron star) with a mass M collapses within a cylinder of the radius r less than the Schwarzschild’s radius rS. This cylinder is called the Schwarzschild black hole, as shown in Fig. 5.13. According to Eq. (5.250), the Schwarzschild’s radius rS of the Sun is about 2.95 Â 103 m compared to its radius R of 6.95 Â 108 m (approximately 3 235,000 times larger than rS). Therefore, our Sun with the mass m ~ R would produce a huge energy E ¼ mc2 before it disappears into the black hole in the very far future. Using the Hamiltonian H consisting of the generalized angular momentum pi, _ active variable q i, and the Lagrangian L, the total conserved energy of the photon trajectory in the gravitational ﬁeld is calculated as 244 5 Applications of Tensors and Differential Geometry

Fig. 5.13 Black hole in the time-like ct Schwarzschild spacetime

black hole t → ∞ ≤ r rS = r rS

0 space-likes (r,ϕ,θ) t = 0

X X ∂ ; ; _ L _ ; _ ; HqðÞ¼i pi t piq i À L ¼ ∂ _ q i À LqðÞi qi t i i q i p2 ¼ T þ V ¼ þ VqðÞ 2m EP where the Lagrangian L is deﬁned as

L T À V; the generalized angular momentum and active variable can be written as

∂ ∂ L ; _ H pi ¼ ∂ _ q i ¼ ∂ q i pi

The total conserved energy of the photon is approximately written according to [17] as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2MG E / 1 À ð5:253Þ P r c2r

In fact, the Hamiltonian is the total conserved energy of the kinetic and potential energy T and V of the system. Differentiating Eq. (5.253) with respect to r and calculating the second derivative at the extreme, the radius rP results as the maximum energy of the photon trajectory occurs at the zero first derivative and negative second derivative. 5.11 Schwarzschild Black Hole 245 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ∂ ∂ 3MG À r EP 1 2MG qc2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / 1 À 2 ¼ ¼ 0 ∂r ∂r r c r 3 2MG r 1 À c2r

Thus,

3MG 3 r ¼ ¼ r ð5:254Þ P c2 2 S

The second derivative of the photon energy is calculated as pffiffiffi ∂2E À1 À 3 P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ< / ¼ 3 0 ∂r2 3 2MG 3MG r 1 À 2 2 rP P c rP c

Thus, the maximum energy results at r ¼ rP, as shown in Fig. 5.14. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2MG c2 EP, max / 1 À / pffiffiffi ð5:255Þ r c2r 3 3MG r¼rP

The photon with the maximum energy at the radius rP locates in a bi-stable state. Through a small disturbance at this bi-stable state, either the photon trajectory is rejected outwards the photon sphere (r > rP) or attracted towards the black hole (r ! rS) in order to reach the stable state of minimum energy, as shown in Fig. 5.14. The sphere with the radius of rP ¼ (3/2) rS is called the photon sphere of the

black hole photon sphere

vP v P photons

light beam τ r

attracting zone

EP bi-stable state c2 E ∝ P,max 1 2MG 3 3MG E ∝ 1− P r c2r

0 2MG 3MG r r = r = S c2 P c2

Fig. 5.14 Total energy of photons in the gravitational field of a curved space 246 5 Applications of Tensors and Differential Geometry gravitational field. Furthermore, the photon energy equals zero at the Schwarzschild’s radius rS according to Eqs. (5.250) and (5.253). Considering a photon moving from the outside (r rP) near to the photon sphere, the photon energy reaches a maximum at r ¼ rP. Between the photon sphere and the black hole (rS < r < rP), the photon energy reduces from the maximum to zero at the Schwarzschild’s radius rS of the black hole. If the photon trajectory (light beam) moves outside the photon sphere, the photon trajectory is accelerated with a bending radius around the black hole, as shown in Fig. 5.14. In another case, the photon trajectory enters the photon sphere with a velocity vP. There are two possibilities for the light beam depending on the moving direction of vP [17]. First, if the velocity vP is tangent to the photon sphere surface, the light beam moves on the photon sphere surface in a stable condition. Second, if the radial component of vP moves towards the black hole center (called the time line τ), the light beam becomes unstable and collapses itself into the black hole after a number of cycles because the balance between the energy created by the strong force and gravitational force fails. The energy of the photon trajectory reduces to zero as it moves towards the black hole (r ! rS) under the very strong gravitational field of the black hole. This is to blame for the attraction of the photon trajectory to the black hole. In this case, the light beam has never left the black hole back to the outside. The neutron star will be contracted in a infinitesimal point particle in the black hole under its huge gravitational field. Finally, the neutron star collapses in the black hole, crushing all its atoms into a highly dense ball of neutrons.

References

1. Klingbeil, E.: Tensorrechnung fur€ Ingenieure (in German). B.I.-Wissenschafts-verlag, Mann- heim (1966) 2. Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York, NY (1982) 3. Iben, H.K.: Tensorrechnung (in German). B.G. Teubner, Stuttgart (1999) 4. Nguyen-Scha¨fer, H.: Aero and Vibroacoustics of Automotive Turbochargers. Springer, Berlin (2013) 5. Crocker, M.J.: Noise and Vibration Control. Wiley, Hoboken, NJ (2007) 6. Fahy, F., Gardonio, P.: Sound and Structural Vibration, 2nd edn. Academic Press, London (2007) 7. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 8. Chen, N.: Aerothermodynamics of Turbomachinery. Analysis and Design. Wiley, Somerset, NJ (2010) 9. Schobeiri, M.: Turbomachinery Flow Physics and Dynamic Performance, 2nd edn. Springer, Berlin (2012) 10. Green, A.E., Zerna, W.: Theoretical Elasticity. Oxford University Press (OUP), Oxford (1968) 11. Grifﬁths, D.: Introduction to Electrodynamics, 3rd edn. Prentice Hall, Englewood Cliffs, NJ (1999) 12. Lawden, D.F.: Introduction to Tensor Calculus. Relativity and Cosmology, 3rd edn. Dover, Mineola, NY (2002) References 247

13. Landau, D., Lifshitz, E.M.: The Classical Theory of Fields. Addison-Wesley, Reading, MA (1962) 14. Penrose, R.: The Road to Reality. Alfred A. Knopf, New York, NY (2005) 15. Cahill, K.: Physical Mathematics. Cambridge University Press, Cambridge (2013) 16. Susskind, L., Lindesay, J.: An Introduction to Black Holes, Information and the String Theory Revolution. World Scientiﬁc Publishing, Singapore (2005) 17. Susskind, L.: Lectures of General Relativity. Stanford University (2012) 18. Lurie, A.I.: Theory of Elasticity. Springer, Berlin (2005) Chapter 6 Tensors and Bra-Ket Notation in Quantum Mechanics

6.1 Introduction

Classical mechanics have been used to study Newtonian mechanics including conventional mechanics, electrodynamics, cosmology, and relativity physics of Einstein. Compared to quantum mechanics, classical mechanics is deterministic, in which the processes are well determined and foreseeable. On the contrary, processes in quantum mechanics are undeterministic, uncertain, nonlocal, and intrinsically random. Therefore, quantum mechanics deals with probability and uncertainty principles to study the atomic and subatomic world. Due to their extremely small sizes of the quantum particles, the capabilities of apparatuses are quite limited to measure the undeterministic and uncertain behaviors of the particles. Therefore, interactions between these very small particles in the quantum world, such as electrons and photons are simulated using mathematical abstractions that will be discussed in the following sections. Mathematics of quantum physics describes nonlocal correlations between the particles with the intrinsic randomness (e.g. Schrodinger€ ’s cat) in the combined system. In quantum mechanics, Hilbert space is defined as any finite or infinite N- dimensional vector space in which abstract concepts of vectorial and functional analysis are carried out to study the interactions of the quantum particles. An element of the Hilbert space H can be expressed in coordinates of an orthonormal basis that is analogous with Cartesian coordinate system. Quantum mechanics (also quantum physics) is a peculiar physics so that Richard Feynman had said, “nobody understands it after all”. Contrary to classical physics, quantum mechanics has some special characteristics, such as entanglement, quantum teleportation, and nonlocality that lead to an intrinsic randomness at the measurement of a quantum state [1]. At first, such characteristics must be clearly understood before we try to use abstract mathematics to describe them.

© Springer-Verlag Berlin Heidelberg 2017 249 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5_6 250 6 Tensors and Bra-Ket Notation in Quantum Mechanics

6.2 Quantum Entanglement and Nonlocality

Erwin Schrodinger€ had said that quantum entanglement is necessary for quantum mechanics that is also based on the Heisenberg’s uncertainty principle. Firstly, entanglement of two entangled particles in a composite system could be understood that there is somehow an interaction between two particles in two different worlds at a long distance. Einstein called this action as the spooky action at a distance (in German: “spukhafte Fernwirkung”) due to some hidden variables. Systems that contain no entanglement are defined as separable systems (non-entangled). Secondly, quantum nonlocality describes the nonlocal correlations of two entangled subsystems in a composite system SAB in the combined Hilbert space HA Â HB. The composite system consists of two or many subsystems. Each subsystem has objects with the matter and physical states. On the contrary, the concept of locality states that an object is only directly influenced by its surround- ings. The action between two points is carried out by means of waves through a field that transports the information through the space between them. According to the special relativity theory, the transmitting velocity must be in any case less than the light speed. In the Bell’s game, two players A (Alice) in the subsystem of SA and B (Bob) in the subsystem SB have two similar game boxes. They play with the boxes at the same time in a very long distance between two planets without any physical connections (e.g., internet LAN, WLAN or any telecommunication techniques). Both game players did not know the strategy with that they intent to play in the game in advance. Note that the game is repeated in many times at a given time interval. Both results of the quantum state vectors |ΨAi and |ΨBi are instantaneously measured in the subsystems SA and SB, as shown in Fig. 6.1. At the game end, the results show the average outcome of A is somehow correlated with the average outcome of B even if they had agreed in advance.

entanglement

Ψ Ψ A B

⊂ ⊂ Subsystem S A H A Subsystem SB H B

≡ ⊗ ⊂ × Composite system S AB S A SB H A H B

Fig. 6.1 Entanglement between two particles in a composite system 6.2 Quantum Entanglement and Nonlocality 251

Both players A and B have a truly chance-like event in the Bell’s game. This is called the intrinsic randomness that only exists in the nonlocal correlations (nonlocality in the composite system) without any physical communication between A and B. However, there is somehow any telepathic communication, which is simultaneously caused by quantum entanglement between the current states of the players [1–3]. Such correlations are nonlocal and intrinsically random in a composite system. The results denote the general statement “quantum entanglement is necessary but not sufficient for nonlocal correlations of the entangled particles”. To prove it, the first part of the statement is proved if the following proposition is true [4]. If the quantum entanglement does not occur in a composite system, then nonlocal correlations do not take place in it. This proposition is always true. Hence, the first part of the statement “quantum entanglement is necessary for nonlocal correlations of the entangled particles” has been proved. Next, a quantum entanglement could however occur with a local correlation between the entangled particles in a composite system of two very close subsystems. This proposition is true at least for this case. Therefore, the second part of the statement “quantum entanglement is not sufficient for nonlocal correlations of the entangled particles” is correct [4]. As a result, the general statement is proved. Using the quantum entanglement of the states, quantum physics tries to predict the nonlocal correlations between the entangled particles of the different subsystems at a distance. Note that quantum bit (called Qubit) is a unit of quantum information that describes the quantum state of a system that is equivalent to the von-Neumann entropy. Therefore, the Qubit state is the current state of an object; and it is a linear superposition of the basis vectors |0 i for spin down and |1i for spin up at the same time. Note that particles in a composite system are entangled and correlated due to quantum entanglement in which time is its side effect. In general, the more uncertain the quantum system is, the more entangled the system becomes at increasing time [S. Lloyd]. In the composite system, quantum teleportation, teleporting entanglement, and intrinsic randomness are taken into account. Quantum teleportation is defined as a direct teleportation of the Qubit state of the object (i.e. just only the state of quantum information, not the whole object of mass and energy) from one location to another location at a distance without passing through any medium. That means the state of the object must disappear from its original location; and its teleported state will appear in the new location independently of any distance [1, 5]. Teleporting entanglement is a quantum entanglement of the states between two entangled particles at measurements. The nonlocal correlations always occur in the composite system during the entanglement take place between two entangled particles. Therefore, the nonlocal correlations are a sufficient condition for the quantum entanglement of the two-particle state. Such nonlocal correlations are well determined in the composite system. How- ever, the quantum states of the subsystems are undetermined by the uncertainty principle. The Heisenberg’s uncertainty principle states that the product of the uncertainties of the position and momentum of a particle must be always larger than 252 6 Tensors and Bra-Ket Notation in Quantum Mechanics a half of the reduced Planck constant (also Dirac constant) [6, 7]. As an example, the relative distance between two electrons is well determined in the composite system, but the positions of the electrons in the subsystems are not definite due to entanglement and the uncertainty principle. Furthermore, the quantum entanglement between two entangled particles at an infinite distance leads to another issue at which the teleporting velocity of the quantum states between the particles is larger than the light speed. This new issue violates the Einstein special relativity theory, which is valid for quantum mechanics. The above interpretations and discussions are quite confused; therefore, it is quite difficult to understand quantum mechanics. There are two contrary opinions about the quantum entanglement [1, 7]. Firstly, Einstein, Schrodinger,€ and De Broglie endorsed the hidden-variable theory with the “spooky action at a distance” and were against the interpretation of quantum entanglement. Secondly, Heisen- berg and Bohr accepted the entanglement interpretation and rejected the hidden- variable theory.

6.3 Alternative Interpretation of Quantum Entanglement

Results of the recent measurements have shown that there are neither hidden variables nor the spooky action at a distance in quantum mechanics and the teleporting velocity between two entangled particles is less than the light speed. However, they do not explain how the quantum entanglement really works. The ﬁrst author of this book [Nguyen-Scha¨fer] attempts to interpret the quantum entanglement in the new way that is based on conservation laws of the quantum states. Let A and B be two entangled particles in the subsystems SA and SB, respectively. The quantum state vectors |ΨAi and |ΨBi are measured using the apparatuses MA and MB in the subsystems. The composite system SAB is a combined system of the subsystems SA and SB and belongs to the Hilbert space HA Â HB (see Fig. 6.2). Most of processes occurring in physics are almost completely symmetric. According to Noether’s theorems [8], symmetric processes obey conservation laws; e.g., the conservation of angular momentum is also valid in a composite system that consists of two or many entangled subsystems. Note that conservation laws are the fundamental principles of physics. As a result, quantum mechanics strongly depends on the conservation of probability of the expectation values (average values) at the measurements. Note that the quantum entanglement is broken when the entangled particles decohere by means of interaction with the environment, such as at measurement of an observable. In this case, the outcome of the measurement collapses onto the entangled state of the whole system. However, the conservation of the von-Neumann entropy is always held. The von-Neumann entropy that consists of the quantum information describes the disorder of the entangled quantum system. 6.3 Alternative Interpretation of Quantum Entanglement 253

Conservation laws (Law of physics)

Ψ Ψ MA A MB B

SubsystemS ⊂ H Subsystem S ⊂ H A A Entanglement B B

≡ ⊗ ⊂ × Composite system S AB S A SB H A H B

Fig. 6.2 Alternative interpretation of quantum entanglement

This entropy is used to measure the quantum entanglement. Generally, the larger the von-Neumann entropy is, the more entangled the composite system becomes. The angular momentum of the quantum state (spin state) of a particle is conserved in a composite system but must not be necessarily invariant in its subsystems. According to the conservation law of the angular momentum, the current states of the subsystems are nonlocally correlated in the composite system instantaneously and independently of any distance between two entangled subsystems. At measurements in each subsystem, the instant state of A is correlated with the instant state of B in advance due to conservation of the quantum states in the composite system. In this case, the conservation of the quantum states (quantum information) is the root cause (upper hand) for the quantum entanglement. As a result, quantum entanglement is just the logical consequence of the conservation law of quantum information; and it is not caused by quantum teleportation or teleporting entanglement. In this point of view, the conservation of the quantum states in the composite system is the root cause for the quantum entanglement. An example is given to make this interpretation more plausible. It is assumed that the initial angular momentum of the quantum states of the composite system equals zero. The angular momentum of the composite system is always conserved and therefore invariant with respect to time. In the case if the quantum state of A (Alice) is changed into the spin up |ui in the subsystem SA, then the quantum state of B (Bob) must be obviously changed into the spin down |di in the subsystem SB in order to satisfy the conservation of spin momentum in the composite system. At the game end, the average outcomes of A and B at measurements are correlated with each other. In this interpretation, nonlocal correlations of the entangled particles occur in any composite system due to conservation laws of entropy. As a result, the statement “quantum entanglement is necessary but not sufﬁcient for nonlocal correlations of the entangled particles” has been always valid. Therefore, the quantum entanglement is related to nonlocality of the entangled particles. This statement opposes the EPR (Einstein, Podolsky, and Rosen) paradox, which is 254 6 Tensors and Bra-Ket Notation in Quantum Mechanics based on the locality of the interacted particles in a composite system. Recent experiments have proved that Einstein and his colleagues were wrong on the EPR effect. Furthermore, this interpretation of quantum entanglement could overcome the problems: – Quantum entanglement is independent of any distance between the subsystems. Therefore, the teleporting velocity v > c (light speed) is not involved in quantum entanglement; – Quantum teleportation and teleporting entanglement are not involved; they are the consequences of the entanglement and not the causes; – Hidden variables of the EPR paradox (Einstein, Podolsky, and Rosen) are irrelevant to quantum entanglement; – The spooky action at a distance proposed by Einstein is not needed.

6.4 The Hilbert Space

Square integrable functions create the Hilbert space H(R1) that has three properties: – Being an infinite-dimensional complex vector space C1; – Having Hermitian positive definite scalar product of two complex vectors; – Containing the Hilbert bases of the Hermite functions ϕn(x). A complex function f(x) of a real variable x 2 R is defined as a square integrable function if it satisfies the condition

þ1ð jjfxðÞ2dx < 1 À1

Furthermore, each square integrable function f(x) can be written in a linear expression of the orthonormal Hermite functions ϕn(x)as

X1 ϕ fxðÞ¼ Cn nðÞ2x HðÞR n¼0 where Cn is the component of f(x) on the Hermite function ϕn(x) that is calculated as

1ð ϕ ϕ* Cn ¼ hinjf ¼ nðÞx fxðÞdx 0 6.5 State Vectors and Basis Kets 255

The orthonormal Hermite functions ϕn(x) are used for the bases of the Hilbert space H(R). They are called the Hilbert bases. As a result, each quantum state vector in the Hilbert space can be expressed using the orthonormal Hilbert bases as

X1 X1 ψ ϕ ji¼ Cnjin Cnjin n¼0 n¼0

Obviously, Cn is the component of the square integrable function ψ(x) on the Hilbert basis |ni of the Hilbert space. ϕ ψ ψ Cn ¼ hinj ¼ hinj

Due to orthonormality, the Hermitian scalar product of two arbitrary Hilbert bases can be written in Kronecker delta

himjn ¼ hinjm ¼ δnm

6.5 State Vectors and Basis Kets

State vector denotes the quantum state of the object that is a set of numbers of Qubit. There are two possibilities of a coin: head (H) and tail (T). Similar to the coin, each object has two basis states |0 i and |1i, which are called Qubits. Thus, the possible number of a state vector results as

n NS ¼ 2 ð6:1Þ where n is the number of objects. 2 As an example, there are four possible combinations (NS ¼ 2 ) for two objects a and b of the combined state vector (ket) that is superimposed by two basis states.

jiab ¼ ji00 ; ji01 ; ji10 ; ji11 :

In quantum mechanics, the spins up and down |ui and |di are used for the basis states according to L. Susskind [7]. As a result, the possible combination of the combined state vector |abi is written as

jiab ¼ jiuu ; jiud ; jidu; jidd ð6:2Þ

In Chap. 1, we dealt with the eigenvalues and eigenvectors of the Hermitian operator. At the special eigenvalues of +1 and À1, their corresponding eigenvectors are chosen as the basis kets (called basis spin or eigenbasis) in quantum mechanics. The normalized eigenvectors are orthonormal in Cartesian coordinate system 256 6 Tensors and Bra-Ket Notation in Quantum Mechanics

Fig. 6.3 Orthogonal basis z kets (eigenkets) in quantum mechanics λ = + u u 1 y i λ = + i 1 l r x λ = − λ = + l 1 0 r 1

o λ = − o 1 λ = − d d 1

(Euclidean space); therefore, the basis spins are also orthonormal in the Hilbert space. All possible basis spins (u: up, d: down, r: right; l: left; i: inner, and o: outer) are displayed in Fig. 6.3. The state vector is deﬁned as a linear superposition of the basis vectors. Using the bra-ket notation (cf. Chap. 1), the state ket of a single system (non-entangled system) can be written in the eigenkets of the spins up and down as X jiΨ ¼ αijiλi ¼ αujiu þ αdjid ð6:3Þ i where αi is the component of the state vector on the eigenvector; |λii is the orthonormal eigenvector (basis ket, eigenket) relating to its eigenvalue λi. According to the Born rule, the probability of the measurement of the state 2 vector in the eigenvector |λii is the squared amplitude |αi| . The component of the state vector (complex number) is deﬁned as its projection on the basis ket according to Eq. (1.58).

α λ jΨ ¼ ψλðÞ2C i hii i : 2 2 ð6 4Þ ) jjαi ¼ jjψλðÞi

At the measurement, the state vector could be collapsed onto a single term on any basis. In this case, the state vector becomes after collapsing on the basis ket |λji. X jiΨ ¼ αijiλi 2 H i ÀÁ ÀÁ ð6:5Þ ) Ψcollapse ¼ Ψ λj ¼ αj λj ¼ ψλj λj

Schrodinger€ ’s cat [6] in an imaginary box with a radioactive trigger is an example for the collapse of quantum state. 6.5 State Vectors and Basis Kets 257

During the measurement at closing the box, the cat is neither alive nor dead but rather in a blend state of them (i.e. half alive and half dead in the sense of the statistical interpretation with a probability 50 % alive to 50 % dead). Therefore, his quantum state before opening the box is undeterministic in one half of the alive state |Ψalivei and one half of the dead state |Ψdeadi since nobody knows whether the cat is alive or dead inside the box. 1 1 Ψbefore ¼ pffiffiffi jiΨalive þ pffiffiffi jiΨdead : 2 2

At the measurement end, the cat has only a deterministic state either being alive or dead at the time of opening the box. As a result, the quantum state of the cat immediately collapses onto a single deterministic state either |Ψalivei or |Ψdeadi. Ψcollapse ¼þ1jiΨalive or þ 1 jiΨdead :

According to the Born rule, the relative probability of the state vector in the basis ket |λji is written as ÀÁ λ λ Ψ 2 λ Ψ * λ Ψ P j ¼ jj ¼ jj ÀÁjj ÀÁ Ψ λ λ Ψ ψ * λ ψλ ð6:6Þ ¼ j j jj ¼ ÀÁj Á j α*α α 2 ψλ 2 > ¼ j j ¼ j ¼ j 0 where the component ψ*(λj) is the complex conjugate of its component ψ(λj). The calculating rules of adjoints (also transpose conjugates) of entities are given in Table 6.1 (cf. Chap. 1). They are very useful to calculate the operators in quantum mechanics. Let A be an operator (matrix or second-order tensor). Its transpose is written as

T A ¼ Aij ) A ¼ Aji

The complex conjugate of A is deﬁned as

* * A ¼ Aij

The transpose conjugate of A is called the adjoint A{ that is calculated as

Table 6.1 Calculating rule of adjoints of entities Calculating rules of adjoints (transpose conjugates) Entity i Ψi AAB A Ψi Φ Ψ Φ A Ψ { { { { { Adjoint Ài Ψ A B A ΨA Ψ Φ ΨA Φ { If the operator A is Hermitian, then A ¼ A 258 6 Tensors and Bra-Ket Notation in Quantum Mechanics

ÀÁ ÀÁ * * { AT ¼ A ¼ A* A{ ¼ A ji ji ij * * * { , A ¼ Aji ¼ A ji ÀÁ ij * , AT ¼ A{

If A is hermitian, then A is self-adjoint; i.e.,

{ { : Aij ¼ Aij , A ¼ A

As an example, the probability of the spin up |ui in the state vector |Ψi|ri,as shown in Eqs. (6.22a) and (6.22b), at the eigenvalue λu ¼ +1 is calculated as (see Fig. 6.3) 1 1 1 1 PðÞ¼λu hirju Á hiujr ¼ pffiffiffiðÞ11 Á ðÞ10pffiffiffi 2 0 2 1 1 1 1 ¼ pffiffiffi Á pffiffiffi ¼ 2 2 2

The probability of the spin down |di in the state vector |Ψi|ri at the eigenvalue λd ¼À1 is calculated as 1 0 1 1 PðÞ¼λd hirjd Á hidjr ¼ pffiffiffiðÞ11 Á ðÞ01pffiffiffi 2 1 2 1 1 1 1 ¼ pffiffiffi Á pffiffiffi ¼ 2 2 2

Thus,

X 1 1 PðÞ¼λi PðÞþλu PðÞ¼λd þ ¼ 1 i 2 2

Thus, the average measured value of the basis states |ui and |di in the state vector |Ψi |ri results as (see Fig. 6.8) X P hiP ¼ PðÞλi λi ¼ PðÞλu λu þ PðÞλd λd i 1 1 ¼ ðÞþ1 ðÞ¼À1 0 2 2

This result shows that the measured spins |ui and |di have a chance of 50 % for each spin in the state vector |ri under consideration of quantum mechanics. However, the average value after many times of measurement is zero because the state vector |ri is perpendicular to both spins up and down. 6.5 State Vectors and Basis Kets 259

The outer product (projection operator) of the basis ket |λji is written as using Eq. (1.68).

XN λ λ : Pλj ¼ j j ) Pλj ¼ I ðÞðidentity operator 6 7Þ j

The absolute probability (probability density) of the state vector |Ψi in the basis ket |λji is deﬁned as ÀÁ ÀÁ P λ Ψjλ λ jΨ hjΨ Pλ jiΨ λ j j j j p j ¼ Ψ Ψ ¼ Ψ Ψ ¼ Ψ Ψ Dhij E hij hij ð6:8Þ Ψ0 Ψ0 Pλj

The normalized state vector is defined as E X DE X X 0 jiΨ 0 Ψ pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 ¼ ¼ λijΨ jiλi ¼ ψ jiλi α jiλi hiΨjΨ i i i E i i 0 ) Ψ ¼ 1 ð6:9Þ

Thus, the total absolute probability should be equal to 1 using Eq. (6.9). X ÀÁ X D E λ Ψ0 Ψ0 p j ¼ Pλj j Xj DEDED X E 0 0 0 0 ¼ Ψ jλj λjjΨ ¼ Ψ λj λj Ψ ð6:10aÞ DEj DE Ej 2 0 0 0 0 0 ¼ Ψ jIΨ ¼ Ψ jΨ ¼ Ψ ¼ 1 ðÞq:e:d:

Proof E E 0 0 I Ψ ¼ Ψ ð6:10bÞ

The LHS of Eq. (6.10b) is written as ! E X E X X Ψ0 λ λ Ψ0 λ λ α0 λ I ¼ j j ¼ j j ijii Xj Xj i α0 λ λ λ α0 δ λ ¼ i jj i j ¼ i ij j Xi, j E i, j α0 λ Ψ0 : : : ¼ ijii ¼ ðÞq e d i

In continuum medium, the normalized state ket is written in the normalized wave function ψ’(x) of a particle with the eigenket |xi as [6, 9] 260 6 Tensors and Bra-Ket Notation in Quantum Mechanics ð ð E DE 0 0 0 Ψ ¼ xjΨ jix dx ¼ ψ ðÞx jix dx ð6:11Þ where the normalized wave function of the particle is deﬁned as DE 0 ψ 0 ðÞ¼x xjΨ

Using Eq. (6.11), one obtains the same result of Eq. (6.10b)as ð ð E E DE 0 0 0 I Ψ ¼ jix hjx dx Ψ ¼ jix xjΨ dx ð E 0 0 ¼ ψ ðÞx jix dx ¼ Ψ

According to the Born rule in quantum mechanics, the absolute probability (probability density) for ﬁnding the particle in position x at instant t is the squared amplitude of the particle normalized wave function.

pxðÞ¼; t kkψ 0 ðÞx; t 2

Generally, the absolute probability pab for ﬁnding the particle in a position between a and b at time t is written as (see Fig. 6.4)

ðb ψ 0 ; 2 pab ¼ kkðÞx t dx a

According to the statistical interpretation, the integral over the entire space of the probability density function of the normalized wave function ψ’(x) of the particle equals 1 at any time t.

Fig. 6.4 Absolute ψ ′ 2 probability of the particle b = ψ ′ 2 occurring between a and b pab ∫ dx a

a b x 6.5 State Vectors and Basis Kets 261

þ1ð þ1ð pxðÞ; t dx ¼ kkψ 0 ðÞx; t 2dx ¼ 1: À1 À1

The total absolute probability in a continuum medium is calculated as

þ1ð X ÀÁ pxj; t pxðÞ; t dx j À1 þ1ð þ1ð DEDE D E 0 0 0 0 ¼ Ψ jx xjΨ dx ¼ Ψ Px Ψ dx À1 À1 D þ1ð E DE E 2 0 0 0 0 0 ¼ Ψ Pxdx Ψ ¼ Ψ jIΨ ¼ Ψ ¼ 1 ðÞq:e:d: À1

In another way, one also obtains the total absolute probability

þ1ð þ1ð pxðÞ; t dx ¼ ψ 0*ðÞÁx; t ψ 0 ðÞx; t dx À1 À1 þ1ð ¼ kkψ 0 ðÞx; t 2dx ¼ 1 ðÞq:e:d: À1

Therefore,

þ1ð X ÀÁ p λj ¼ pxðÞ; t dx ¼ 1 j À1 : þ1ð ð6 12Þ ) pðÞr; t d3r ¼ 1 À1

The inner product of bra and ket in Eq. (1.60a) can be written as ð ð hiΦjΨ ¼ hiΦjx hixjΨ dx ¼ ϕ*ðÞÁx ψðÞx dx 2 R ð6:13Þ in which 262 6 Tensors and Bra-Ket Notation in Quantum Mechanics

* ϕ*ðÞ¼x hiðxjΦ ¼ hiΦjx ð ¼ hiΦjξ hiξjx dξ ¼ ϕ*ðÞξ hiξjx dξ; ψðÞx ðhixjΨ ð ¼ hixjξ hiξjΨ dξ ¼ hixjξ ψξðÞdξ:

Using Eq. (6.13), the norm of the state ket is calculated in a continuum medium as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ð kkjiΨ ¼ hiΨjΨ ¼ ψ *ðÞÁx ψðÞx dx ð6:14Þ

The Dirac delta function is deﬁned as (see Fig. 6.5) & 0 forx6¼ x0 δðÞ¼x À x0 hixjx0 ¼ ð6:15Þ 1 forx¼ x0

The closure or completeness relation of the Dirac delta function is written as X δðÞ¼x À x0 hixjx0 ¼ xjΨj Ψjjx0 X j ψ ψ * ¼ jðÞx Á j ðÞx0 j

The Dirac delta function can be written in the Gaussian function as ! 2 1 ÀðÞx À x0 δðÞ¼x À x0 lim pffiffiffi exp ε!0 ε π ε2

Using Eq. (6.15) and δ(xÀx0) ¼ 0 for any x 6¼ x0, the position wave function of the particle at the position x0 is calculated as

Fig. 6.5 Dirac delta δ − (x x0 ) function ∞

0 x0 x 6.5 State Vectors and Basis Kets 263

þ1ð

ψðÞ¼x0 hix0jΨ ¼ hix0jx hixjΨ ¼ hix0jx Á ψðÞx dx À1 þ1ð þ1ð * ¼ δ ðÞx À x0 ψðÞx dx ¼ ψðÞx0 δðÞx À x0 dx À1 À1

This result denotes that the integral of the Dirac delta function over dx must be equal to 1.

þ1ð

δðÞx À x0 dx ¼ 1 ð6:16Þ À1

Generally, the position wave function of a particle at any position x can be written as

þ1ð ψðÞ¼x hixjΨ ¼ hixjξ hiξjΨ ¼ hixjξ Á ψξðÞdξ À1 þ1ð ¼ δðÞx À ξ ψξðÞdξ À1

Therefore, any function f(x) can be written using the Dirac delta function as

þ1ð þ1ð fxðÞ¼ hixjξ f ðÞξ dξ ¼ δðÞx À ξ f ðÞξ dξ À1 À1

The Dirac and wave formalisms in quantum mechanics are shown in Table 6.2.

Table 6.2 Dirac and wave formalisms Dirac Formalism Wave formalism Notation ψi ψ(r) State vector $ Wave function ψðÞit ψ(r, t) Ket of state vector ψðÞt ψ*(r, t) Bra of state vector Ð ψ ψ ψ *ðÞr ψ ðÞr d3r Hermitian scalar product 1 2 Ð 1 2 k ψk2 ¼ ψ ψ ψðÞr 2d3r Squared norm of ψ(r) Ð ψ L ψ ψ *ðÞr Lψ ðÞr d3r Linear operator L 1 2 Ð 1 2 3 L¼hi¼L ψ L ψ ψ *ðÞr LψðÞr d r Expectation value of L 264 6 Tensors and Bra-Ket Notation in Quantum Mechanics

6.6 The Pauli Matrices

The Pauli matrices are spin operators that are used to calculate the Hamiltonian, the possible energy of the system. Let σ be the spin operator of the subsystem A; τ be the spin operator of the subsystem B; ω be the angular frequency. The Hamiltonian of the composite system of A and B is written as [7] ω ω ÀÁ H ¼ ðÞ¼σ Á τ σ τ þ σ τ þ σ τ ð6:17Þ 2 2 x x y y z z

In the following section, the Pauli matrices are derived in a Hilbert space. The spin operator σz in the coordinate z is a (2 Â 2) Pauli matrix with the eigenvectors of spins |ui and |di at the eigenvalues λu ¼ +1 and λd ¼À1, respectively (see Fig. 6.3). The orthonormal eigenvectors in the direction z are deﬁned as 1 0 jiu jizþ ¼ ; jid jizÀ ¼ 0 1 ) hiujd ¼ hiz þjzÀ ¼ 0; hidju ¼ hiz Àjzþ ¼ 0

The equation of the spin operator σz is written as σ σ 1 1 σ jiu ¼þ1 jiu , z,11 z,12 ¼ ; z σ σ 0 0 z,21 z,22 ð6:18Þ σz,11 σz,12 0 0 σz jid ¼À1 jid , ¼À σz,21 σz,22 1 1

Solving Eq. (6.18), one obtains the spin operator σz and τz. 10 σ σ ¼ ¼ τ τ ð6:19aÞ z 3 0 À1 z 3

Analogously, the other spin operators result as 01 σ σ ¼ ¼ τ τ ; x 1 10 x 1 ð6:19bÞ 0 Ài σ σ ¼ ¼ τ τ y 2 i 0 y 2

The Pauli matrices have the following properties that are easily proved: 6.7 Combined State Vectors 265

σ2 σ2 σ2 : x ¼ y ¼ z ¼ I unitary matrix σxσy þ σyσx ¼ 0; σyσz þ σzσy ¼ 0; σzσx þ σxσz ¼ 0 ÂÃσxσy ¼ iσz; σyσzÂÃ¼ iσx; σzσx ¼ iσy σx; σy ¼ 2iσz; σy; σz ¼ 2iσx; ½¼σz; σx 2iσy

Equations (6.19a) and (6.19b) delivers the terms that result from the spin operators act on the basis kets of |ui and |di.

σzjiu ¼ jiu ; σzjid ¼Àjid σxjiu ¼ jid ; σxjid ¼ jiu ð6:20Þ σyjiu ¼ idji; σyjid ¼Àiuji

6.7 Combined State Vectors

To study quantum entanglement between two entangled subsystems, it is necessary to deﬁne a composite system (whole system) that involves in the subsystems. In the composite system, the combined state vectors of two state vectors of the particles are used. Similarly, the basis vectors of the composite system are the combined basis vectors of the subsystems. The following symbols are used by L. Susskind [7]. Let |ai,|bi be the basis vectors of the subsystems SA and SB, respectively. Both subsystems SA and SB belong to the Hilbert spaces HA and HB. It is impossible to describe the entangled state vector of the composite system in a pure basis state either of the subsystem SA or the subsystem SB. In this case, the outcome of the measurement is random with each possibility having a probability of 50 % of both collapsed states of the composite system. Particles with integer spins of h,2h,3h, and so on that are called bosons (e.g. photons) comply with the Bose-Einstein statistics, which involves counting particles, not counting wave frequencies. On the contrary, the Fermi-Dirac statistics, which describes the energy distribution of identical particles with half-integer spins of (1/2)h, (3/2)h, (5/2)h, and so on in a system at thermodynamic equilibrium is generally applied to quantum mechanics. Such particles are called fermions, e.g. electrons, protons, and neutrons that obey the Fermi-Dirac statistics. Using the Fermi-Dirac statistics, one of the entangled normalized states of the composite system can be expressed in the superimposed collapsed states using the basis vectors of each subsystem.

1 ÀÁ jiΨ ¼ pffiffiffi jid jiu À jiu jid 2 HA Â HB AB 2 A B A B

This entangled state vector of the composite system indicates that

– if the measured state of SA is |diA, the entangled state collapses onto |diA |uiB; 266 6 Tensors and Bra-Ket Notation in Quantum Mechanics

– if the measured state of SA is |uiA, the entangled state collapses onto |uiA |diB. The combined basis vector in the composite system is deﬁned as the tensor product of two basis vectors of its subsystems.

jiab jia jib 2 HA Â HB ð6:21Þ

Figure 6.3 shows the orthonormal bases in a Hilbert space H. They are used for the orthonormal basis kets (eigenkets) in quantum mechanics. – Basis kets up and down: 1 jiu jizþ ¼ ; 0 ð6:22aÞ 0 jid jizÀ ¼ 1

– Basis kets right and left: 1 1 1 jir jixþ ¼ pffiffiffiðÞ¼jiu þ jid pffiffiffi ; 1 2 2 ð6:22bÞ 1 1 1 jil jixÀ ¼ pffiffiffiðÞ¼jiu À jid pffiffiffi 2 2 À1

– Basis kets inner and outer: 1 1 1 jii jiyþ ¼ pffiffiffiðÞ¼jiu þ idji pffiffiffi ; i 2 2 ð6:22cÞ 1 1 1 jio jiyÀ ¼ pffiffiffiðÞ¼jiu À idji pffiffiffi 2 2 Ài

The combined state vector of two single state kets of |ai and |bi is calculated as

jiab ¼ jia jib 0 1 a1b1 B C a1 b1 B a1b2 C ð6:23Þ ¼ ¼ @ A; 8ai, bj 2 C a2 b2 a2b1 a2b2

The combined state vector has four components (N2 ¼ 22) in a two-dimensional Hilbert space (N ¼ 2) with two indices i and j, cf. Eq. (2.60). Using Eqs. (6.22a) and (6.23), the combined basis kets of |ui and |di in the Hilbert space H result as 6.7 Combined State Vectors 267 0 1 0 1 0 1 0 1 1 0 0 0 B 0 C B 1 C B 0 C B 0 C jiuu ¼ B C; jiud ¼ B C; jidu ¼ B C; jidd ¼ B C ð6:24Þ @ 0 A @ 0 A @ 1 A @ 0 A 0 0 0 1

Generally, the entangled state vector can be expressed in the combined basis kets as 0 1 0 1 0 1 0 1 0 1 a1b1 1 0 0 0 B C B C B C B C B C B a1b2 C B 0 C B 1 C B 0 C B 0 C jiab ¼ @ A ¼ a1b1@ A þ a1b2@ A þ a2b1@ A þ a2b2@ A a2b1 0 0 1 0 a2b2 0 0 0 1

¼ a1b1jiuu þ a1b2jiud þ a2b1jidu þ a2b2jidd X2 X2 ¼ aibj ei ej i¼1 j¼1 ð6:25Þ

Furthermore, let A and B be two operators of second-order tensors in a two-dimensional Hilbert space. The tensor product of two tensors is calculated as (cf. Chap. 2).

i j k l A B ¼ ÀÁAijg g Bklg g : i j k l ð6 26Þ ¼ AijBkl g g g g

Using Eq. (6.26), the tensor product A B results as a fourth-order tensor with 16 tensor elements (N4 ¼ 24) in a two-dimensional Hilbert space (N ¼ 2) with four indices i, j, k, and l. A11 A12 B11 B12 A B ¼ 2 HA Â HB 0A21 A22 B21 B22 1 A11B11 A11B12 A12B11 A12B12 B C ð6:27Þ B A11B21 A11B22 A12B21 A12B22 C ¼ @ A A21B11 A21B12 A22B11 A22B12 A21B21 A21B22 A22B21 A22B22

Furthermore, the tensor product of two operators that act on the state vectors is calculated using Eq. (6.27)as

Aaji Bbji¼ ðÞA B jiab ð6:28Þ

Proof Using Eq. (6.23), the RHS of Eq. (6.28) results as 268 6 Tensors and Bra-Ket Notation in Quantum Mechanics 0 10 1 A11B11 A11B12 A12B11 A12B12 a1b1 B CB C B A11B21 A11B22 A12B21 A12B22 CB a1b2 C ðÞA B jiab ¼ @ A@ A A21B11 A21B12 A22B11 A22B12 a2b1 0 A21B21 A21B22 A22B21 A22B22 a2b12 A11B11a1b1 þ A11B12a1b2 þ A12B11a2b1 þ A12B12a2b2 B C B A11B21a1b1 þ A11B22a1b2 þ A12B21a2b1 þ A12B22a2b2 C ¼ @ A A21B11a1b1 þ A21B12a1b2 þ A22B11a2b1 þ A22B12a2b2 A21B21a1b1 þ A21B22a1b2 þ A22B21a2b1 þ A22B22a2b2

The single terms on the LHS of Eq. (6.28) are calculated as A A a A a þ A a Aaji¼ 11 12 1 ¼ 11 1 12 2 ; A21 A22 a2 A21a1 þ A22a2 B B b B b þ B b Bbji¼ 11 12 1 ¼ 11 1 12 2 B21 B22 b2 B21b1 þ B22b2

Using Eq. (6.23), the tensor product of two above terms on the LHS results as 0 1 ðÞÁA11a1 þ A12a2 ðÞB11b1 þ B12b2 B C B ðÞÁA11a1 þ A12a2 ðÞB21b1 þ B22b2 C Aaji Bbji¼ @ A ðÞÁA21a1 þ A22a2 ðÞB11b1 þ B12b2 ðÞÁA21a1 þ A22a2 ðÞB21b1 þ B22b2

Having compared the RHS to the LHS of Eq. (6.28), one obtains Aaji Bbji¼ ðÞA B jiab ðÞq:e:d:

Let σz be an observable (¼ measured quantity) operator in the subsystem SA in which |di is a state vector. In the subsystem SB,|ui is a state vector. If the subsystem SB does not affect the state vector |di of the subsystem SA, the combined entangled state vector of the composite system SAB is derived in the following section, as shown in Fig. 6.6. The observable operator σz acts on the state vector |di in the subsystem SA; therefore, the measured result of the entangled state vector |di in the composite system SAB is calculated using Eqs. (6.20) and (6.28)as

Fig. 6.6 Entangled state σ ⊗ ( z I) du vector without effect of SB in a composite system σ z d I u S S A B 6.7 Combined State Vectors 269

ðÞσ I jidu ¼ σ jid Ijiu z z ð6:29Þ ¼Àjid jiu ¼Àjidu where I is the identity operator. At ﬁrst, the observable operator that acts on the combined basis state is calculated using Eqs. (6.19a) and (6.27)as 0 1 10 0 0 10 10 B 01 0 0C σ I ¼ ¼ B C z 0 À1 01 @ 00À10A 00 0 À1

Using Eq. (6.24), one obtains 0 1 0 1 0 1 10 0 0 0 0 B 01 0 0C B 0 C B 0 C ðÞσ I jidu ¼ B C Á B C ¼ÀB C ¼Àjidu z @ 00À10A @ 1 A @ 1 A 00 0 À1 0 0

Similarly, one calculates the following terms:

ðÞσ I jiuu ¼ jidu; ðÞσ I jiud ¼ jidd ; x x ð6:30aÞ ðÞσ I jidu ¼ jiuu ; ðÞσ I jidd ¼ jiud ÀÁx ÀÁx σ I jiuu ¼ iduji; σ I jiud ¼ iddji; ÀÁy ÀÁy ð6:30bÞ σy I ji¼Àdu iuuji; σy I ji¼Àdd iudji ðÞσ I jiuu ¼ jiuu ; ðÞσ I jiud ¼ jiud ; z z ð6:30cÞ ðÞσz I jidu ¼Àjidu; ðÞσz I jidd ¼Àjidd

The observable operators σz and τx act on the state vectors |d i and |u i of the subsystems, respectively (see Fig. 6.7). Using Eq. (6.28), the combined entangled state vector in the composite system SAB is calculated as

σzjid τxjiu ¼ ðÞσz τx ðÞjid jiu ð6:31Þ ¼ ðÞσz τx ji¼Àdu jidd

Proof According Eq. (6.20), the LHS of Eq. (6.31) is written as

σzjid τxjiu ¼Àjid jid ¼Àjidd ðÞq:e:d:

Using Eqs. (6.19a), (6.19b) and (6.27), the observable operator that acts on the combined basis state is calculated as 270 6 Tensors and Bra-Ket Notation in Quantum Mechanics

Fig. 6.7 Combined (σ ⊗τ ) du entangled state vector in a z x composite system σ τ z d x u S S A B

0 1 01 0 0 10 01 B 10 0 0C σ τ ¼ ¼ B C z x 0 À1 10 @ 00 0 À1 A 00À10

Using Eq. (6.24), one obtains 0 1 0 1 0 1 01 0 0 0 0 B 10 0 0C B 0 C B 0 C ðÞσ τ jidu ¼ B C Á B C ¼ÀB C ¼Àjidd z x @ 00 0 À1 A @ 1 A @ 0 A 00À10 0 1

A general combined quantum state vector in the composite space is expressed as

Ψ Ψ Ψ ji¼ jiÀÁA jiB 2 HA Â HÀÁB ψ ψ ψ ψ ¼ ÀÁA, ujiu þ A, djid ÀÁB,ujiu þ B,djid ψ ψ ψ ψ : ¼ ÀÁA,u B, u jiu jiu þ ÀÁA, u B,d jiu jid ð6 32Þ ψ ψ ψ ψ þ A, d B, u jid jiu þ A,d B,d jid jid ψ ψ ψ ψ uujiuu þ udjiud þ dujidu þ ddjidd

Due to normalized state vector, the relation between the state components is written as

ψ * ψ ψ * ψ ψ * ψ ψ * ψ : uu Á uu þ ud Á ud þ du Á du þ dd Á dd ¼ 1 ð6 33Þ

6.8 Expectation Value of an Observable

Quantities, such as position, momentum, spin, polarization, energy, and entropy that can be measured by an apparatus, are called observables. The observables of entangled particles are correlated with each other in the whole system. An action on any particle has strong inﬂuences on the other particles and vice versa in the composite system. Quantum mechanics is intrinsically random; therefore, the expectation value is used for measurement of observables in the statistical quantum world. 6.8 Expectation Value of an Observable 271

Let L be an observable operator that has the eigenvalues λi for i ¼ 1,...,N and their corresponding eigenkets |λii. The expectation value (also average value) of the observable L in a normalized state vector |Ψi at the measurement is deﬁned as (see Fig. 6.8)

þ1ð hiL L ¼ hjΨ LjiΨ ¼ ψ *ðÞÁx; t L Á ψðÞx; t dx À1 : þ1ð ð6 34Þ ¼ L Á kkψðÞx; t 2dx À1 where the probability density of the normalized state vector |Ψi satisﬁes

þ1ð kkψðÞx; t 2dx ¼ 1: À1

Note that the expectation value of an observable L is the average value of repeated measurements on a set of many identically prepared systems. In general, the more the repeated measurements are, the better the expectation value is achieved. Some examples of expectation value of an observable are given here. – The expectation value of the particle position x is written as

þ1ð hix ¼ x ¼ x Á kkψðÞx; t 2dx À1

– The expectation value of the particle momentum p is written as

þ1ð hip ¼ p ¼ p Á kkψðÞx; t 2dx À1

Trace of the observable operator L is deﬁned as the sum of all diagonal elements of the operator.

Fig. 6.8 Expectation value Observable Apparatus Expectation value of an observable L in a state L Ψ | i Ψ λ = Ψ Ψ i L L λ i 272 6 Tensors and Bra-Ket Notation in Quantum Mechanics X X TrL ¼ hji Lji¼i Lii ð6:35Þ i i

According to Eq. (6.35), trace of the outer product of the state bra and ket and the operator L is written as X X Tr½¼jiΨ hjΨ L hji jiΨ hjΨ Lji¼i hjΨ Ljii hiijΨ Xi i : ¼ hjΨ LIijiΨ ¼ hjΨ LIjiΨ ð6 36Þ i ¼ hjΨ LjiΨ ¼ hiL ¼ L

Equation (6.36) describes the expectation value or average value of the outcome of the observable L in a state vector |Ψi at the measurement. Let |Ψi be a state vector in a Hilbert space H. It can be written as X jiΨ ¼ αijiλi i

Multiplying both sides by the observable L, one obtains X LjiΨ ¼ αiLjiλi i

Using the characteristic equation of the observable L at the eigenvalue λi, one obtains

Ljiλi ¼ λijiλi

The expectation value of L of the state vector |Ψi at the measurement results as

þ1ð hiL ¼ hjΨ LjiΨ ¼ L Á kkψðÞx; t 2dx X À1 X X ¼ hjΨ α Ljiλ ¼ α* λ α Ljiλ i i j j i i : X i jX i ð6 37Þ α*α λ λ λ α*α λ δ ¼ j i i jj i ¼ j i i ji Xi, j X i, j α*α λ ψ ; 2λ ¼ i i i ¼ kkiðÞx t i i i where ψ(x,t) is a wave function of which amplitude is calculated as sðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kkψðÞx; t ¼ ψ *ðÞÁx; t ψðÞx; t dx in which the complex conjugate of ψ(x,t) is defined as ψ*(x,t). 6.9 Probability Density Operator 273

Substituting the relative probability P(λi) in Eq. (6.6) into Eq. (6.37), one obtains X X α*α λ λ λ : hiL ¼ i i i ¼ PðÞi i ð6 38Þ i i

Finally, the expectation value of L in a state vector |Ψi is summarized as X hiL ¼ L ¼ hjΨ LjiΨ ¼ TrjiΨ hjΨ L ¼ PðÞλi λi ð6:39Þ i

6.9 Probability Density Operator

The probability density operator associates with an observable to generate the expectation value (average value) of the observable in a state vector.

6.9.1 Density Operator of a Pure Subsystem

Let |Ψi be a state vector in a pure subsystem Sa in a Hilbert space HA. The state vector is written in the basis ket |ani as

X þ1ð þ1ð jiΨ ¼ αnjian ¼ jix hjx Á jiΨ dx ¼ jix ψðÞx dx ð6:40Þ n À1 À1

The density operator ρ is deﬁned as the outer product of ket and bra of the state vector |Ψi.

ρ jiΨ hjΨ ð6:41Þ

Substituting Eq. (6.40) into Eq. (6.41), the density operator is rewritten as X ρ Ψ Ψ α*α jihj¼ n njian hjan X À Á n ð6:42Þ ¼ P an jian hjan n 274 6 Tensors and Bra-Ket Notation in Quantum Mechanics

in which P(an) is the relative probability of the state vector in the basis ket is deﬁned in Eq. (6.6).

α*α PaðÞ¼n n n

Using Eqs. (6.39) and (6.41), the expectation value of an observable L in the state vector |Ψi is calculated as

hiL ¼ L ¼ TrjiΨ hjΨ L ¼ TrðÞρL ð6:43Þ

Therefore, the density operator is very important for calculating the expectation value of an observable. For this reason, the density operator is dealt with in the following section. Due to orthogonality of the basis kets, the elements of the density operator are calculated using Eq. (1.78) as ρ ρ Ψ Ψ ij ¼ hjai aj ¼ÀÁhiaij jaj : ψ ψ * α α* α*α ð6 44Þ ¼ ðÞÁai aj ¼ i j ¼ j i 2 R

An example is given here to show how to calculate the density operator. Let |Ψi be a normalized state vector that is written as 1 1 ji¼Ψ pffiffiffi ) kk¼jiΨ 1 2 À1

The orthonormal basis kets in the direction z result from Eq. (6.22a)as 1 0 jiu jizþ ¼ ; jid jizÀ ¼ 0 1

Using Eq. (6.44), the elements of the density operator are calculated as

ρ ¼ hju ρjiu ¼ hiujΨ hiΨju uu ! ! 1 1 1 1 ¼ ðÞ10pffiffiffi Á pffiffiffiðÞ1 À1 2 À1 2 0 1 1 1 ¼ pffiffiffiðÞÁ1 þ 0 pffiffiffiðÞ¼1 þ 0 2 2 2

Similarly, one obtains 6.9 Probability Density Operator 275

1 ρ ¼ hju ρjid ¼ hiujΨ hiΨjd ¼À ; ud 2 1 ρ ¼ hjd ρjiu ¼ hidjΨ hiΨju ¼À ; du 2 1 ρ ¼ hjd ρjid ¼ hidjΨ hiΨjd ¼ dd 2

The density operator ρ in the pure subsystem results as ρ ρ 1 ρ uu ud þ1 À1 : ¼ ρ ρ ¼ du dd 2 À1 þ1

6.9.2 Density Operator of an Entangled Composite System

Quantum entanglement occurs in a composite system SAB that consists of the entangled subsystems SA and SB. The calculation of the density operator of an entangled state vector of a composite system is quite complicated. However, we have it done step by step. The entangled normalized state vector of a composite system SAB is deﬁned as X jiΨab ¼ ψðÞab jia jib Xab ð6:45Þ ¼ ψðÞab jiab 2 SA SB ab where |abi is the basis kets in the composite system; ψ(ab) is the wave function of the state vector in the composite system. The expectation value of an observable L in the entangled state vector |Ψabi in the composite system is calculated using Eq. (6.39)as L ¼ L ¼ Ψ 0 0 L Ψ hiSAB SAB a b jiab X D 0 0 0 0 ¼ ψ * a b a b Ljiab ψðÞab 0 a0 b , ab ð6:46Þ X * 0 0 ψ 0 ψ a b ÁLa0 b , ab Á ðÞab 0 a0 b , ab where ha0b0| is the basis bra in the composite system that is deﬁned as D D D 0 0 0 0 a b a b 276 6 Tensors and Bra-Ket Notation in Quantum Mechanics

In the following section, the observable L only acts on the subsystem SA; and the 0 subsystem SB does not affect the observable L. In this case, the basis bra hb |is identical to hb|. Therefore, the expectation value of L in the entangled normalized state vector |Ψabi becomes Ψ 0 Ψ hiL S ¼ LSA ¼ a b Ljiab A X 0 : ψ * ψ ð6 47Þ ¼ a b ÁLa0 ,a Á ðÞab a0 , ab

The probability of the entangled state vector |Ψabi in the composite system SAB is calculate as

PabðÞ¼ψ *ðÞÁab ψðÞab X X : ) PabðÞ¼ ψ *ðÞÁab ψðÞ¼ab 1 ð6 48Þ ab ab

The density operator ρ in the composite system SAB is deﬁned as ρ Ψ Ψ 0 0 6:49 SAB jiab a b ð Þ

Similar to Eq. (6.44), the elements of the density operator of the composite system are calculated using Eq. (6.49)as[7] X E X E 0 0 0 0 ρ 0 0 ab ρ a b ab Ψ Ψ 0 0 a b a b , ab ¼ hjSAB ¼ hjjiab a b 0 0 a0 b , ab a0 b , ab X DEX ð6:50Þ 0 0 * 0 0 Ψ Ψ 0 ψ ψ ¼ hiabj ab a0 b ja b ¼ a b Á ðÞab 0 0 a0 b , ab a0 b , ab

If only the observable L acts on the subsystem SA, the elements of the reduced density operator of the one-half mixed state of SA results from Eq. (6.50)as ρ jiΨ Ψ 0 ) SA Xak a k DEX ρ Ψ Ψ 0 ψ * 0 ψ ð6:51Þ a0 a ¼ hiakj ak a0 kja k ¼ a k Á ðÞak k k in which k is the integrating variable that is deﬁned as

hjk ¼ hju , hjd ; jik ¼ jiu , jid 0 ; a ¼ hju , hjd jia ¼ jiu , jid 0 ) a k ¼ hjuu; hjud; hjdu; hjdd ) jiak ¼ jiuu ; jiud ; jidu; jidd 6.9 Probability Density Operator 277

An example is given here to show how to calculate the density operator in the composite system. Let |Ψaki be a normalized entangled state vector that is written as 0 1 0 B C 1ffiffiffi 1ffiffiffiB 1 C jiΨak ¼ p ðÞ¼jiud À jidu p @ A 2 2 À1 0 * 1 ) Ψ 0 ¼ jiΨak ¼ pffiffiffiðÞ01À10 a k 2

According to Eq. (6.24), one calculates the combined basis kets in SAB as 0 1 1 B 0 C jiuu ¼ B C ) hjuu ¼ ðÞ1000; @ 0 A 0 0 1 0 B 1 C ji¼ud B C ) hj¼ud ðÞ0100; @ 0 A 0 0 1 0 B 0 C jidu ¼ B C ) hjdu ¼ ðÞ0010; @ 1 A 0 0 1 0 B 0 C jidd ¼ B C ) hjdd ¼ ðÞ0001. @ 0 A 1 The wave functions are calculated. 0 1 0 B C 1 B 1 C 1 ψðÞ¼uu ðÞÁ1000 pffiffiffi @ A ¼ pffiffiffi Á 0 ¼ 0; 2 À1 2 0 0 1 1 B C * 1 B 0 C 1 ψ ðÞ¼uu pffiffiffi ðÞÁ01À10 @ A ¼ pffiffiffi Á 0 ¼ 0: 2 0 2 0

Analogously, one obtains the other terms 278 6 Tensors and Bra-Ket Notation in Quantum Mechanics

1 1 ψðÞ¼ud pffiffiffi; ψ *ðÞ¼ud pffiffiffi; 2 2 1 1 ψðÞ¼Àdu pffiffiffi; ψ *ðÞ¼Àdu pffiffiffi; 2 2 ψðÞ¼dd 0; ψ *ðÞ¼dd 0:

Using Eq. (6.51), the elements of the reduced density operator are calculated as X ρ ψ * ψ uu ¼ ðÞuk ðÞuk k 1 ¼ ψ *ðÞuu ψðÞþuu ψ *ðÞud ψðÞ¼ud ; X 2 ρ ψ* ψ ud ¼ ðÞuk ðÞdk k ¼ ψ*ðÞuu ψðÞþdu ψ *ðÞud ψðÞ¼dd 0; X ρ ψ* ψ du ¼ ðÞdk ðÞuk k ¼ ψ*ðÞdu ψðÞþuu ψ *ðÞdd ψðÞ¼ud 0; X ρ ψ * ψ dd ¼ ðÞdk ðÞdk k 1 ¼ ψ *ðÞdu ψðÞþdu ψ *ðÞdd ψðÞ¼dd : 2

Finally, the probability reduced density operator of the subsystem SA in the combined system SAB results as ρ ρ 1 10 ρ ¼ uu ud ¼ SA ρ ρ du dd 2 01

Thus, the element of the squared density operator results as 0 1 1 0 ρ2 @ 4 A ρ ρ2 1 < S ¼ 6¼ a0 a ) Tr S ¼ 1 A 0 1 A 2 4

This result of the elements of the reduced density operators shows that the entangled state vector is an one-half mixed state in the subsystem SA [7]. Using Eqs. (6.43) and (6.51), the expectation value (average value) of an observable L in the entangled state vector |Ψabi in the subsystem SA results as ! ÀÁ X ρ Ψ Ψ 0 : hiL ¼ L ¼ Tr SA L ¼ Tr jiak a k L ð6 52Þ k 6.10 Heisenberg’s Uncertainty Principle 279

6.10 Heisenberg’s Uncertainty Principle

Heisenberg’s uncertainty principle (HUP) shows the relationship between the position of a particle and its momentum state. The more precisely measured the particle position is, the less precisely is its momentum state at the measurements. The momentum of a particle in quantum mechanics results from De Broglie formula as π h 2 Àh : p ¼ λ λ ð6 53Þ where λ is the wavelength of the particle; h (read h bar) is the reduced Planck’s constant or Dirac constant, which is deﬁned as

h Àh ¼ 1:054 Â 10À34Js 2π

The uncertainty principle is based on the standard variation that is recapitulated at ﬁrst. The individual deviation of the particle position from its average value is written as (see Fig. 6.9).

Δx ¼ x À hix ¼ x À x ð6:54Þ

The expectation value (average value) of the particle position x is calculated as X x ¼ hix hjΨ xjiΨ ¼ xiPxðÞi; t i þ1ð þ1ð ð6:55Þ ¼ x Á ψ *ðÞx; t ψðÞx; t dx ¼ x Á kkΨðÞx; t 2dx À1 À1 where x is the operator of the particle position, as deﬁned in Eq. (6.69); P(xi,t) is the relative probability of the particle position at xi at time t; and it is written as

2 PxðÞ¼i; t kkΨðÞxi; t

Using the product rule of differentiation, one obtains ÀÁ ∂kkψ 2 ∂ ψ *ψ ∂ψ ∂ψ * ¼ ¼ ψ * þ ψ ∂t ∂t ∂t ∂t 280 6 Tensors and Bra-Ket Notation in Quantum Mechanics

Fig. 6.9 Probability P(x) function of the particle Δ position x x

x = x x x

The Schrodinger€ equation (cf. Sect. 6.12) gives the following relations

∂ψ iÀh ∂2ψ i ¼ À Vψ ∂t 2m ∂x2 Àh ∂ψ * ÀiÀh ∂2ψ * i ) ¼ þ Vψ * ∂t 2m ∂x2 Àh

Using the product rule of integration, the expectation value of the particle velocity is calculated [6]as

þ1ð dxhi ∂kkψ 2 hi¼v ¼ x dx dt ∂t À1 þ1ð iÀh ∂ ∂ψ ∂ψ * ¼ x ψ * À ψ dx ð6:56aÞ 2m ∂x ∂x ∂x À1 þ1ð ÀiÀh ∂ψ ∂ψ * ¼ ψ * À ψ dx 2m ∂x ∂x À1

The product rule of integration of the function d(ψ*ψ)/dx gives

þ1ð þ1ð ∂ψ ∂ψ * ψ * dx ¼À ψ dx ð6:56bÞ ∂x ∂x À1 À1

Substituting Eq. (6.56b) into Eq. (6.56a), one obtains

þ1ð dxhi ÀiÀh ∂ψðÞx; t hi¼v ¼ ψ *ðÞx; t dx dt m ∂x À1

Therefore, the expectation value (average value) of the particle momentum hpi results as 6.10 Heisenberg’s Uncertainty Principle 281

þ1ð ∂ψðÞx; t p ¼ hip ¼ mvhi¼ÀiÀh ψ *ðÞx; t dx ð6:56cÞ ∂x À1

The squared standard deviation of the particle position x is deﬁned as the expectation value of squared Δx [6, 7], cf. Appendix H. DEX À Á X À Á σ2 Δ 2 Δ 2 2 x ðÞx ¼ xi PxðÞ¼i xi À x PxðÞi X i X À Á iX À Á 2 2 6:57a ¼ xi PxðÞi À 2x xiP xi þ x P xi ð Þ i i i ¼ x2 À 2x2 þ x2 ¼ x2 À ðÞx 2

Therefore, the standard deviation of the particle position x is written as rDEffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 σx ðÞΔx ¼ hix À ðÞx ð6:57bÞ

Analogously, the standard deviation of the particle momentum p results as DE σ2 Δ 2 2 2 : p ðÞp ¼ p À ðÞp ; ð6 58aÞ rDEffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 σp ðÞΔp ¼ hip À ðÞp ð6:58bÞ

In general, the standard deviation of an observable Hermitian operator A is written using Eq. (6.57a)as DEDEÀÁ 2 σ2 ðÞΔA 2 ¼ A À A A ÀÁ ÀÁÀÁ 2 ð6:59aÞ ¼ hjΨ A À A jiΨ ¼ A À A Ψj A À A Ψ hif Aj f A in which the function fA is deﬁned as ÀÁ Ψ : f A A À A ðÞx ð6 59bÞ

Similarly, the standard deviation of a Hermitian operator B is written as ÀÁÀÁ σ2 Ψ Ψ : B ¼ B À B j B À B hif Bj f B ð6 60aÞ with the function fB is deﬁned as ÀÁ Ψ : f B B À B ðÞx ð6 60bÞ 282 6 Tensors and Bra-Ket Notation in Quantum Mechanics

Using the Schwarz inequality, the product of the standard deviations in written as

σ2 σ2 2 : A B ¼ hif Aj f A hif Bj f B kkhif Aj f B ð6 61Þ

At ﬁrst, the RHS of Eq. (6.61) is calculated as ÀÁÀÁ ÀÁÀÁ Ψ Ψ Ψ Ψ hif Aj f B ¼ AÀÁÀ A j B À B ¼ hjA À A B À B ji ¼ hjΨ AB À AB À AB þ A B jiΨ ¼ hjΨ ABjiΨ À BhjΨ AjiΨ À AhjΨ BjiΨ þ hjΨ ABjiΨ ð6:62Þ ¼ hiAB À B A À A B þ A B ¼ hiAB À B A ¼ hiÀAB hiB hiA

Calculating the adjoint of Eq. (6.62), it results ÀÁ * * * hi¼f Bj f A hif Aj f B ¼ hiAB À B A ¼ hiBA À A B ð6:63Þ ¼ hiBA À hiA hiB

Subtracting Eq. (6.62) from Eq. (6.63), one obtains

hif j f À hif j f ¼ hiAB À hiBA ¼ hiAB À BA A B B A ð6:64Þ hi½A; B

The term [A, B] is called the commutator of two Hermitian operators A and B that results from Eq. (6.64)as

½¼A; B AB À BA 6¼ 0 ð6:65Þ

The commutator is sometimes called the Lie bracket (cf. Sect. 3.12.2). If [A, B] 6¼ 0, a complete set of mutual eigenkets of A and B does not exist. In this case, both operators are not commutative with each other. Conversely, if [A, B] ¼ 0, a complete set of mutual eigenkets of A and B exists, in which the operators A and B are commutative. The commutator of the Hermitian operators has the following properties:

½¼ÀA; B ½B; A : anticommutativity ½¼A; A 0 ½¼A þ B, C ½þA; C ½B; C ð6:66Þ ½þA; ½B; C ½þB; ½C; A ½¼C; ½A; B 0 : Jacobi identity ½¼A; BC ½A; B C þ BA½; C ½¼AB; C AB½þ; C ½A; C B

The relation between the commutator (Lie bracket) applied to quantum mechanics and Poisson bracket used in classical mechanics was recognized by Dirac. 6.10 Heisenberg’s Uncertainty Principle 283

½$A; B iÀhAfg; B ð6:67Þ

Next, the norm (absolute value) of the inner product of Eq. (6.62) is calculated to derive Heisenberg’s uncertainty principle (HUP). The inner product is a complex number that can be written as

α β z hif Aj f B ¼ þ i 2 C

Its transpose conjugate is calculated as

* * α β z hif Aj f B ¼ hif Bj f A ¼ À i 2 C

Using Eq. (6.64), one obtains

* ; β z À z hif Aj f B À hif Bj f A ¼ hi½A B ¼ 2i hi½A; B ) β ¼ 2i

Using Eqs. (6.61) and (6.64), one obtains the inequality equation

σ2 σ2 α2 β2 A B kkz kkhif Aj f B ¼ þ 2 β2 hi½A;B ¼ 2i

Therefore,

hi½A; B σ σ 0 ð6:68Þ A B 2i

Finally, let A be the position operator x of particle and B be its momentum operator px. Note that both operators x and px are Hermitian operators. The position operator x of particle at a position x is deﬁned as

þ1ð x xxjihjx dx ð6:69Þ À1

The momentum operator px of particle at a position x is deﬁned using the imaginary unit i as

∂ p ÀiÀh ð6:70Þ x ∂x

In the following section, the transformation of position and momentum coordinates x and p is carried out. 284 6 Tensors and Bra-Ket Notation in Quantum Mechanics

The position wave function of the particle is written using Eq. (6.15)as

þ1ð ψ Ψ Ψ Ψ xðÞ¼x hixj ¼ hixjp hipj ¼ hixjp hipj dp À1 þ1ð þ1ð ψ ψ δ ¼ pðÞp hixjp dp ¼ pðÞp ðÞx À p dp À1 À1

Analogously, the momentum wave function of the particle results from Eq. (6.15) as

þ1ð ψ Ψ Ψ Ψ pðÞ¼p hipj ¼ hipjx hixj ¼ hipjx hixj dx À1 þ1ð þ1ð ψ ψ δ ¼ xðÞx hipjx dx ¼ xðÞx ðÞp À x dx À1 À1

The momentum operator px acting on the state vector |Ψ(x)i is expressed as ∂ hjx p jiΨ ¼ p ψðÞ¼x ÀiÀh ψðÞx x x ∂x

Hence,

∂ψðÞx hjx p jiΨ ¼ÀiÀh x ∂x

Generally, this above equation is valid for any state vectors |Φi and |Ψi. Using partially integrating the RHS term of the below equation, one obtains ð ∂ψ Φ Ψ ϕ* ðÞx hjpxji¼ÀiÀh ðÞx dx ∂x ð ÂÃ * þ1 ∂ϕ ðÞx ¼ÀiÀh ϕ*ψ þ iÀh ψðÞx dx ð À1 ∂x ∂ϕ*ðÞx ¼ iÀh ψðÞx dx ∂x

The above-calculated term is equal to its transpose conjugate (cf. Chap. 1). 6.10 Heisenberg’s Uncertainty Principle 285

ð ∂ϕ*ðÞx hjΦ p jiΨ ¼ iÀh ψðÞx dx x ∂x Ψ Φ { Φ { Ψ ¼ hjpxji¼ hjpx ji { ) px ¼ px ! Hermitian

This result indicates that momentum operator px is a Hermitian operator. Using Eq. (6.65), the commutator of x and px results as ∂ψ ∂ ψ ; ψ ðÞx ½x px ðÞ¼x x ÀiÀh ÀÀiÀh ∂x ∂x ∂ψ ∂ψ ∂x ð6:71Þ ¼Àx iÀh þ x iÀh þ iÀh ψðÞx ∂x ∂x ∂x ¼ iÀhψðÞx

The result of Eq. (6.71) gives the very useful commutator for HUP

; : ½x px ðÞ¼xpx À pxx iÀh ð6 72Þ

This is the fundamental equation of matrix mechanics in quantum physics (Hei- senberg, Born, and Jordan). However, this equation is very peculiar in the everyday experience. Having squared both sides of Eq. (6.72), one obtains using Eq. (6.65) the squared value of the commutator

; 2 2 2 < ½x px ¼ ðÞxpx À pxx ¼ÀÀh 0

This results shows that the squared value of a quantum quantity is negative.Thatis the quantum mathematical language that opens a new view of the quantum world, which differs from the mathematical world of classical physics. In this case, both Hermitian operators x and px are called the canonical conjugate entities that have a canonical commutation relation. Generally, the commutator (Lie bracket) of differential Hermitian operators rk and pl of position and momentum of a particle in an N-dimensional space is written using Kronecker delta as

; δ ; ... : ½¼rk pl iÀh kl k, l ¼ 1, 2, , N

Substituting Eq. (6.72) into Eq. (6.68), the well-known Heisenberg’s uncertainty principle is derived as

hi½x; p hiiÀh Àh Àh σ σ x ¼ ¼ , Δx Á Δp ð6:73Þ x p 2i 2i 2 2 where Δx and Δp are the uncertainties of the position and momentum of the particle, respectively. 286 6 Tensors and Bra-Ket Notation in Quantum Mechanics

In 1927, W. Heisenberg stated that the more precisely measured the position of a particle is, the less precisely is its quantum momentum state at the measurements. The Heisenberg’s uncertainty principle in quantum mechanics is statistically based on the repeated measurements that are carried out in a set of many identically prepared systems

6.11 The Wave-Particle Duality

A quantum object (e.g. photon, light quanta) has both characteristics of wave and particle as well. That is called the wave-particle duality of a quantum object. Sometimes the object behavior is like a wave; and another time, like a particle, but the object itself is the same. In fact, the quantum object with a wave-like or particle-like characteristic depends on how it is considered like in topology and differential geometry (cf. Sect. 4.1). If we consider the object at a distance as a whole without considering it in detail like topologists do, the object responses as wave-like. On the contrary, if we consider it much closer and concern it much more in detail with all smallest scales of it like geometers do, the object reacts as particle-like. In quantum mechanics, Born and Heisenberg preferred the particle-like characteristic for a quantum object; and Einstein, Schrodinger,€ and De Broglie preferred rather the wave-like characteristic for the object [3, 5]. On the one hand, the wave- like properties are found in the phenomena of interference behind two slots and diffraction of light. On the other hand, the particle-like properties are used to explain the photoelectric and Compton effects. Furthermore, Fourier analysis shows that the wave and particle duality is an intrinsic mathematical equivalence that always occurs in all quantum entities [3]. In fact, a quantum object can be considered as either a particle-like or a wave- like behavior. This point of view depends on how to interpret the possibility of the quantum-mechanical formalism. The existence of light quanta denotes that the light wave is quantized into many quanta portions that move together in a waveform. Thus, this quantized wave theory goes well with the quantized particle theory. Both interpretations of the wave-particle duality do not contradict to each other. Further- more, electrons cannot be considered as single particles because their positions could not be exactly determined. In this case, they are regarded as rather a cloud of many electrons than single particles in quantum mechanics. The wave-particle duality is based on wave mechanics (Schrodinger)€ and matrix mechanics (Heisenberg), respectively. They are only a special case of quantum algebra that was developed by Dirac. The quantum algebra involves additions and multiplications of quantum variables, such as canonical coordinates qr and momenta ps of the particles in three-dimensional Hilbert spaces. The Bohr’s complementarity is a fundamental part of nature, in which a quantum object has both particle and wave properties that depend on how one observes it at measurements. As a result, the process of measurement triggers the current 6.11 The Wave-Particle Duality 287 wavefunction from the superposition of eigenstates into one of the component eigenstates. This process is called the wavefunction collapse at the measurement. The unity of the Bohr’s complementarity, Heisenberg’s uncertainty principle, and wavefunction collapse triggered by measurements has been known as the “Copen- hagen interpretation of quantum mechanics”. The fundamental quantum conditions of the quantum variables satisfy the Dirac quantum-algebra equations using Kronecker delta δrs [10]: ; qrqs À qsqr ¼ 0 ; prps À pspr ¼ 0 ; δ : qrps À psqr ¼ iÀhqfgr ps ¼ iÀh rs

The Poisson bracket (PB) of two dynamical variables u and v is deﬁned as, cf. Eq. (6.67) X3P Ài ∂u ∂v ∂u ∂v fgu; v ¼ ðÞuv À vu À h ∂q ∂p ∂p ∂q À r¼1 r r r r where r is the index from 1 to three times of the number of particles P.

6.11.1 De Broglie Wavelength Formula

De Broglie wavelength formula gives the relation between wave and particle characteristics of a free particle by means of its momentum and wavelength. Let a particle be moving with a velocity v < c (light speed). According to the special relativity theory, its relativistic mass m results as

m0 m ¼ m0γ qffiffiffiffiffiffiffiffiffiffiffiffi ð6:74Þ v2 1 À c2 where m0 is the mass at rest (v ¼ 0); γ is the Lorentz factor, which is defined as

1 γ qffiffiffiffiffiffiffiffiffiffiffiffi 1 ð6:75Þ v2 1 À c2

Equation (6.74) denotes that the higher the velocity of the moving mass is, the heavier its relativistic mass (i.e. moving mass) becomes. At rest, its mass equals the mass at rest m0; its relativistic mass goes to inﬁnity as the velocity reaches the light speed. 288 6 Tensors and Bra-Ket Notation in Quantum Mechanics

Multiplying both sides of Eq. (6.74)byc2 and squaring them, one obtains the energy-momentum relation 2 2 4 v 2 4 m c 1 À ¼ m0c c2 : 2 4 2 4 2 2 2 ð6 76Þ , m c ¼ m0c þ ðÞm v c 2 2 2 2 , E ¼ E0 þ p c

Let the particle be a photon that has no mass; i.e. m0 ¼ 0. Hence, E0 ¼ 0. Equation (6.76) is written for a photon as

E ¼ pc ð6:77Þ

The momentum of the particle results as from Eq. (6.77)

E p ¼ ð6:78Þ c

The energy of the light quanta only depends on the light frequency.

E ¼ hv ÀhðÞ¼À2πv hω ð6:79Þ in which h is the Planck constant (¼6.63 Â 10À34 J s); h is the reduced Planck constant or Dirac constant, which is deﬁned as

h Àh 2π

v is the particle frequency; ω is the particle angular frequency (ω ¼ 2πv). Substituting Eq. (6.79) into Eq. (6.78), the particle momentum results as

hv p ¼ ð6:80Þ c

The wavelength λ of the particle is written as c λ ¼ ð6:81Þ v

Substituting Eqs. (6.80) and (6.81), the particle wavelength is calculated as

h h λ ¼ ) p ¼ ð6:82Þ p λ 6.11 The Wave-Particle Duality 289

This is the De Broglie wavelength formula of a free particle. The momentum of a particle with a mass m m0 is deﬁned in a non-relativistic system (v <

p ¼ m0v mv ð6:83Þ

Using Eq. (6.83), the kinetic energy of the particle with a mass m m0 is calculated in a non-relativistic system as 2 2 p2 1 2 1 p p : Eke ¼ m0v ¼ m0 ¼ ð6 84Þ 2 2 m0 2m0 2m

6.11.2 The Compton Effect

The Compton Effect is used to demonstrate that the quantum object has a particle- like behavior rather than a wave-like behavior. The experiment is carried out with a single photon (i.e. light quanta with m ¼ 0) at a frequency of v that collides with a rest electron. Due to the collision, the incident photon is scattered with a scattering angle θ from the incident direction at a frequency of v’. The electron with a mass at rest me is deﬂected from the incident direction of the photon with a velocity v (see Fig. 6.10). Note that all considered particles are relativistic in this experiment. Due to energy conservation, the balance of the relativistic particle energy of the system before and after collision is written as

0 E0,eÀ þ E ¼ EeÀ þ E : 2 2 0 ð6 85aÞ , mec þ hv ¼ ðÞmeγ c þ hv where me is the mass at rest of the electron; c is the light speed¸ and γ is the Lorentz factor in Eq. (6.75). Note that E0,eÀ is the quantum energy of the electron even at rest; and EeÀ is the quantum energy of the electron at moving according to Einstein’s special relativity theory. Equation (6.85a) can be written as 0 2 ÀÁ 2γ2 hv hv hΔv 2 me ¼ me þ c2 À c2 me þ c2 2m hΔv h2ðÞΔv 2 2m hΔv ¼ m2 þ e þ m2 þ e e c2 c4 e c2

Multiplying both sides of the equation by c2, one obtains 290 6 Tensors and Bra-Ket Notation in Quantum Mechanics

scattered photon ′ 2 hν E − = m c ′ = 0,e e p c p − = (m γ )v e e E′ = hν ′ incident photon θ − hν = ν e p = E h c 2 E − = (m γ )c e e

deflecting electron e−

Fig. 6.10 The Compton Effect of relativistic particles

2γ2 2 2 2 Δ : me c mec þ 2meh v ð6 85bÞ in which

0 Δv v À v

The conservation of momentum before and after collision is expressed as

~ ~0 ~ : p ¼ p þ peÀ ð6 86aÞ

Using the law of cosines for a triangle, one obtains the relation of the momenta.

0 2 0 hv 2 hv hv hv ðÞm γv 2 ¼ þ À 2 Á Á cos θ ð6:86bÞ e c c c c

The RHS of Eq. (6.86b) is simpliﬁed as 0 2 2 h 0 hv hv RHS ¼ v À v þ 2 Á Á ðÞ1 À cos θ c2 c c 2h2vv0 ðÞ1 À cos θ ð6:86cÞ c2

Using the relation ÀÁ 1 v2 c2 γ2 1 c2 1 γ2v2 À ¼ v2 À ¼ v2 ¼ 1 À c2 1 À c2 and Eq. (6.85b), the LHS of Eq. (6.86b) is written as 6.11 The Wave-Particle Duality 291

2γ2 2 2 2 γ2 2 2γ2 2 2 me v ¼ mÀÁec ðÞ¼À 1 me c À me c 2 2 Δ 2 2 : mec þ 2meh v À me c ð6 87Þ 2mehΔv

Substituting Eqs. (6.87) and (6.86c) into Eq. (6.86b), one obtains the scattered frequency of the photon.

2h2vv0 2m hΔv ðÞ1 À cos θ e c2

Using the trigonometric formula, one obtains the scattered photon frequency.

0 0 θ Δ 0 hvv θ 2hvv 2 v v À v 2 ðÞ¼1 À cos 2 sin mec mec 2 0 v v ð6:88Þ ) v ¼ θ ¼ π θ v 2hv 2 4 Àhv 2 1 þ 2 sin 1 þ 2 sin mec 2 mec 2

Substituting Eq. (6.81) into Eq. (6.88), it gives the scattered photon wavelength.

cc 2h 0 θ 0 1 1 λλ 2 v À v ¼ c Á À 0 2 sin λ λ mec 2 π θ 4 Àh 2 ¼ 0 sin meλλ 2

Thus,

0 4πÀh θ h Δλ λ À λ ¼ sin 2 ¼ ðÞ1 À cos θ mec 2 mec ð6:89Þ 0 4πÀh θ h ) λ ¼ λ þ sin 2 ¼ λ þ ðÞ1 À cos θ λ mec 2 mec

The quantity of h/(mec) in Eq. (6.89) is called the Compton wavelength of the À12 electron. This wavelength is equal to about 2.43 Â 10 m; the quantity of h/(mec), the reduced Compton wavelength of the electron equals ca 3.9 Â 10À13 m. In non-relativistic systems, the kinetic energy of any moving mass m m0 at a velocity v c is calculated as the difference between the quantum energies of the moving object and the rest object. 292 6 Tensors and Bra-Ket Notation in Quantum Mechanics

2 2 Eke ¼ E À E0 ¼ ðÞmγ c 0À mc 1 B c2 C ¼ mc2ðÞ¼γ À 1 m @qffiffiffiffiffiffiffiffiffiffiffiffi À c2A v2 1 À c2 0 1 2 2 1 v2 Bc À c 1 À 2 C B 2 c C 1 2 m @ A mv 1 v2 2 1 À 2 c2

The Compton Effect indicates that the frequency of scattered photon is less than its incident frequency due to the necessary energy that needs to deﬂect the electron from the rest position. Obviously, the lower the wave frequency is, the longer the wavelength of the particle is, as shown in Eqs. (6.88) and (6.89). This effect could be easily explained by Einstein’s light quantum theory (photoelectric effect) based on the particle-like theory. In this case, the wave-like theory fails to explain this Compton Effect. However, the particle-like theory cannot explain such phenomena of the light interference and diffraction that could be explained by means of the wave-like theory [3]. According to the wave-like approach, the energy of the quantum object is on the one hand written in its frequency as

E ¼ hv

On the other hand, the energy of the quantum object with the mass at rest m0 can be expressed in the particle-like approach in a relativistic system as, cf. Eq. (6.76) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 4 2 E ¼ p c þ m0c ¼ Eke þ m0c

The momentum p of the quantum object with the mass at rest m0 is defined in a relativistic system as m v p ¼ qffiffiffiffiffiffiffiffiffiffiffiffi0 v2 1 À c2

Using the Taylor series, the kinetic energy Eke of the quantum object with the mass at rest m0 is calculated in a relativistic system as 6.11 The Wave-Particle Duality 293 "# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 4 2 2 p E ¼ p c þ m c À m0c ¼ m0c 1 þ À 1 ke 0 m0c "# 2 4 2 1 p 1 p m0c 1 þ À ÆÁÁÁÀ1 2 m0c 8 m0c p2 p4 À 3 2 ÆÁÁÁ 2m0 8m0c

The second term on the RHS of the above equation is the relativistic correction term of the kinetic energy that is considered in the relativistic system.

6.11.3 Double-Slit Experiments with Electrons

The double-slit experiments (Young’s experiment) are carried out in which electrons are generated by an electron generator, as shown in Fig. 6.11. The electrons go through two slits A and B. At ﬁrst, only the slit A is opened and the slit B is closed, the probability function PA(z) of the electrons from the slit A is counted on the screen after a long observing time. The same experiment is repeated at closing the slit A and opening the slit B, the probability function PB (z) of the electrons from the slit B is displayed on the screen. Note that the maximum of the probability distribution on the screen locates in the middle of the slit axis for each experiment. In this case, no interference pattern that is caused by the interference of the waves of the electrons from A and B is found on the screen (see Fig. 6.11a). The probability function Pi(z) of the electrons for ﬁnding the electron at position z in a long observing time is the squared amplitude of the wave function ψi(z), cf. Eq. (6.6).

ψ ; 2 PiðÞz kkiðÞz t for i ¼ A, B

The experiment with opening both slits A and B is repeated. After a long observing time, the probability function PAB(z) of the electrons is very different from the earlier experiments (see Fig. 6.11b). The interference pattern (interference stripes) of the diffracting electrons from both slits occurs on the screen, in which no electron hits in some certain positions. The results show that the electrons behave like waves behind the slits where the waves of the electrons coming from A and B interfere with each other, add the amplitudes of waves together at the crests (bright spots), and cancel them at the troughs (dark spots) of the interference fringes (Young’s experiment). The amplitudes of the electrons are superimposed on the screen in form of the interference pattern. 294 6 Tensors and Bra-Ket Notation in Quantum Mechanics

Fig. 6.11 Double-slit (a) (b) experiments with an P (z) electron generator PA(z) AB z Slit A

Electron Dark generator spot

Bright spot

Interference Slit B pattern

PB (z) Screen

In case of the interference pattern, the probability function of the electrons for ﬁnding the electron at position z in a long observing time is the squared amplitude of the sum of the wave functions ψA(z) and ψB(z). The probability function PAB(z) at position z is calculated as, cf. Appendix F.

ψ ; 2 ψ ; ψ ; 2 PABðÞz kkABðÞz t ¼ kkAðÞþz tÀÁBðÞz t ψ 2 ψ 2 ψ ψ * ¼ kkA þ kkB þ 2ReÀÁA B ψ ψ * ¼ PAðÞþz PBðÞþz 2ReÀÁA B ψ * ψ ¼ PAðÞþz PBðÞþz 2Re A B

The third term on the RHS of the above equation generates the quantum- interference pattern, as displayed in Fig. 6.11b. The additional term can be positive or negative. Note that the interference term is mathematically created; and it only exists in quantum mechanics [9]. This quantum-interference term is not taken into account in the probability function PAB(z) in the classical probability theory. So far so good, such experimental results can be explained using the wave theory of classical mechanics. However, some points of view about the particle-like behavior should be discussed in the following section. How about is the light beam (quanta photon) in quantum mechanics? Note that the light beam consists of many quanta photons; each of them has a certain portion of energy (quanta entity) according to Planck and Einstein. How about are the photoelectric and Compton effects, in which the particle-like behavior is preferred? To make sure that the electron beam is made of single particles (quanta), the intensity of the electron generator is reduced so that a single electron is generated one after another. Each electron of them travels through the slit A or B at time. The experiment is repeated at opening both slits A and B. At ﬁrst, no interference pattern occurs on the screen. However, the interference pattern takes places again on the screen after a long observing time [11]. The question emerges in the experiment why a single electron can interfere with itself to cause the interference pattern. 6.11 The Wave-Particle Duality 295

Fig. 6.12 Double-slit Electron PAB,0 (z) experiments with electron detector A z detectors Slit A

Electron generator

No interference pattern Slit B Electron detector B Screen

Would the single electron split into two halves? Each of them goes through both slits at the same time. Then, these two halves of the electron interfere with each other to generate the interference pattern. To answer this question, two detectors are installed at both opening slits A and B. Every electron travels through the slit is detected by the electron detectors using the light beam at a frequency vd acting upon the moving electron. As a result, no split of the electrons occurs; and the interference pattern does not happen on the screen at a long observing time. The probability function PAB,0(z) of the electrons is shown in Fig. 6.12. In case without the interference pattern, the probability function PAB,0(z) of the electrons for ﬁnding the electron at position z in a long observing time is the sum of the squared amplitudes of the wave functions ψA(z) and ψB(z).

ψ ; 2 ψ ; 2 PAB,0ðÞz kkAðÞz t þ kkBðÞz t

This result shows that such detecting the electrons that travel through the slits may destroy the interference pattern. Possibly, the detection of the exact positions of the electrons prevents their interference ability. The experiment is repeated once again with reducing the intensities of the detector light beams so that both detectors have not affected the electrons travelling through the slits any longer. To lower the intensity of the detecting light in W/m2 or J/(m2s), either the number of light quanta or the light frequency vd acting upon the electron must be reduced since the energy of the light beam in J equals n times hvd in which n is the number of the light quanta. Even reducing one photon for each electron passing through the slit, the interference pattern does not occur on the screen. In another way, the detecting light frequency vd must be reduced at least its wavelength λd is larger than the distance between two slits. In this case, we do not know which electron travels through which slit; i.e., the positions of the electrons are not determined at the measurement. That means the detecting light does not affect the electrons travelling through the slits. After an observing time, the interference pattern returns on the screen. The same result is obtained if both electron detectors at the slits are removed at the same time. Note that if the electrons are not observed or detected, they could be able to 296 6 Tensors and Bra-Ket Notation in Quantum Mechanics interfere with each other to generate the interference fringes that lead to the interference pattern on the screen. Using the Heisenberg’s uncertainty principle (HUP), the phenomena occurring in the double-slit experiments with the electron detectors could be explained. In the case of detecting the travelling electrons through each slit, the positions of the electrons are quite determined, which electron travels through the slit A or the slit B. Therefore, their momenta in direction z cannot be exactly determined according to the HUP, cf. Eq. (6.73)as

Àh Δp z 2Δz

The momentum uncertainty of the electron travelling through the slit is usually equal to the momentum uncertainty of the photon beaming on the electron that depends on the De-Broglie wavelength λd of the detecting light according to Eq. (6.82). Thus, the momentum uncertainty of the electron is written as

Δ Δ h pz pd ¼ λd

Therefore, the velocity uncertainty of the electron on the screen is calculated as Δ pz h Δvz ¼ meÀ meÀ λd

At detecting the electrons, the position uncertainty Δz of the electrons in direction z is in the order of the wavelength of the detecting light λd (Δz ~ λd). Therefore, the momentum uncertainty of the electrons in direction z satisﬁes the HUP. π Δ h 2 Àh pz ¼ λd λd Àh Àh ðÞq:e:d: 2λd 2Δz

At a measurement using a wavelength of the detecting violet light λd ¼ 400 nm, the À31 À34 electron mass meÀ ¼ 9.11 Â 10 kg, and the Planck’s constant h ¼ 6.63 Â 10 J s, one obtains the velocity uncertainty of the electrons

: À34 Δ h ÀÁ6 63 Â 10ÀÁ: 3m vz ¼ À31 À7 1 82 Â 10 meÀ λd 9:11 Â 10 Á 4 Â 10 s

The HUP is a statistical average over a set of many identical measurements. In this case, the velocity uncertainty of the electrons is too high so that we do not know where these electrons locate after 1 s, in the large uncertain range between 0 and 1.8 km from the measurement. As a result, the interference phenomenon does not 6.12 The Schrodinger€ Equation 297 take place at the measurement. On the contrary, if the wavelength of the detecting light λd is long enough that must be much larger than the distance ls between two slits (ls 0.7 mm), the velocity uncertainty of the electrons becomes much smaller. In this case, the light detectors do not affect the electrons moving through the slits. As a result, the electrons locate in the measurement and interfere with each other. Hence, the interference pattern returns on the screen. This result indicates that the measured results are strongly inﬂuenced by the apparatus or the measurement method in the quantum world. Therefore, the electrons behave sometimes like waves and in another time like particles depending on the measurement and observing method [11]. If the experiment is set up to study the wave-like behavior of the electrons, the wave phenomena of the interference and diffraction that are interacted between the electrons and the observing method are found. In another case, the particle-like behaviors, such as the photoelectric and Compton effects are given if the experiment is set up to detect the particle characteristic of the electrons. It would be better off letting the nature laws organize themselves, especially in the quantum world. In general, physics teaches us how to understand the nature laws, not how to manipulate them. In the double-slit experiment with detectors at each slit, each observer sees the electron go through just one slit. In accordance with the Copenhagen interpretation of many worlds, this experiment of the real world splits in two worlds that are separate and non-interacting with each other. Both worlds have the same probability of electron going through the slit (i.e. 50 % to 50 %). In fact, they are not parallel but perpendicular to each other in the many-worlds interpretation (MWI). As a result, no interference between the separate and non-interacting worlds takes place on the screen. In fact, the quantum objects with a wave-like or particle-like characteristic depend on how they are considered like in topology and differential geometry (cf. Sect. 4.1). If we consider the objects at a distance as a whole without considering them in detail (i.e. without detecting the electron positions) like topologists do, the measured objects response as wave-like behavior with the interference pattern (see Fig. 6.11b). On the contrary, if we consider them much closer and concern them much more in detail with all smallest scales of them (i.e. detecting the electron positions) like geometers do, the measured objects react as particle-like behavior (see Fig. 6.12).

6.12 The Schrodinger€ Equation

Schrodinger€ equation describes the behavior of non-relativistic quantum objects that are based on the wave-like behavior of wave mechanics. 298 6 Tensors and Bra-Ket Notation in Quantum Mechanics

6.12.1 Time Evolution in Quantum Mechanics

Time evolution is derived from two main principles of the state vector of the quantum object [7]. The ﬁrst principle states that the time-dependent state vector is consists of a function of time and its state vector at time t ¼ 0.

jiΨðÞt ¼ UtðÞjiΨðÞ0 ð6:90aÞ

Thus,

UðÞ¼0 1 at t ¼ 0:

The formalism in Eq. (6.90a) is the time-dependent state vector in quantum mechanics. The formulation of the complex number is used here to describe the time dependence of the state vector. If the Hamiltonian H is independent of time, the time-evolution operator is deﬁned as ÀÁ X1 ÀiHt k 2 ÀiHt Àh iHt 1 ÀiHt UtðÞe Àh ¼ 1 À þ þÁÁÁ ð6:90bÞ k! h 2! h k¼0 À À

In the other case, if the Hamiltonian H is dependent on time, the time-evolution operator is deﬁned as 0 1 ðt Ài UtðÞexp@ HðÞτ dτA ð6:90cÞ Àh 0

The second principle states that the below relation is always valid for any time.

hiΦðÞjt ΨðÞt ¼ hiΦðÞj0 ΨðÞ0 for 8t 0 ð6:91Þ

The bra hΦ(t)| is calculated from Eq. (6.90a)as

jiΦðÞt ¼ UtðÞjiΦðÞ0 ) hjΦðÞt ¼ U{ðÞt hjΦðÞ0

Using Eqs. (6.90a) and (6.91), the inner product of bra and ket is written as

hiΦðÞjt ΨðÞt ¼ hjΦðÞ0 U{ðÞÁt UtðÞjiΨðÞ0 ¼ hiΦðÞj0 ΨðÞ0 ð6:92Þ ) U{ðÞÁt UtðÞ¼1

This result is called the uniquetarity on time. 6.12 The Schrodinger€ Equation 299

For time t ¼ ε 1, the time function U(t) can be written using Eq. (6.90b)as

iε UðÞ¼ε 1 À H Àh ð6:93Þ iε { ) U{ðÞ¼ε 1 þ H Àh

Using Eqs. (6.92) and (6.93), one calculates iε { iε U{ðÞÁε UðÞ¼ε 1 þ H Á 1 À H Àh Àh 2 iε iε { ε { ¼ 1 À H þ H þ 2 H H ÀhÀÁÀh Àh iε { 1 þ H À H ¼ 1 Àh

{ Thus, H ¼ H: This result denotes that the Hamiltonian H is a Hermitian operator (cf. Chap. 1). The time derivative of the state vector is deﬁned using Eq. (6.90a)as

djiΨðÞε UðÞε jiΨðÞ0 À jiΨðÞ0 jiΨðÞ0 ðÞUðÞÀε 1 ¼ lim ¼ lim dε ε!0 ðÞε À 0 ε!0 ε ε Ψ i ÀjiðÞ0 H iH ¼ lim Àh ¼À jiΨðÞ0 ; 8ε << 1 ε!0 ε Àh

Multiplying both sides of the above equation by U(t), one obtains

∂jiΨðÞt iH ¼À jiΨðÞt ∂t Àh ∂jiΨðÞt ) iÀh ¼ HjiΨðÞt ∂t

This above equation is called the fundamental Schrodinger€ equation for non-relativistic particles. Its general solution of the state vector results from Eqs. (6.90a) and (6.90b)as

ÀiHt jiΨðÞt ¼ UtðÞjiΨðÞ0 ¼ e Àh jiΨðÞ0 iHt ) hjΨðÞt ¼ e Àh hjΨðÞ0 300 6 Tensors and Bra-Ket Notation in Quantum Mechanics

6.12.2 The Schrodinger€ and Heisenberg Pictures

In the following section, the Schrodinger€ and Heisenberg pictures are introduced for the expectation value of the time-dependent operator A. The Schrodinger€ picture is deﬁned as the expectation value of a time-dependent observable A that is a Hermitian linear operator.

Ψ Ψ ; hiA t ¼ hjSðÞt ASjiSðÞt ÀiHt jiΨSðÞt ¼ e Àh jiΨðÞ0

€ € where AS is an operator in the Schrodinger picture (Schrodinger representation). The Heisenberg picture is deﬁned using the time-dependent state vector as

iHt ÀiHt Ψ Ψ Ψ Àh Àh Ψ hiA t ¼ hjSðÞt ASjiSðÞt ¼ hjðÞ0 e ASe jiðÞ0 hjΨðÞ0 AðÞt jiΨðÞ0

Thus, the time-dependent operator A in the Heisenberg picture results as

iHt ÀiHt AðÞ¼t e Àh ASe Àh

Replacing AS by ∂AS/∂t considered as an observable operator in the Heisenberg picture [12], one obtains the corresponding Heisenberg operator

∂AðÞt iHt ∂AS ÀiHt ¼ e Àh e Àh ∂t ∂t

Time differentiating the time-dependent operator A(t), one obtains the Heisenberg equation of motion. dA iH iHt ÀiHt iHt ∂AS ÀiHt ¼ Á e Àh ASe Àh þ e Àh e Àh dt Àh∂t iHt ÀiHt iH þ e Àh A e Àh ÁÀ S Àh i iHt ∂AS ÀiHt i ∂A ¼ ðÞþHA À AH e Àh e Àh ¼ ½þH; A Àh ∂t Àh ∂t where [H, A] is the commutator of the operators H and A, cf. Eq. (6.65). Note that both operators usually do not commute. However, there are some properties in the Heisenberg picture of a non-relativistic particle: • The state vector in the Schrodinger€ picture is time-dependent. 6.12 The Schrodinger€ Equation 301

ÀiHt jiΨSðÞt ¼ e Àh jiΨðÞ0

€ • The operator AS in the Schrodinger picture is time-independent.

∂A ∂A S ¼ ¼ 0 ∂t ∂t

• The state vector in the Heisenberg picture is time-independent.

iHt iHt ÀiHt jiΨðÞt ¼ e Àh jiΨSðÞt ¼ e Àh e Àh jiΨðÞ0 ¼ jiΨðÞ0

Therefore, the Heisenberg equation of motion can be written as

dA iÀh ¼ ½A; H dt

Having replaced the operator A by the Hamiltonian H in the Heisenberg equation, it gives using Eq. (6.66)

dH iÀh ¼ ½¼H; H 0 ) HðÞ¼t HðÞ0 H ¼ const: dt

This result indicates that the Hamiltonian H (Hermitian operator) does not depend on time explicitly, cf. Eq. (6.90b). In summary, we obtain • Solutions of the Schrodinger€ equation:

djiΨðÞt ÀiHt iÀh ¼ HjiΨ ) jiΨ ðÞt ¼ e Àh jiΨðÞ0 dt S

The state vector in the Schrodinger€ picture is determined by the initial condition |ψ (0)i. • Solutions of the Heisenberg equation of motion:

dA iHt ÀiHt iÀh ¼ ½A; H ) AðÞt ¼ e Àh A e Àh dt S

The Heisenberg picture A(t) is the solution of the motion equation. Using either the Schrodinger€ picture or the Heisenberg picture, the expectation value of the time-dependent operator A results as the same value. 302 6 Tensors and Bra-Ket Notation in Quantum Mechanics

Ψ Ψ Ψ Ψ hiA t ¼ hjSðÞt ASjiSðÞt ¼ hjðÞ0 AðÞt jiðÞ0

6.12.3 Time-Dependent Schrodinger€ Equation (TDSE)

For simplicity, the Schrodinger€ equation is written in a one-dimensional space (1-D) using the wave function instead of the state vector as

∂ΨðÞx; t iÀh ¼ HΨðÞx; t ð6:94Þ ∂t

The Hamiltonian H is deﬁned in classical mechanics as the sum of the kinetic energy T and potential energy V using Eq. (6.84)as

p2 H ¼ T þ V ¼ þ VxðÞ 2m

Using the momentum operator in Eq. (6.70), one obtains

∂ ∂2 p ¼ÀiÀh ) p2 ¼ÀÀh2 ð6:95Þ ∂x ∂x2

Thus, the RHS of Eq. (6.94) is expressed as p2 HΨðÞ¼x; t þ VxðÞ ΨðÞx; t 2m ! 2 ð6:96Þ ÀÀh2 ∂ ¼ þ VxðÞ ΨðÞx; t 2m ∂x2

Substituting Eq. (6.96) into Eq. (6.94), one obtains the time-dependent Schrodinger€ equation (TDSE) in a one-dimensional space for a single non-relativistic particle.

∂ΨðÞx; t ÀÀh2 ∂2ΨðÞx; t iÀh ¼ þ VxðÞΨðÞx; t ð6:97Þ ∂t 2m ∂x2

In the case that the Hamiltonian H is not an explicit function of time t; i.e., the potential energy V only depends on x, the solution of the general wave function of Eq. (6.97) can be written using the separation method of variables as

ΨðÞx; t ψðÞÁx φðÞt ð6:98Þ 6.12 The Schrodinger€ Equation 303

Thus,

∂ΨðÞx; t dφðÞt ψðÞx ; ∂t dt ∂2ΨðÞx; t d2ψðÞx φðÞt : ∂x2 dx2

Substituting them into Eq. (6.97), one obtains

dφðÞt ÀÀh2 d2ψðÞx iÀhψðÞx ¼ φðÞt þ VxðÞψðÞÁx φðÞt dt 2m dx2

Dividing both sides of the above equation by ψ(x)φ(t), it gives

iÀh dφðÞt ÀÀh2 d2ψðÞx ¼ þ VxðÞ φðÞt dt 2mψðÞx dx2

Both terms of the above equation are independent of each other; therefore, each term is independent of each other and must be therefore an invariant.

iÀh dφðÞt ÀÀh2 d2ψðÞx ¼ þ VxðÞE φ t dt 2mψ x dx2 ðÞ ðÞ ð6:99Þ iÀh dφðÞt ) ¼ E φðÞt dt

The solution of the above equation results from integrating over dt as ð ð ð dφ E iE ÀiEt ¼ dt ¼ À dt ) φðÞ¼t e Àh ð6:100Þ φ iÀh Àh

Multiplying both sides of Eq. (6.99)byψ(x), one obtains ! 2 2 ÀÀh d ψ ψ 2 þ VxðÞ ðÞx E ðÞx 2m dx ð6:101Þ , HψðÞ¼x EψðÞx

Equation (6.101) is called the time independent Schrodinger€ equation (TISE). It is obvious that E is the energy eigenvalue at the energy eigenstate wave function ψ(x) of the Hamiltonian operator H. The energy eigenstate solution is called the stationary state solution (also standing wave solution) at the energy eigenvalue E. The stationary state solution is the basis solution for the general solution of the wave function of the TDSE. The energy eigenvalue E results from the expectation value (average value) of the Hamiltonian operator H using Eq. (6.101)as 304 6 Tensors and Bra-Ket Notation in Quantum Mechanics

þ1ð hiH ¼ H ¼ ψ *ðÞx HψðÞx dx À1 þ1ð þ1ð ¼ ψ *ðÞx EψðÞx dx ¼ E ψ *ðÞx ψðÞx dx À1 À1 ¼ E

Inserting Eq. (6.100) into Eq. (6.98), the general solution of the wave function of the TDSE results as

ÀiEt ÀiHt ΨðÞ¼x; t ψðÞx e Àh ¼ ψðÞx e Àh ð6:102Þ

In general, the time-dependent Schrodinger€ equation (3-D TDSE) is written in a three-dimensional space (3-D) in Cartesian coordinates as

∂ΨðÞr; t iÀh ¼ HΨðÞr; t ∂t

The momentum operator p of the particle in a Hilbert space is deﬁned as

p ÀiÀh∇ ) p2 ¼ÀðÞiÀh∇ 2 ¼ÀÀh2∇2 ¼ÀÀh2Δ

Thus, the Hamiltonian operator results from the kinetic and potential energies as

p2 Àh2 H ¼ þ VxðÞ¼À Δ þ VxðÞ 2m 2m

Thus, the 3-D time-dependent Schrodinger€ equation becomes

∂Ψ ; 2 ðÞr t ÀÀh ∇2Ψ ; Ψ ; iÀh ∂ ¼ ðÞþr t VðÞr ðÞr t t 2m ð6:103Þ ÀÀh2 ¼ ΔΨðÞþr; t VðÞr ΨðÞr; t 2m where r is the position vector of the particle, r(x, y, z) 2 R3. The general solution of Eq. (6.103) is written using Fourier series as X ÀiEnt Ψ ; ψ Àh : ðÞ¼r t cn nðÞr e ð6 104Þ n

The coefﬁcients cn are used to satisfy the initial conditions at t ¼ 0. 6.12 The Schrodinger€ Equation 305

X1 Ψ ; ψ f ðÞr ðÞ¼r 0 cn nðÞr n¼1

Applying Dirichlet’s theorem to function f(r), one obtains the coefﬁcient cm using Kronecker delta due to orthonormality of ψn(r)

þ1ð þ1ð X1 ψ * ψ * ψ mðÞr f ðÞr dr ¼ cn mðÞr nðÞr dr n¼1 À1 À1 X1 ¼ cnδmn ¼ cm n¼1

Thus, the coefﬁcients cn in Eq. (6.104) are calculated as

þ1ð ψ * : cn ¼ nðÞr f ðÞr dr ð6 105Þ À1

Analogously, the 3-D time independent Schrodinger€ equation (3-D TISE) of Eq. (6.101) results as

ÀÀh2 ∇2ψðÞþr VðÞr ψðÞ¼r EψðÞr ð6:106Þ 2m

For N non-interacting particles in 3-D, the Hamiltonian operator H is written as X ! pnpn ; ; H ¼ þ VnðÞxn yn zn n 2mn

The momentum operator is deﬁned as

∇ pn ÀiÀh n ! ∂2 ∂2 ∂2 ) p p ¼ p2 ¼ÀÀh2∇2 ¼ÀÀh2 þ þ n n n n ∂ 2 ∂ 2 ∂ 2 xn yn zn

Therefore, the 3-D time independent Schrodinger€ equation (3-D TISE) for N non- interacting particles results as ! XN Àh2 À ∇2 þ VðÞr ; r ; ...; r ψðÞr ; r ; ...; r 2 n 1 2 N 1 2 N ð6:107Þ n¼1 mn ¼ EψðÞr1; r2; ...; rN

The general solution of the wave function of N non-interacting particles results as 306 6 Tensors and Bra-Ket Notation in Quantum Mechanics

YN ÀiEt ΨðÞ¼r1; r2; ...; rN; t e Àh ψðÞrn ð6:108Þ n¼1

6.12.4 Discussions of the Schrodinger€ Wave Functions

The quantum wave functions in Eqs. (6.104) and (6.108) of the Schrodinger€ equations have the amplitudes in complex numbers that always contain the imaginary number i. As a result, one can never measure their amplitudes of complex numbers in reality. Note that only the amplitudes of real numbers can be measured in experiments. As having known, quantum mechanics is inherently a probabilistic theory. Thus, the squared amplitude of the Schrodinger€ wave function is interpreted as the probability of ﬁnding the particle at any point in space and at any particular time according to Born’s interpretation as

PðÞ¼r; t kkΨðÞr; t 2 ¼ hiΨjΨ ¼ Ψ*ðÞÁr; t ΨðÞ2r; t R

In fact, the Schrodinger€ wave functions can never be measured by experiments in quantum mechanics. However, the squared amplitudes of the wave functions are measurable probabilistic quantities that describe the ﬁnding probabilities of the particle at any point in space and at any given time. Moreover, these particle probabilities strongly obey the Heisenberg uncertainty principle according to the Born’s interpretation. Nowadays, the probabilistic theory of the Schrodinger€ wave function has been accepted as the nature of physical reality and therefore plays a fundamental component of modern physics [13].

6.13 The Klein-Gordon Equation

The Schrodinger€ wave equation is valid for non-relativistic particles in quantum mechanics. In fact, the Klein-Gordon equation is a relativistic variance of the Schrodinger€ equation. The relativistic energy equation for a free particle is written as [2, 3, 13]

E2 ¼ p2c2 þ m2c4 ð6:109Þ

The energy operator is deﬁned as 6.13 The Klein-Gordon Equation 307

∂ E iÀh ∂ t : ∂2 ð6 110Þ ) E2 ¼ÀÀh2 ∂t2

The momentum operator is deﬁned as

p ÀiÀh∇ ð6:111Þ ) p2 ¼ÀÀh2∇2

Substituting Eqs. (6.110) and (6.111) into Eq. (6.109), one obtains the Klein- Gordon equation

1 ∂2Ψ m2c2 À ∇2Ψ þ Ψ 2 ∂ 2 2 c t Àh ð6:112Þ 1 ∂2Ψ À ∇2Ψ þ μ2Ψ ¼ 0 c2 ∂t2 where the factor μ is deﬁned as mc μ Àh

In the spacetime, the Minkowski metric with four spacetime coordinates (t, x, y, z) results as (cf. Chap. 5) 0 1 2 Àc 000 B 0100C g ¼ g ¼ B C μv @ 0010A 0001

In this case, the Klein-Gordon equation is written in the contravariant notation as ∂μ∂v Ψ μ2Ψ : À gμv þ ¼ 0 ð6 113Þ

The general solution of Eq. (6.112) is the wave function of

ω ΨðÞ¼r; t Ae½iðÞkÁrÀ t ð6:114Þ where r is the position vector of the particle; k is the wave vector of the particle; ω is the angular frequency of the particle. The position vector r is written as 308 6 Tensors and Bra-Ket Notation in Quantum Mechanics

r ¼ ðÞ2x, y, z R3

The wave vector k is written as ÀÁ 3 k ¼ kx, ky, kz 2 R in which the wavenumber k is deﬁned as

2π p E k ¼ ¼ ¼ λ Àh Àhc

The angular frequency ω is deﬁned as π ω π 2 c cp ¼ 2 v ¼ λ ¼ ¼ ck ω Àh ) k ¼ c

The 4-wavevector in special relativity is written as ω k ¼ , k , k , k 2 R4 c x y z

Partially differentiating with respect to t and r the wave function in Eq. (6.114), one obtains

∂2ΨðÞr; t ¼Àω2ΨðÞr; t ; ∂ 2 t ð6:115Þ ∂2ΨðÞr; t ¼Àk2ΨðÞr; t : ∂r2

Substituting Eq. (6.115) into Eq. (6.112), it gives the dispersion relation ω 2 ω 2 m2c2 À kkk 2 ¼ À k2 þ k2 þ k2 ¼ ð6:116Þ c c x y z Àh2

6.14 The Dirac Equation

The Schrodinger€ wave equation does not include spin vectors or rotations that happen in the everyday sense. Furthermore, they are very important in the Pauli matrices, cf. Sect. 6.6. Let Ψ(r,t) be the state vector (state ket) of the wave function. The wave state ket is written in the non-relativistic case as 6.14 The Dirac Equation 309 0 1 ψ ; 1ðÞr t B ψ ; C Ψ ; B 2ðÞr t C : jiðÞr t ¼ @ ... A ð6 117Þ ψ ; NðÞr t

The sth component of the state vector on the spin vector s is deﬁned as

ψ ; ψ : sðÞr t hirsj ðÞt ð6 118Þ where r is the orbital variable vector, and s are the spin variables (s ¼ 1, 2,...,N)of the state vector. The probability function is the squared magnitude of the wave function, cf. Sect. 6.11.3.

XN ; Ψ ; 2 ψ 2 : PðÞ¼r t kkðÞr t ¼ s ð6 119Þ s¼1

Thus, the wave function is expressed as

∂ΨðÞr; t iÀh ¼ H ΨðÞr; t ð6:120Þ ∂t D where HD is the Dirac Hamilton operator that is also a Hermitian operator of the state-vector space. The Dirac Hamilton operator HD is deﬁned as

HD α Á p þ βm ð6:121Þ where α ¼ (αx, αy, αz) is the spin operator with three spin variables; p is the momentum operator; β is the spin variable; m is the rest mass of the particle. The energy operator is written using Eq. (6.110)as

∂ E iÀh ∂t

The momentum operator is written using Eq. (6.111)as

p ÀiÀh∇

Substituting Eq. (6.121) and both operators into Eq. (6.120), the wave function becomes 310 6 Tensors and Bra-Ket Notation in Quantum Mechanics ∂ iÀh À H ΨðÞ¼r; t ½E À α Á p À βm ΨðÞ¼r; t 0 ð6:122Þ ∂t D

Equation (6.122) is called the Dirac equation. Multiplying the LHS of Eq. (6.122) by the term [E + α Á p + βm], one obtains [12] 0 X X 1 2 α2 2 2β2 α α α α E À k pk À m À ðÞk l þ l k pkpl B < C @ X k k l AΨðÞ¼r; t 0 ð6:123Þ α β βα À ðÞk þ k mpk k

The Klein-Gordon equation (6.112) can be written in forms of the energy and momentum operators as

1 ∂2Ψ m2c2 À ∇2Ψ þ Ψ ¼ 0 c2 ∂t2 Àh2 ð6:124Þ ÂÃ , E2 À p2c2 À m2c4 ΨðÞ¼r; t 0

Equation (6.123) is identical to the Klein-Gordon equation if the spin variables in Eq. (6.123) are anticommute and their amplitudes satisfy the following conditions

ðÞ¼αkαl þ αlαk 0fork 6¼ l; ðÞ¼αkβ þ βαk 0; ð6:125Þ α2 β 2 k ¼ c

The spin variables result from the Pauli’s matrices in case of the light speed is chosen c 1.

σx ¼Àiαyαz; σy ¼Àiαzαx; σz ¼Àiαxαy τ ¼ σ α ¼Àiα α α ; 1 z z x y z ð6:126Þ τ2 ¼Àiτ1τ3 ¼Àβαxαyαz; τ3 ¼ β

The Pauli’s matrices in Eq. (6.126) are deﬁned as, cf. Sect. 6.6 01 σx τ1 ¼ ; 10 0 Ài σy τ2 ¼ ; ð6:127Þ i 0 10 σ τ ¼ : z 3 0 À1 References 311

References

1. Gisin N (2014) Quantum chance – nonlocality, teleportation and other quantum marvels. Springer, Berlin 2. Esposito G et al (2015) Advanced concepts in quantum mechanics. Cambridge University Press, Cambridge 3. Longair M (2013) Quantum concepts. Cambridge University Press, Cambridge 4. Bradley R, Swartz N (1979) Possible worlds – an introduction to logic and its philosophy. Hackett Publishing Company, Indianapolis, IN 5. Laloe F (2012) Do we really understand quantum mechanics? Cambridge University Press, Cambridge 6. Griffiths D (2005) Introduction to quantum mechanics, 2nd edn. Pearson Education, Upper Saddle River, NJ 7. Susskind L, Friedman A (2014) Quantum mechanics – the theoretical minimum. Basic Books, New York, NY 8. Kosmann-Schwarzbach Y (2006) Les Theóre`mes de Noether – Invariance et Lois de Conser- vation au XXe sie`cle, Deuxie`me edition. Les e´ditions de l’ećole polytechnique 9. Binney J, Skinner D (2015) The physics of quantum mechanics. Oxford University Press, Oxford 10. Dirac PAM (2011) The principles of quantum mechanics, 4th edn. Oxford Science Publica- tions, OUP, Oxford 11. Calle CI (2010) Superstrings and other things, 2nd edn. CRC Press, Taylor & Francis Group, Boca Raton, FL 12. Messiah A (2014) Quantum mechanics. Dover Publications, New York, NY 13. Lederman LM, Hill CT (2004) Symmetry and the beautiful universe. Prometheus Books, New York, NY Appendix A: Relations Between Covariant and Contravariant Bases

The contravariant basis vector gk of the curvilinear coordinate of uk at the point P is perpendicular to the covariant bases gi and gj, as shown in Fig. A.1. This contravariant basis gk can be deﬁned as

∂r ∂r α gk g Â g ¼ Â ðA:1Þ i j ∂ui ∂uj where α is the scalar factor; gk is the contravariant basis of the curvilinear coordinate of uk. g3 u3

g3 x3 g1 P g e 2 3 g2 2 1 u 0 u g1 e e1 2 x2 x1

Fig. A.1 Covariant and contravariant bases of curvilinear coordinates

© Springer-Verlag Berlin Heidelberg 2017 313 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 314 Appendix A: Relations Between Covariant and Contravariant Bases

Multiplying Eq. (A.1) by the covariant basis gk, the scalar factor α results in ÀÁ α k αδk α gi Â gj Á gk ¼ g Á gk ¼ k ¼ hi ðA:2Þ α ; ; ) ¼ gi Â gj Á gk gi gj gk

The scalar triple product of the covariant bases can be written as ffiffiffi α ; ; p : ¼ ½¼g1 g2 g3 ðÞÁg1 Â g2 g3 ¼ g ¼ J ðA 3Þ where Jacobian J is the determinant of the covariant basis tensor G. The direction of the cross-product vector in Eq. (A.1) is opposite if the dummy indices are interchanged with each other in Einstein summation convention. There- fore, the Levi-Civita permutation symbols (pseudo-tensor components) can be used in expression of the contravariant basis. pffiffiffi g gk ¼ J gk ¼ g Â g ¼À g Â g i j j i ε ε ðA:4Þ ijk gi Â gj ijk gi Â gj ) gk ¼ pffiffiffi ¼ g J where the Levi-Civita permutation symbols are defined by 8 < þ1ifðÞi, j, k is an even permutation; ε ; ijk ¼ : À1ifðÞi, j, k is an odd permutation 0ifi ¼ j,ori ¼ k; or j ¼ k ðA:5Þ 1 , ε ¼ ðÞÁi À j ðÞÁj À k ðÞk À i for i, j, k ¼ 1, 2, 3 ijk 2

Thus, the cross product of the covariant bases gi and gj results from Eq. (A.4): pffiffiffiffi g Â g ¼ ε g gk ¼ ε Jgk ^ε gk i j ijk ijk ijk ε ijk gi Â gj k ^ε ijk ðA:6Þ ) g ¼ pffiffiffi ¼ gi Â gj g ^ε ) ijk ¼ gi Â gj Á gk

The covariant permutation symbols in Eq. (A.6) can be deﬁned as Appendix A: Relations Between Covariant and Contravariant Bases 315

8 ffiffiffi < p ; þ ffiffiffig if ðÞi, j, k is an even permutation ^ε p ; : ijk ¼ : À g if ðÞi, j, k is an odd permutation ðA 7Þ 0ifi ¼ j,ori ¼ k; or j ¼ k

The contravariant permutation symbols in Eq. (A.6) can be defined as 8 > 1 > þ pffiffiffi if ðÞi, j, k is an even permutation; < g ^ε ijk 1 : ¼ > ffiffiffi ; ðA 8Þ > Àp if ðÞi, j, k is an odd permutation :> g 0ifi ¼ j,ori ¼ k; or j ¼ k

k The covariant basis vector gk of the curvilinear coordinate of u at the point P is perpendicular to the contravariant bases gi and gj, as shown in Fig. A.1. Therefore, the cross product of the contravariant bases gi and gj can be written as ε ε i j ijk ijk ^ε ijk ðÞ¼g Â g pffiffiffiffigk ¼ gk gk g J ðA:9Þ ) ^ε ijk ¼ ðÞÁgi Â gj gk

Thus, the covariant basis results from Eq. (A.9): pffiffiffi ÀÁÀÁ g ¼ ε g gi Â gj ¼ ε J gi Â gj k ijk ÀÁ ijk : i j ðA 10Þ ¼ ^ε ijk g Â g

Obviously, there are some relations between the covariant and contravariant permutation symbols:

^ε ijk^ε ijk ¼ 1ðÞ no summation : ijk 2 ðA 11Þ ^ε ijk ¼ ^ε J ðÞno summation

The tensor product of the covariant and contravariant permutation pseudo-tensors is a sixth-order tensor. 8 < þ1; ðÞi, j, k and ðÞl; m; n even permutation ^ε ijk^ε δijk ; ; ; : pqr ¼ pqr ¼ : À1 ðÞi, j, k and ðÞl m n odd permutation ðA 12Þ 0; otherwise

The sixth-order Kronecker tensor can be written in the determinant form: 316 Appendix A: Relations Between Covariant and Contravariant Bases δ i δ i δ i p q r ^ε ijk^ε δijk δ j δ j δ j : pqr ¼ pqr ¼ p q r ðA 13Þ δ k δ k δ k p q r

Using the tensor contraction rules with k ¼ r, one obtains δ i δ i δ i δ i δ i δ i p q r p q r δij δijr δ j δ j δ j δ j δ j δ j pq ¼ pqr ¼ p q r ¼ p q r δ r δ r δ r p q r 001 ðA:14Þ δ i δ i ^ε ij^ε δij : p q δ i δ j δ i δ j ) pq ¼ pq ¼ 1 δ j δ j ¼ p q À q p p q

Further contraction of Eq. (A.14) with j ¼ q gives

^ε iq^ε δ i δ q δ i δ q pq ¼ p q À q p ðA:15Þ δ i δ q δ i δ i δ i δ i ¼ p q À p ¼ 2 p À p ¼ p for i, p ¼ 1, 2

From Eq. (A.15), the next contraction with i ¼ p gives

^ε pq^ε δ p pq ¼ p ðÞsummation over p ðA:16Þ δ1 δ2 ¼ 1 þ 2 ¼ 2forp ¼ 1, 2

Similarly, contracting Eq. (A.13) with k ¼ r; j ¼ q, one has for a three-dimensional space.

^ε iq^ε δ i δ q δ qδ i pq ¼ p q À p q ðA:17Þ δ i δ q δ i δ i δ i δ i ¼ p q À p ¼ 3 p À p ¼ 2 p for i, p ¼ 1, 2, 3

Contracting Eq. (A.17) with i ¼ p, one obtains

^ε pq^ε ¼ 2δ p ðÞsummation over p pq ÀÁp δ1 δ2 δ3 : ¼ 2 1 þ 2 þ 3 ðA 18Þ ¼ 21ðÞ¼þ 1 þ 1 6forp ¼ 1, 2, 3

The covariant metric tensor M can be written as 2 3 g11 g12 g13 4 5 : M ¼ g21 g22 g23 ðA 19Þ g31 g32 g33 where the covariant metric coefﬁcients are deﬁned by gij ¼ gi Á gj. Appendix A: Relations Between Covariant and Contravariant Bases 317

The contravariant metric coefficients in the contravariant metric tensor MÀ1 result from inverting the covariant metric tensor M. 2 3 g11 g12 g13 MÀ1 ¼ 4 g21 g22 g23 5 ðA:20Þ g31 g32 g33 where the contravariant metric coefficients are defined by gij ¼ gi Á gj. Thus, the relation between the covariant and contravariant metric coefficients can be written as

ik ik δ i À1 À1 : g gkj ¼ gkjg ¼ j , M M ¼ MM ¼ I ðA 21Þ

ik ik In case of i 6¼ j, all terms of g gkj equal zero. Thus, only nine terms of g gki for i ¼ j remain in a three-dimensional space R3:

ik 1k 2k 3k g gki ¼ g gk1 þ g gk2 þ g gk3 for i, k ¼ 1, 2, 3 δ1 δ2 δ3 δ i : ¼ 1 þ 2 þ 3 ¼ i for i ¼ 1, 2, 3 ðA 22Þ ¼ 1 þ 1 þ 1 ¼ 3

The relation between the covariant and contravariant bases in the general curvilinear coordinates results in

i ik δ i; g Á gj ¼ g gkj ¼ j for i j ) i 1 2 3 : g Á gi ¼ g Á g1 þ g Á g2 þ g Á g3 for i ¼ 1, 2, 3 ðA 23Þ ik δ i ¼ g gki ¼ i for i, k ¼ 1, 2, 3

ik According to Eq. (A.23), nine terms of g gki for k ¼ 1,2,3 result in 8 < 1 1k 11 12 13 δ1 ; g Á g1 ¼ g gk1 ¼ g g11 þ g g21 þ g g31 ¼ 1 for i ¼ 1 2 2k 21 22 23 δ2 ; : : g Á g2 ¼ g gk2 ¼ g g12 þ g g22 þ g g32 ¼ 2 for i ¼ 2 ðA 24Þ 3 3k 31 32 33 δ3 : g Á g3 ¼ g gk3 ¼ g g13 þ g g23 þ g g33 ¼ 3 for i ¼ 3

The scalar product of the covariant and contravariant bases gives gðÞi Á g ¼ gðÞi Á g cos gðÞi , g ðÞi ðÞi ðÞi pffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi ðA:25Þ ðÞii p ðÞi ¼ g Á gðÞii cos g , gðÞi ¼ 1 where the index (i) means no summation is carried out over i. Equation (A.25) indicates that the product of the covariant and contravariant basis norms generally does not equal 1 in the curvilinear coordinates. 318 Appendix A: Relations Between Covariant and Contravariant Bases qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 ðÞii : g Á gðÞii ¼ 1 ðA 26Þ ðÞi ; cos g gðÞi

(i) In orthogonal coordinate systems, g is parallel to g(i). Therefore, Eq. (A.26) becomes qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 1 ðÞii ðÞii : g Á gðÞii ¼ 1 ) g ¼ pffiffiffiffiffiffiffiffi ¼ ðA 27Þ gðÞii hi Appendix B: Physical Components of Tensors

The physical component of a tensor can be defined as the tensor component on its unitary covariant basis. Therefore, the covariant basis of the general curvilinear coordinates has to be normalized. Dividing the covariant basis by its vector length, the unitary covariant basis (covariant normalized basis) results in * gi gi * : gi ¼ ¼ pffiffiffiffiffiffiffiffi ) gi ¼ 1 ðB 1aÞ jjgi gðÞii

The covariant basis norm |gi| can be considered as a scale factor hi without summation over (i). pffiffiffiffiffiffiffiffi : hi ¼ jjgi ¼ gðÞii ðB 1bÞ

Thus, the covariant basis can be related to its unitary covariant basis by the relation pffiffiffiffiffiffiffiffi * * : gi ¼ gðÞii gi ¼ higi ðB 2Þ

The contravariant basis can be related to its unitary covariant basis using Eqs. (2.47) and (B.2).

i ij ij * : g ¼ g gj ¼ g hjgj ðB 3Þ

The contravariant second-order tensor can be written in the unitary covariant bases using Eq. (B.2).

© Springer-Verlag Berlin Heidelberg 2017 319 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 320 Appendix B: Physical Components of Tensors ÀÁ ij ij * * *ij * * : T ¼ T gigj ¼ T hihj gi gj T gi gj ðB 4Þ

Thus, the physical contravariant tensor components denoted by star result in

*ij ij T hihjT ðB:5Þ

The covariant second-order tensor can be written in the unitary contravariant bases using Eq. (B.3). ÀÁ i j ik jl * * * * * : T ¼ Tijg g ¼ Tijg g hkhl gkgl Tijgkgl ðB 6Þ

Similarly, the physical covariant tensor components denoted by star result in

* ik jl : Tij g g hkhlTij ðB 7Þ

The mixed tensors can be written in the unitary covariant bases using Eqs. (B.2) and (B.3) ÀÁ T ¼ T ig gj ¼ T ig gjkg j i j ÀÁi ÀÁk ¼ T i h g* gjkh g* j i i k k i jk * * ðB:8Þ ¼ Tj g hihk gi gk * i * * Tj gi gk

Thus, the physical mixed tensor components denoted by star result in * i jk i : Tj g hihkTj ðB 9Þ

Analogously, the contravariant vector can be written using Eq. (B.2).

i i * v ¼ v gi ¼ ðÞv hi gi *i : *i * v ðB 10Þ v gi ¼ gi hi

* Thus, the physical component of the contravariant vector v on the unitary basis gi is defined as pffiffiffiffiffiffiffiffi *i i i : v hiv ¼ gðÞii v ðB 11Þ

The contravariant basis can be normalized dividing by its vector length without summation over (i). Appendix B: Physical Components of Tensors 321

gi gi g*i ¼ ¼ pffiffiffiffiffiffiffiffi ðB:12Þ jjgi gðÞii where g(ii) is the contravariant metric coefficient that results from Eq. (A.20). Using Eq. (B.12), the covariant vector v can be written as qffiffiffiffiffiffiffiffi i ðÞii *i * *i : v ¼ vig ¼ vi g g vi g ðB 13Þ

Thus, the physical component of the covariant vector v results in qffiffiffiffiffiffiffiffi * ðÞii : vi ¼ vi g ðB 14Þ

According to Eq. (A.27), Eq. (B.14) can be rewritten in orthogonal coordinate systems: qffiffiffiffiffiffiffiffi 1 1 * ðÞii : vi ¼ g vi ¼ pffiffiffiffiffiffiffiffi vi ¼ vi ðB 15Þ gðÞii hi

Using Eq. (B.3), the covariant vector can be written in

v ¼ v gi ¼ v gijg ÀÁi i j ¼ v gijh g* v*g* i j j j j ðB:16Þ * vj i ¼ ij g g hj

* Thus, the physical contravariant vector component of v on the unitary basis gj can be deﬁned as

* ij : vj g hjvi ðB 17Þ

Furthermore, the vector v can be written in both covariant and contravariant bases.

j i v ¼ÀÁvjg ¼ v gi ) v gj Á g ¼ ðÞÁvig g j k i k ðB:18Þ δ j i ) vj k ¼ v gik i ) vk ¼ v gik

Interchanging i with j and k with i, one obtains

j : vi ¼ v gij ðB 19Þ

Only in orthogonal coordinate systems, we have 322 Appendix B: Physical Components of Tensors

; 2 : gij ¼ 0fori 6¼ j gðÞii ¼ hi ðB 20Þ

Thus, one obtains from Eq. (B.19)

v vjg v1g v2g ... vNg i ¼ ij ¼ i1 þ i2 þ þ iN : i i 2 ðB 21Þ ¼ v gðÞii ¼ v hi

Substituting Eq. (B.11) into Eq. (B.21), one obtains Eq. (B.22) that is equivalent to Eq. (B.15). *i i 2 v 2 *i : vi ¼ v hi ¼ hi ¼ hiv ðB 22Þ hi Appendix C: Nabla Operators

Some useful Nabla operators are listed in Cartesian and general curvilinear coordinates: 1. Gradient of an invariant f • Cartesian coordinate {xi}

∂f ∂f ∂f ∇f ¼ e þ e þ e ðC:1Þ ∂x x ∂y y ∂z z

• General curvilinear coordinate {ui}

∂f ∂f ∇f ¼ f gi ¼ gi ¼ gijg j ðC:2Þ , i ∂ui ∂ui j

2. Gradient of a vector v • General curvilinear coordinate {ui} ∇ k jΓ k i k i v ¼ v,i þ v ij gkg ¼ v igkg ðC:3Þ Γ j k i k i ¼ vk,i À vj ik g g ¼ vkjig g

3. Divergence of a vector v • Cartesian coordinate {xi}

© Springer-Verlag Berlin Heidelberg 2017 323 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 324 Appendix C: Nabla Operators

∂v ∂v ∂v ∇ Á v ¼ x þ y þ z ðC:4Þ ∂x ∂y ∂z

• General curvilinear coordinate {ui} ∇ i i jΓ i Á v ¼ v ji v,i þ v ij ÀÁpffiffiffi 1 ∂ gvi 1 ÀÁpffiffiffi ¼ pffiffiffi ¼ pffiffiffi gvi ðC:5Þ g ∂ui g , i ∇ ki Γ j ki Á v ¼ vkij g ¼ vk, i À vj ik g

4. Gradient of a second-order tensor T • General curvilinear coordinate {ui} for a covariant second-order tensor m m i j k ∇T ¼ Tij, k À Γ Tmj À Γ Tim g g g ik jk ðC:6Þ i j k ¼ Tijj kg g g

• General curvilinear coordinate {ui} for a contravariant second-order tensor ∇ ij Γ i mj Γ j im k T ¼ T,k þ kmT þ kmT gigjg ðC:7Þ ij k ¼ T jkgigjg

• General curvilinear coordinate {ui} for a mixed second-order tensor ∇ i Γ i m Γ m i j k T ¼ Tj,k þ kmTj À jk Tm gig g ðC:8Þ i j k ¼ Tj jkgig g

5. Divergence of a second-order tensor T • General curvilinear coordinate {ui} for a covariant second-order tensor ÀÁ m m i j k ∇ Á T ¼ Tij, k À Γ Tmj À Γ Tim g g Á g ik jk ðC:9Þ jk i Tijj kg g

• General curvilinear coordinate {ui} for a contravariant second-order tensor ∇ Á T ¼ Tij þ Γ i Tmj þ Γ j Tim δ kg , k km km i j ij Γ i mj Γ j im ðC:10Þ ¼ T,i þ imT þ imT gj ij T jigj Appendix C: Nabla Operators 325

• General curvilinear coordinate {ui} for a mixed second-order tensor ∇ Á T ¼ T i þ Γ i T m À Γ mT i δ kgj j,k km j jk m i i Γ i m Γ m i j ðC:11Þ ¼ Tj,i þ imTj À ji Tm g i j i jk Tj jig ¼ Tj jig gk

6. Curl of a vector v • Cartesian coordinate {xi} ex ey ez ∇ Â v ¼ ∂ ∂ ∂ ðC:12Þ ∂x ∂y ∂z vx vy vz

The curl of v results from calculating the determinant of Eq. (C.12). ∂v ∂v ∂v ∂v ∂v ∂v ∇ Â v ¼ z À y e þ x À z e þ y À x e ðC:13Þ ∂y ∂z x ∂z ∂x y ∂x ∂y z

• General curvilinear coordinate {ui}

∇ ^ε ijk : Â v ¼ vj,igk ðC 14Þ

The contravariant permutation symbol is defined by 8 > 1 > þpffiffiffi if ðÞi, j, k is an even permutation; < g ^ε ijk 1 : ¼ > ffiffiffi ; ðC 15Þ > À p if ðÞi, j, k is an odd permutation :> g 0ifi ¼ j, or i ¼ k; or j ¼ k

where J is the Jacobian.

7. Laplacian of an invariant f • Cartesian coordinate {xi}

∂2f ∂2f ∂2f ∇2f Δf ¼ þ þ ðC:16Þ ∂x2 ∂y2 ∂z2

• General curvilinear coordinate {ui} ∇2 Δ Γ k ij f f ¼ f ,ij À f , k ij g ðC:17Þ Γ k ij ij ¼ vi,j À vk ij g vijg 326 Appendix C: Nabla Operators

where the covariant vector component and its derivative with respect to uk are deﬁned by

∂f ∂f ∂2f v ¼ f ¼ ; v ¼ f ¼ ; v ¼ f ¼ ðC:18Þ i ,i ∂ui k , k ∂uk i, j , ij ∂ui∂uj

8. Calculation rules of the Nabla operators

Div Grad f ¼ ∇ Á ðÞ¼∇f ∇2f ¼ Δf ðÞLaplacian ðC:19Þ Curl Grad f ¼ ∇ Â ðÞ¼∇f 0 ðC:20Þ Div Curl v ¼ ∇ Á ðÞ¼∇ Â v 0 ðC:21Þ ΔðÞ¼fg f Δg þ 2∇f Á ∇g þ gΔf ðC:22Þ Curl Curl v ¼ ∇ Â ðÞ¼∇ Â v ∇∇ðÞÀÁ v Δv ðÞðCurl identity C:23Þ

The Laplacian of v in Eq. (C.23) is computed in the tensor formulation for general curvilinear coordinates.

Div Grad v ¼ Laplacian v ¼ Δv ∇ ∇ ∇2 ¼ ÁðÞv ¼ v p C:24 vi vi Γ vp Γ i gjkg ð Þ ¼ j,k À p jk þ j pk i i jk v jkg gi

9. Essential Vector and Nabla Identities Let f and g be arbitrary scalars in R; A, B, and C arbitrary vectors in RN.

Products of vectors

A Â B ¼ÀB Â A A Á ðÞ¼B Â C B Á ðÞ¼C Â A C Á ðÞA Â B : even permutation A Á ðÞ¼ÀB Â C C Á ðÞ¼ÀB Â A B Á ðÞA Â C : odd permutation A Â ðÞ¼B Â C BAðÞÀÁ C CAðÞÁ B

Gradient ∇()

∇ðÞ¼f þ g ∇f þ ∇g 2 RN ∇ðÞ¼fg f ∇g þ g∇f 2 RN ∇ðÞ¼A Á B ðÞA Á ∇ B þ ðÞB Á ∇ A þ A Â ðÞþ∇ Â B B Â ðÞ2∇ Â A RN ∇f ∇ f 2 RN : 1st order tensor, vector ∇A ∇ A 2 RN Â RN : 2nd order tensor Appendix C: Nabla Operators 327

Divergence ∇Á()

∇ Á ðÞ¼A þ B ∇A þ ∇B 2 R ∇ Á ðÞ¼f A f ∇ Á Α þ A Á ∇f 2 R ∇ Á ðÞ¼A Â B B Á ðÞÀ∇ Â A A Á ðÞ2∇ Â B R 1 ðÞA Á ∇ A ¼ ∇A2 À A Â ðÞ2∇ Â A RN 2

Curl ∇Â()

∇ Â ðÞ¼A þ B ðÞþ∇ Â A ðÞ2∇ Â B RN ∇ Â ðÞ¼f A f ðÞÀ∇ Â Α A Â ∇f 2 RN ∇ Â ðÞ¼A Â B AðÞÀ∇ Á B ðÞA Á ∇ B À BðÞþ∇ Á A ðÞB Á ∇ A 2 RN ∇ Â ðÞ¼∇ Â A ∇∇ðÞÀÁ A ∇2A 2 RN 1 ðÞA Á ∇ A ¼ ∇A2 À A Â ðÞ2∇ Â A RN 2

Laplacian Δ(),∇2()

Δf ∇2f ¼ ðÞ∇ Á ∇ f ¼ ∇ Á ðÞ∇f : Laplacian f 2 R ΔðÞfg ∇2ðÞ¼fg f ∇2g þ 2∇f Á ∇g þ g∇2f 2 R ΔA ¼ ∇2A ∇∇ðÞÀÁ A ∇ Â ðÞ∇ Â A : Laplacian A 2 RN ΔðÞf A ∇2ðÞ¼f A A∇2f þ 2ðÞ∇f Á ∇ A þ f ∇2A 2 RN ΔðÞA Á B ∇2ðÞ¼A Á B A Á ∇2B À B Á ∇2A þ 2∇ Á ½2ðÞB Á ∇ A þ B Â ðÞ∇ Â A R

Differentiations

∇ Â ðÞ∇f ∇ Â ðÞ¼∇ f 0 ∇ Á ðÞ¼∇ Â A 0 ∇ Á ðÞf ∇g ¼ f ∇2g þ ∇f Á ∇g 2 R ∇ Á ðÞ¼f ∇g À g∇f f ∇2g À g∇2f 2 R

10. Summary of essential Nabla operators

Operator Laplacian Operand ( ) Grad ∇( ) Div ∇Á( ) Curl ∇Â() Δ() Function f 2 R (0th Vector 2 RN – – Scalar 2 R order tensor) Vector v 2 RN (1st 2nd order tensor Scalar Vector Vector order tensor) 2 RN Â RN 2 R 2 RN 2 RN Tensor T 2 RN Â RN 3rd order tensor Vector –– (2nd order tensor) 2 RN Â RN Â RN 2 RN Order of results One order higher One order Order Order lower unchanged unchanged Appendix D: Essential Tensors

Derivative of the covariant basis

Γ k : gi,j ¼ ijgk ðD 1Þ

Derivative of the contravariant basis

i ∂g i g i ¼ Γ^ gk ¼ÀΓ i gk ðD:2Þ ,j ∂uj jk jk

Derivative of the covariant metric coefﬁcient

Γ p Γ p : gij, k ¼ ikgpj þ jkgpi ðD 3Þ

First-kind Christoffel symbol 1 Γ ¼ g þ g À g ¼ g Γ l ) Γ l ¼ glkΓ ðD:4Þ ijk 2 ik, j jk, i ij, k lk ij ij ijk

Second-kind Christoffel symbol based on the covariant basis

Γ k k klΓ : ij ¼ gi, j Á g ¼ g ijl ðD 5Þ ∂uk ∂2xp Γ k ¼ Á ¼ Γ k ðD:6Þ ij ∂ p ∂ i∂ j ji x u u 1 Γ k ¼ gkp g þ g À g ðD:7Þ ij 2 ip,j jp,i ij,p

© Springer-Verlag Berlin Heidelberg 2017 329 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 330 Appendix D: Essential Tensors

1 ∂J ∂ðÞlnJ Γ i ¼ ¼ ij ∂ j ∂ j J u pffiffiffi u ÀÁpffiffiffi : 1 ∂ g ∂ ln g ðD 8Þ ¼ pffiffiffi ¼ g ∂uj ∂uj

Second-kind Christoffel symbol based on the contravariant basis

Γ^ i Γ i Γ^ i : kj ¼À kj ¼ jk ðD 9Þ

Covariant derivative of covariant ﬁrst-order tensors Γ k : Tij¼ Ti,j À ijTk ¼ T,j Á gi ðD 10Þ

Covariant derivative of contravariant ﬁrst-order tensors i i Γ i k i : T j ¼ T, j þ jkT ¼ T,j Á g ðD 11Þ

Covariant derivative of covariant and contravariant second-order tensors

T j ¼ T À Γ mT À Γ mT ij k ij,k ik mj jk im : ij ij Γ i mj Γ j im ðD 12Þ T jk ¼ T, k þ kmT þ kmT

Covariant derivative of mixed second-order tensors

i i i m m i T jk ¼ T þ Γ T À Γ T j j, k km j jk m ðD:13Þ j j Γ j m Γ m j Ti jk ¼ Ti, k þ kmTi À ik Tm

Second covariant derivative of covariant ﬁrst-order tensors Γ m Γ m Tikj¼ Ti, jk À ik, jTm À ik Tm,j Γ m Γ mΓ n : À ij Tm, k þ ij mkTn ðD 14Þ Γ m Γ mΓ n À jk Ti, m þ jk imTn

Second covariant derivative of the contravariant vector k k k Γ p p Γ k : v jlm v jl,m À v p lm þ v jl pm ðD 15Þ where ÀÁ k k k n Γ k nΓ k : v jl, m ¼ v jl ,m v, lm þ v, m nl þ v nl, m ðD 16Þ Appendix D: Essential Tensors 331 k k nΓ k : v p v, p þ v np ðD 17Þ p p nΓ p : v jl v, l þ v nl ðD 18Þ

Riemann-Christoffel tensor

n Γ n Γ n Γ mΓ n Γ mΓ n : Rijk ik, j À ij,k þ ik mj À ij mk ðD 19Þ

Riemann curvature tensor

n : Rlijk gnlRijk ðD 20Þ

First-kind Ricci tensor

∂Γ k ∂Γ k R ¼ ik À ij À Γ rΓ k þ Γ r Γ k ij ∂uj ∂uk ij rk ik rj ÀÁ ffiffiffi : 2 pffiffiffi ∂ p Γ k ðD 21Þ ∂ ln g 1 g ij ¼ À pffiffiffi þ Γ r Γ k ∂ui∂uj g ∂uk ik rj

Second-kind Ricci tensor

R i gikR j 0kj 1 ÀÁffiffiffi pffiffiffi ∂2 p ∂ gΓ m : @ ln g 1 kj m n A ðD 22Þ ¼ gik À pffiffiffi þ Γ Γ ∂uk∂uj g ∂um kn mj

Ricci curvature 0 1 ÀÁffiffiffi pffiffiffi ∂2 p ∂ gΓ k ij@ ln g 1 ij k r A R ¼ g À pffiffiffi þ Γ Γ ðD:23Þ ∂ui∂uj g ∂uk ir kj

Einstein tensor

1 G i R i À δ iR ¼ gikG j j 2 j kj k 1 : Gij ¼ gikGj ¼ Rij À gijR ðD 24Þ 2 i : Gj ¼ 0 ðD 25Þ i

First fundamental form 332 Appendix D: Essential Tensors

2 2; I ¼ Eduþ 2Fdudv þ!Gdv ! EF ru Á ru ru Á rv ðD:26Þ M ¼ gij ¼ FG rv Á ru rv Á rv

Second fundamental form

II ¼ Ldu2 þ 2 Mdudv þ!Ndv 2; ! ÀÁ LM ruu Á nruv Á n ðD:27Þ H ¼ hij ¼ MN ruv Á nrvv Á n

Gaussian curvature of a curvilinear surface ÀÁ 2 LN À M dethij K ¼ κ1κ2 ¼ ¼ ðD:28Þ EG À F2 det gij

Mean curvature of a curvilinear surface

1 EN ÀÁÀ 2MF þ LG H ¼ ðÞ¼κ1 þ κ2 ðD:29Þ 2 2 EG À F2

Unit normal vector of a curvilinear surface g Â g r Â r r Â r n ¼ 1 2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu v ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu v ðD:30Þ 2 jjg1 Â g2 EG À F det gij

Differential of a surface area qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dA ¼ rjjg1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÂ g2 dudv ¼ g11g22 À ðÞg12 dudv pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD:31Þ 2 ¼ det gij dudv ¼ EG À F dudv

Gauss derivative equations

g Γ kg h g Γ kg h n i, j ¼ ij k þ ij 3 ¼ ij k þ ij : Γ k ðD 32Þ , gi j gi,j À ijgk ¼ hijn

Weingarten’s equations ÀÁ j jk : ni ¼Àhi gj ¼À hikg gj ðD 33Þ

Codazzi’s equations Appendix D: Essential Tensors 333

Kij, k ¼ Kik, j ) ðÞðK11,2 ¼ K12,1 ; K21,2 ¼ K22,1 D:34Þ

Gauss equations ÀÁ j 1 2 1 2 K ¼ det Ki ¼ K1K2 À K2K1 2 ðD:35Þ K11K22 À K12 R1212 ¼ 2 ¼ g11g22 À g12 g Appendix E: Euclidean and Riemannian Manifolds

In the following appendix, we summarize fundamental notations and basic results from vector analysis in Euclidean and Riemannian manifolds. This section can be written informally and is intended to remind the reader of some fundamentals of vector analysis in general curvilinear coordinates. For the sake of simplicity, we abstain from being mathematically rigorous. Therefore, we recommend some literature given in References for the mathematically interested reader.

E.1 N-Dimensional Euclidean Manifold

N-dimensional Euclidean manifold EN can be represented by two kinds of coordinate systems, Cartesian (orthonormal) und curvilinear (non-orthogonal) coordinate systems with N dimensions. Lines, curves, and surfaces can be considered as subsets of Euclidean manifold. Two lines or two curves can generate a ﬂat (planes) and curvilinear surface (cylindrical and spherical surfaces), respectively. Both kinds of surfaces can be embedded in Euclidean space.

E.1.1 Vector in Cartesian Coordinates

Cartesian coordinates are an orthonormal coordinate system in which the bases (i,j, k) are mutually perpendicular (orthogonal) and unitary (normalized vector length). The orthonormal bases (i,j,k) are ﬁxed in Cartesian coordinates. Any vector could be described by its components and the relating bases in Cartesian coordinates. The vector r can be written in Euclidean space E3 (three-dimensional space) in Cartesian coordinates (cf. Fig. E.1).

© Springer-Verlag Berlin Heidelberg 2017 335 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 336 Appendix E: Euclidean and Riemannian Manifolds

z z

P(x,y,z) zk r zk k 0 yj y xi j y i x

x (xi + yj)

Fig. E.1 Vector r in Cartesian coordinates

r ¼ xi þ yj þ zk ðE:1Þ where x, y, z are the vector components in the coordinate system (x,y,z); i, j, k are the orthonormal bases of the corresponding coordinates. The vector length of r can be computed using the Pythagorean theorem as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjr ¼ x2 þ y2 þ z2 0 ðE:2Þ

E.1.2 Vector in Curvilinear Coordinates

We consider a curvilinear coordinate system (u1,u2,u3) of Euclidean space E3, i.e. a coordinate system which is generally non-orthogonal and non-unitary (non-orthonormal basis). By abuse of notation, we denote the basis vector simply basis. In other words, the bases are not mutually perpendicular and their vector lengths are not equal to one [1, 2]. In the curvilinear coordinate system (u1,u2,u3), there are 1 2 3 three covariant bases g1, g2, and g3 and three contravariant bases g , g , and g at the origin 00, as shown in Fig. E.2. Generally, the origin 00 of the curvilinear coordinates could move everywhere in Euclidean space; therefore, the bases of the curvilinear Appendix E: Euclidean and Riemannian Manifolds 337 coordinates only depend on each considered origin 00. For this reason, the bases are not ﬁxed in the whole curvilinear coordinates such as in Cartesian coordinates, as displayed in Fig. E.1. The vector r of the point P(u1,u2,u3) can be written in the covariant and contravariant bases:

1 2 3 r ¼ u g1 þ u g2 þ u g3 : 1 2 3 ðE 3Þ ¼ u1g þ u2g þ u3g where u1, u2, u3 are the contravariant vector components of the coordinates (u1,u2,u3); 1 2 3 g1, g2, g3 are the covariant bases of the coordinate system (u ,u ,u ); 1 2 3 u1, u2, u3 are the covariant vector components of the coordinates (u ,u ,u ); g1, g2, g3 are the contravariant bases of the coordinate system (u1,u2,u3). The covariant basis gi can be deﬁned as the tangential vector to the i corresponding curvilinear coordinate u for i ¼ 1, 2, 3. Both bases g1 and g2 generate a tangential surface to the curvilinear surface (u1u2) at the considered origin 00,as shown in Fig. E.2. Note that the basis g1 is not perpendicular to the bases g2 and g3. 3 However, the contravariant basis g is perpendicular to the tangential surface (g1g2) at the origin 00. Generally, the contravariant basis gk results from the cross product of the other covariant bases (gi Â gj).

α k : g ¼ gi Â gj for i, j, k ¼ 1, 2, 3 ðE 4aÞ where α is a scalar factor (scalar triple product) given in Eq. (1.6). α ¼ gi Â gj Á gk hi ðE:4bÞ ; ; gi gj gk

Thus,

1 g2 Â g3 ; 2 g3 Â g1 ; 3 g1 Â g2 : g ¼ ; ; g ¼ ; ; g ¼ ; ; ðE 4cÞ ½g1 g2 g3 ½g1 g2 g3 ½g1 g2 g3 338 Appendix E: Euclidean and Riemannian Manifolds

g3 P(u1,u2,u3) r g1

0‘ g2 g2 g u1 1 u2

Fig. E.2 Bases of the curvilinear coordinates

E.1.3 Orthogonal and Orthonormal Coordinates

The coordinate system is called orthogonal if its bases are mutually perpendicular, as displayed in Fig. E.1. The dot product of two orthonormal bases is deﬁned as

i Á j ¼ jji Á jjj Á cos ðÞi; j π ¼ ðÞÁ1 ðÞÁ1 cos ðE:5Þ 2 ¼ 0

Thus,

i Á j ¼ i Á k ¼ j Á k ¼ 0 ðE:6Þ

If the length of each basis equals 1, the bases are unitary vectors.

jji ¼ jjj ¼ jjk ¼ 1 ðE:7Þ

If the coordinate system satisﬁes both conditions (E.6) and (E.7), it is called the orthonormal coordinate system, which exists in Cartesian coordinates. Therefore, the vector length in the orthonormal coordinate system results from Appendix E: Euclidean and Riemannian Manifolds 339

jjr 2 ¼ r Á r ¼ ðÞx i þ y j þ z k :ðÞx i þ y j þ z k ¼ x2ðÞþi Á i xyðÞþi Á j xzðÞi Á k ðE:8Þ þyxðÞþj Á i y2ðÞþj Á j yzðÞj Á k þzxðÞþk Á i zyðÞþk Á j z2ðÞk Á k

Due to Eqs. (E.6) and (E.7) the vector length in Eq. (E.8) becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjr 2 ¼ x2 þ y2 þ z2 ) jjr ¼ x2 þ y2 þ z2 ðE:9Þ

The cross product (called vector product) of a pair of bases of the orthonormal coordinate system is (informally) given by means of right-handed rule; i.e., if the right-hand ﬁngers move in the rotating direction from the basis j to the basis k, the thumb will point in the direction of the basis i ¼ j Â k. The bases (i, j, k) form a right-handed triple. 8 < i ¼ j Â k ¼Àk Â j : : j ¼ k Â i ¼Ài Â k ðE 10Þ k ¼ i Â j ¼Àj Â i

The magnitude of the cross product of two orthonormal bases is calculated as

i Â j ¼ jji Á jjj Á sin ðÞi; j k π ðE:11Þ ) jji Â j ¼ jji Á jjj Á sin ðÞi; j jjk ¼ 1 Á 1 Á sin jjk ¼ jjk 2

E.1.4 Arc Length Between Two Points in a Euclidean Manifold

We consider two points P(x1,x2,x3) and Q(x1,x2,x3) in Euclidean space E3 in Cartesian and curvilinear coordinate systems, as shown in Figs. E.3 and E.4. Both 1 2 3 points P and Q have three components x , x , and x in Cartesian coordinates (e1, e2, e3). To simplify some mathematically written expressions, the coordinates x, y, and z in Cartesian coordinates can be transformed into x1, x2, and x3; the bases (i, j, k) turn to (e1, e2, e3). We now turn to the notation of the differential dr of a vector r. The differential dr can be expressed using the Einstein summation convention [1, 3]:

i dr eidx for i ¼ 1, 2, 3 X3 i ðE:12Þ ¼ eidx i¼1 340 Appendix E: Euclidean and Riemannian Manifolds

The Einstein summation convention used in Eq. (E.12) indicates that dr equals the i sum of ei dx by running the dummy index i from 1 to 3. The arc length ds between the points P and Q (cf. Fig. E.3) can be calculated by the dot product of two differentials.

2 ðÞds ¼ dr Á dr ÀÁ ¼ ðÞÁe dxi e dxj ÀÁi j ðE:13Þ i j ¼ ei Á ej dx dx ¼ dxidxi for i ¼ 1, 2, 3: Thus, the arc length in the orthonormal coordinate system results in pffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ qdxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiidxi for i ¼ 1, 2, 3 ðE:14Þ ¼ ðÞdx1 2 þ ðÞdx2 2 þ ðÞdx3 2

The points P and Q have three components u1, u2, and u3 in the curvilinear coordinate system with the basis(g1, g2, g3) in Euclidean 3-space, as displayed in Fig. E.4. The location vector r(u1,u2,u3) of the point P is a function of ui. Therefore, the differential dr of the vector r can be rewritten in a linear formulation of dui.

∂r dr ¼ dui ∂ui ðE:15Þ i gidu i where gi is the covariant basis of the curvilinear coordinate u . x3 x3 P ds dr Q r (C) e3 r + dr 0 x2 e2 x2 e x1 1

x1 Fig. E.3 Arc length ds of P and Q in Cartesian coordinates Appendix E: Euclidean and Riemannian Manifolds 341

g3 P(ui(t ),…) 1 ds

r dr i Q(u (t2),…) r + dr 0‘ g 2 (C) 1 2 u g1 u

Fig. E.4 Arc length ds of P and Q in the curvilinear coordinates

Analogously, the arc length ds between two points of P and Q in the curvilinear coordinate system can be calculated by

ðÞds 2 ¼ dr Á dr ÀÁ i j ¼ gidu Á gjdu ðE:16Þ i j ¼ gi Á gj du du i j ¼ gij du du for i, j ¼ 1, 2, 3

Therefore, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i j ds ¼ gij du du sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt2 i j ðE:17Þ du du ) s ¼ g dt ij dt dt t1 where t is the parameter in the curve C with the coordinate ui(t); gij is defined as the metric coefficient of two non-orthonormal bases: 342 Appendix E: Euclidean and Riemannian Manifolds

δ j : gij ¼ gi Á gj ¼ gj Á gi ¼ gji 6¼ i ðE 18Þ

It is obvious that the symmetric metric coefﬁcients gij vanishes for any i 6¼ j in the orthogonal bases because gi is perpendicular to gj; therefore, the metric tensor can be rewritten as & 0ifi 6¼ j : gij ¼ ðE 19Þ gii if i ¼ j

In the orthonormal bases, the metric coefﬁcients gij in Eq. (E.19) becomes & 0ifi 6¼ j g δ j ¼ ðE:20Þ ij i 1ifi ¼ j

δj where i is called the Kronecker delta.

E.1.5 Bases of the Coordinates

The vector r can be rewritten in Cartesian coordinates of Euclidean space E3.

i r ¼ x ei ðE:21Þ

The differential dr results from Eq. (E.21)in

i dr ¼ eidx ∂r ðE:22Þ ¼ dxi ∂xi

i Thus, the orthonormal bases ei of the coordinate x can be deﬁned as

∂r e ¼ for i ¼ 1, 2, 3 ðE:23Þ i ∂xi

Analogously, the basis of the curvilinear coordinate ui can be calculated in the curvilinear coordinate system of E3. Appendix E: Euclidean and Riemannian Manifolds 343

∂r g ¼ for i ¼ 1, 2, 3 ðE:24Þ i ∂ui

Substituting Eq. (E.24) into Eq. (E.18) we obtain the metric coefﬁcients gij that are generally symmetric in Euclidean space; i.e., gij ¼ gji.

g ¼ g Á g ¼ g 6¼ δ j ij i j ji i ∂r ∂r ∂r ∂xm ∂r ∂xn ¼ Á ¼ Á ∂ui ∂uj ∂xm ∂ui ∂xn ∂uj : ∂xm ∂xn ∂xm ∂xn ðE 25Þ ¼ ðÞ¼e Á e δ n ∂ui ∂uj m n ∂ui ∂uj m ∂xk ∂xk ¼ for k ¼ 1, 2, 3 ∂ui ∂uj

According to Eqs. (E.4a) and (E.4b), the contravariant basis gk is perpendicular to k both covariant bases gi and gj. Additionally, the contravariant basis g is chosen such that the vector length of the contravariant basis equals the inversed vector k: length of its relating covariant basis; thus,g gk ¼ 1. As a result, the scalar products of the covariant and contravariant bases can be written in general curvilinear coordinates (u1,...,uN). & k k δ k ... gi Á g ¼ g Á gi ¼ i for i, k ¼ 1, 2, , N : δ k ... ðE 26Þ gi Á gk ¼ gik ¼ gki 6¼ i for i, k ¼ 1, 2, , N

E.1.6 Orthonormalizing a Non-orthonormal Basis

The basis {gi} is non-orthonormal in the curvilinear coordinates. Using the Gram-Schmidt scheme [4], an orthonormal basis (e1,e2,e3) can be created from the basis (g1,g2,g3). The orthonormalization procedure of the basis (g1,g2,g3) will be derived in this section; and the orthonormalizing scheme is demonstrated in Fig. E.5. 344 Appendix E: Euclidean and Riemannian Manifolds

g 3 g1

g2/1 g3 e e3 e1 2 e1

g3/1

g g3/2 e2 e2 2

Fig. E.5 Schematic visualization of the Gram-Schmidt procedure

The Gram-Schmidt scheme for N ¼ 3 has three orthonormalization steps:

1. Normalize the ﬁrst basis vector g1 by dividing it by its length to get the normalized basis e1.

g1 e1 ¼ jjg1

2. Project the basis g2 onto the basis g1 to get the projection vector g2/1 on the basis g1. The normalized basis e2 results from subtracting the projection vector g2/1 from the basis g2. Then, iteratively, normalize this vector by dividing it by its length to generate the basis e2.

g À g : 2 2=1 g2 À ðÞg2 e1 e1 e2 ¼ ¼ : jjg2 À ðÞg2 e1 e1 g2 À g2=1

3. Subtract projections along the bases of e1 and e2 from the basis g3 and normalize it to obtain the normalized basis e3.

g À g À g : : 3 3=1 3=2 g3 À ðÞg3 e1 e1 À ðÞg3 e2 e2 e3 ¼ ¼ : : jjg3 À ðÞg3 e1 e1 À ðÞg3 e2 e2 g3 À g3=1 À g3=2

Using the Gram-Schmidt scheme, the orthonormal basis {e1,e2,e3} results from the non-orthonormal bases {g1,g2,g3}. Generally, the orthogonal bases {e1,e2,...,eN} for the N-dimensional space can be generated from the non-orthonormal bases {g1,g2,...,gN} according to the Gram- Schmidt scheme as follows: Appendix E: Euclidean and Riemannian Manifolds 345

jXÀ 1 gj À gj Á ei ei i ¼ 1 ... ej ¼ for j ¼ 1, 2, , N XjÀ1 gj À gj Á ei ei i ¼ 1

E.1.7 Angle Between Two Vectors and Projected Vector Component

The angle θ between two vectors a and b can be deﬁned by means of the scalar product (Fig. E.6)

a Á b g Á g aibj cos θ ¼ ¼ i j jja Á jjb jja Á jjb : g aibj g aibj ðE 27Þ ij qffiffiffiffiffiffiffiffiffiffiffiffiffi ij qffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffi i j k l i j gija a Á gklb b a ai Á b bj where

jja 2 ¼ a Á a ðE:28Þ i j ij i ... ¼ gija a ¼ g aiaj ¼ a ai for i, j ¼ 1, 2, , N in which ai, bj are the contravariant vector components; ai, bj are the covariant vector components; ij gij, g are the covariant and contravariant metric coefﬁcients of the bases. The projected component of the vector a on vector b results from its vector length and Eq. (E.27).

ab ¼ jja Á cos θ g aibj g aibj ¼ jja Á ij ¼ ij jja Á jjb jjb ðE:29Þ g aibj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiij for i, j, k, l ¼ 1, 2, ..., N k l gklb b 346 Appendix E: Euclidean and Riemannian Manifolds

a(ui )

θ b(ui )

= θ ab a.cos

Fig. E.6 Angle between two vectors and projected vector component

Examples Given two vectors a and b: pffiffiffi i ; a ¼ 1 Á e1 þ 3 Á e2 ¼ a gi j b ¼ 1 Á e1 þ 0 Á e2 ¼ b gj

Thus, the relating vector components are

; g1 ¼ e1 g2 ¼ ep2 ffiffiffi a1 ¼ 1; a2 ¼ 3 b1 ¼ 1; b2 ¼ 0

The covariant metric coefﬁcients gij in the orthonormal basis (e1,e2) can be calculated according to Eq. (E.18). g11 g12 10 gij ¼ ¼ g21 g22 01

The angle θ between two vectors results from Eq. (E.27).

g aibj cos θ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiij qffiffiffiffiffiffiffiffiffiffiffiffiffi f or i, j, k, l ¼ 1, 2 i j: k l gija a gklb b g a1b1 þ g a1b2 þ g a2b1 þ g a2b2 ¼ 11 q12ffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi21 22 i j: k l gija a gklb b ÀÁpffiffiffi ÀÁpffiffiffi ðÞþ1 Á 1 Á 1 ðÞþ0 Á 1 Á 0 0 Á 3 Á 1 þ 1 Á 3 Á 0 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ 0 þ 0 þ 3 : 1 þ 0 þ 0 þ 0 2

Therefore, Appendix E: Euclidean and Riemannian Manifolds 347 1 π θ ¼ cos À1 ¼ 2 3

The projected vector component can be calculated according to Eq. (E.29).

i j gija b ab ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi for i, j, k, l ¼ 1, 2 g bkbl kl ÀÁpffiffiffi ÀÁpffiffiffi ðÞþ1 Á 1 Á 1 ðÞþ0 Á 1 Á 0 0 Á 3 Á 1 þ 1 Á 3 Á 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 1 þ 0 þ 0 þ 0

E.2 General N-Dimensional Riemannian Manifold

The concept of the Riemannian geometry is a very important fundamental brick in the modern physics of relativity and quantum field theories, theoretical elementary particles physics, and string theory. In contrast to the homogenous Euclidean manifold, the nonhomogenous Riemannian manifold only contains a tuple of fiber bundles of N arbitrary curvilinear coordinates of u1,...,uN. Each of the fiber bundle is related to a point and belongs to the N-dimensional differentiable Rie- mannian manifold. In case of the infinitesimally small fiber lengths in all dimensions, the fiber bundle now becomes a single point. Therefore, the tuple of fiber bundles becomes a tuple of points on the manifold. In fact, Riemannian manifold only contains a point tuple [10]. In turn, each point of the point tuple can move along a fiber bundle in N arbitrary directions (dimensions) in the N-dimensional Riemannian manifold. Generally, a hypersurface of the fiber bundle of curvilinear coordinates {ui} for i ¼ 1, 2,...,N at a certain point can be defined as a differentiable (NÀ1)-dimensional subspace with a codimension of one. This definition can be understood that the (NÀ1)-dimensional subbundle of fibers moves along the one-dimensional remaining fiber.

E.2.1 Point Tuple in Riemannian Manifold

We now consider an N-dimensional differentiable Riemannian manifold RN that contains a tuple of points. In general, each point on the manifold locally has N curvilinear coordinates of u1,...,uN embedded at this point. Therefore, the consid- 1 N ered point Pi can be expressed in the curvilinear coordinates as Pi(u ,...,u ). The notation of Riemannian manifold allows the local embedding of an N-dimensional afﬁne tangential manifold (called afﬁne tangential vector space) into the point Pi,as 348 Appendix E: Euclidean and Riemannian Manifolds

Riemannian manifold RN

uN gN

g2 Pi P (u1,…,uN) 1 i u 2 g1 u RN

Fig. E.7 Bundle of N coordinates at Pi in Riemannian manifold displayed in Fig. E.7. The arc length between any two points of N tuples of coordinates on the manifold does not physically change in any chosen basis. However, its components are changed in the coordinate bases that vary on the manifold. Therefore, these components must be taken into account in the transformation between different curvilinear coordinate systems in Riemannian manifold. To do that, each point in Riemannian manifold can be embedded with the individual metric coefficients gij for the relating point. Note that the metric coefficients gij of the coordinates (u1,...,uN) at any point are symmetric; and they totally have N2 components in an N-dimensional manifold. That means one can embed an affine tangential manifold EN at any point in Riemannian manifold RN in which the metric coefficients gij could be only applied to this point and change from one point to another point. However, the dot product (inner product) is not valid any longer in the affine tangential manifold [1, 10].

E.2.2 Flat and Curved Surfaces

By abuse of notation and by completely abstaining from mathematical rigorous- ness, we introduce the notation of flat and curved surfaces. A surface in Euclidean space is called flat if the sum of angles in any triangle ABC is equal to 180 or, alternatively, if the arc length between any two points fulfills the condition in Eq. (E.13). Therefore, the flat surface is a plane in Euclidean space. On the contrary, an arbitrary surface in a Riemannian manifold, is called curved if the angular sum in an arbitrary triangle ABC is not equal to 180, as displayed in Fig. E.8. Appendix E: Euclidean and Riemannian Manifolds 349

α +β + γ = 180° α +β + γ ≠ 180° A α A γ S β C α B B β γ C P

(a): flat surface P ⊂ E3 (b): curved surface S ⊂ R3

Fig. E.8 (a) Flat surfaces P E3 and (b) curved surfaces S R3

Conditions for the ﬂat and curved surfaces [5]: & α þ β þ γ ¼ 180 for a flat surface ðE:30Þ α þ β þ γ 6¼ 180 for a curved surface

Furthermore, the surface curvature in Riemannian manifold can be used to determine the surface characteristics. Additionally, the line curvature is also applied to studying the curve and surface characteristics.

E.2.3 Arc Length Between Two Points in Riemannian Manifold

We now consider a differentiable Riemannian manifold and calculate the arc length between two points P(u1,...,uN) and Q(u1,...,uN) in the curve C (see Fig. E.9). The arc length is an important notation in Riemannian manifold theory. The coordinates 1 N (u ,...,u ) can be considered as a function of the parameter t that varies from P(t1) to Q(t2). The arc length ds between the points P and Q thus results from ds 2 dr dr ¼ Á ðE:31Þ dt dt dt where the derivative of the vector r(u1,...,uN) can be calculated as

dr dðÞg ui ¼ i dt dt ðE:32Þ _ i ... giu ðÞt for i ¼ 1, 2, , N

Substituting Eq. (E.32) into Eq. (E.31), one obtains the arc length 350 Appendix E: Euclidean and Riemannian Manifolds

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε _ i _ j ds ¼ ðÞÁgiu gju dt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:33Þ ε _ i _ j ... ¼ giju ðÞt u ðÞt dt for i, j ¼ 1, 2, , N where ε (¼Æ1) is the functional indicator that ensures the square root always exists. Therefore, the arc length of PQ is given by integrating Eq. (E.33) from the parameter t1 to the parameter t2.

ðt2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε _ i _ j ... : s ¼ giju ðÞt u ðÞt dt for i, j ¼ 1, 2, , N ðE 34Þ

t1 where the covariant metric coefﬁcients gij are deﬁned by

δ j gij ¼ gi Á gj 6¼ i : ∂xk ∂xk ðE 35Þ ¼ Á for k ¼ 1, 2, ..., N ∂ui ∂uj

We now assume that the points P(u1,u2) and Q(u1,u2) lie on the Riemannian surface S, which is embedded in Euclidean space E3. Each point on the surface only depends on two parameterized curvilinear coordinates of u1 and u2 that are called the Gaussian surface parameters, as shown in Fig. E.9.

Riemannian surface S u2 a2 P Q a1 (C) x3 Euclidean space E3 r(u1,u2) u1

x2 0

Fig. E.9 Arc length between two points in a Riemannian surface Appendix E: Euclidean and Riemannian Manifolds 351

The differential dr of the vector r can be rewritten in the coordinates (u1,u2):

∂r dr ¼ dui ∂ui i ðE:36Þ r, idu i aidu for i ¼ 1, 2

i where ai is the tangential vector of the coordinate u on the Riemannian surface. Therefore, the arc length ds on the differentiable Riemannian parameterized surface can be computed as

ðÞds 2 ¼ dr Á dr i j : ¼ ai Á ajdu du ðE 37Þ i j aijdu du for i, j ¼ 1, 2 whereas aij are the surface metric coefficients only at the point P in the coordinates (u1, u2) on the Riemannian curved surface S. The formulation of (ds)2 in Eq. (E.37) is called the first fundamental form for the intrinsic geometry of Riemannian manifold [6–9]. The surface metric coefficients of the covariant and contravariant components have the similar characteristics such as the metric coefficients:

δ j aij ¼ aji ¼ ai Á aj 6¼ i : ∂xk ∂xk ðE 38aÞ ¼ Á for k ¼ 1, 2, ..., N; ∂ui ∂uj j j δ j : ai ¼ ai Á a ¼ i ðE 38bÞ

Instead of the metric coefﬁcients gij in the curvilinear Euclidean space, the surface metric coefﬁcients aij are used in the general curvilinear Riemannian manifold.

E.2.4 Tangent and Normal Vectors on the Riemannian Surface

We consider a point P(u1,u2) on a differentiable Riemannian surface that is param- 1 2 eterized by u and u . Furthermore, the vectors a1 and a2 are the covariant bases of the curvilinear coordinates (u1,u2), respectively. In general, a hypersurface in an N- dimensional manifold with coordinates {ui} for i ¼ 1, 2,...,N can be deﬁned as a differentiable (NÀ1)-dimensional subspace with a codimension of 1. i The basis ai of the coordinate u can be rewritten as

∂r a ¼ i ∂ui ðE:39Þ r, i for i ¼ 1, 2 352 Appendix E: Euclidean and Riemannian Manifolds

NP tangential surface T 2 r,2 = a2 u nP Riemannian surface θ12 P r,1 = a1 r(u1, u2) (S)

0 u1

Fig. E.10 Tangent vectors to the curvilinear coordinates (u1,u2)

i The covariant basis ai is tangent to the coordinate u at the point P. Both bases a1 and a2 generate the tangential surface T tangent to the Riemannian surface S at the point P, which is defined by the curvilinear coordinates of u1 and u2, as shown in Fig. E.10. The angle of two intersecting Gaussian parameterized curves ui and uj results from the dot product of the bases at the point P(ui,uj). ÀÁ ai Á aj ¼ jjai Á aj cos ai; aj ) ÀÁa Á a cos θ cos ; i j ij ai aj ¼ : jjai Á aj hi ðE 40Þ aij π ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1forθij 2 0, aðÞii Á aðÞjj 2

Note that pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi jjai ¼ ai Á ai ¼ aðÞii , no summation over (ii) where aii and ajj are the vector lengths of ai and aj; aij, the surface metric coefficients. The surface metric coefficients can be defined by

δ j aij ¼ ai Á aj 6¼ i : ∂xk ∂xk ðE 41Þ ¼ Á for k ¼ 1, 2, ..., N ∂ui ∂uj Appendix E: Euclidean and Riemannian Manifolds 353

E.2.5 Angle Between Two Curvilinear Coordinates

We now give a concrete example of the computation of the angle between two curvilinear coordinates. Given two arbitrary basis vectors at the point P(u1,u2), we can write them with the covariant basis {ei}:

a1 ¼ 1 Á e1 þ 0 Á e2; a2 ¼ 0 Á e1 þ 1 Á e2

The covariant metric coefﬁcients aij can be calculated: ÀÁ a11 a12 10 aij ¼ ¼ a21 a22 01

The angle between two base vectors results from Eq. (E.40):

aij cos θij ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi aðÞii Á aðÞjj a12 0 ) cos θ12 ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi ¼ pffiffiffi pffiffiffi ¼ 0 a11 Á a22 1 Á 1

Thus, π À1 a12 À1 θ12 ¼ cos pffiffiffiffiffiffi pffiffiffiffiffiffi ¼ cos ðÞ¼0 a11 Á a22 2

In this case, the curvilinear coordinates of u1 and u2 are orthogonal at the point P on the Riemannian surface S, as shown in Fig. E.10. The tangent vectors a1 and a2 generate the tangential surface T tangent to the Riemannian surface S at the point P. The normal vector NP to the tangential surface T at the point P is given by

∂r ∂r N ¼ Â ¼ r Â r P ∂ui ∂uj , i , j ðE:42Þ ai Â aj for i, j ¼ 1, 2 ¼ α ak where α is the scalar factor; ak is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (E.42) by the covariant basis ak, the scalar factor α results in 354 Appendix E: Euclidean and Riemannian Manifolds ÀÁ ÀÁ α ak Á a ¼ αδk ¼ α ¼ a Â a Á a kÀÁk ÂÃi j k ðE:43Þ ) α ¼ ai Â aj Á ak ai; aj; ak

The scalar factor α equals the scalar triple product that is given in [2]: ÀÁ ÀÁ α ½¼a1; a2; a3 ai Â aj Á ak ¼ ðÞÁak Â ai aj ¼ aj Â ak Á ai 1 1 1 2 2 2 a11 a12 a13 a31 a32 a33 a21 a22 a23 ¼ a21 a22 a23 ¼ a11 a12 a13 ¼ a31 a32 a33 ðE:44Þ a a a a a a a a a q31ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ32 33 21 22 23 11 12 13 ¼ det aij J where Jacobian J is the determinant of the covariant basis tensor. The unit normal vector nP in Eq. (E.42) becomes using the Lagrange identity.

ai Â aj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiai Â aj : nP ¼ ¼ ÀÁ ðE 45Þ ai Â aj 2 aðÞii Á aðÞjj À aij

Note that 2 jjai ¼ ai Á ai ¼ aðÞii , no summation over (ii) The Lagrange identity results from the cross product of two vectors a and b.

jj¼a Â b jjÁa jjb sin ðÞ)a; b jja Â b 2 ¼ jja 2 Á jjb 2 sin 2ðÞa; b ÀÁ ¼ jja 2 Á jjb 2 Á 1 À cos 2ðÞa; b ¼ ðÞjja Á jjb 2 À ðÞjja Á jjb Á cos ðÞa; b 2 ¼ ðÞjja Á jjb 2 À ðÞa Á b 2

Thus, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jja Â b ¼ jja 2 Á jjb 2 À ðÞa Á b 2 ðE:46Þ

Equation (E.46) is called the Lagrange identity.

E.2.6 Surface Area in Curvilinear Coordinates

The surface area S in the differentiable Riemannian curvilinear surface, as displayed in Fig. E.11, can be calculated using the Lagrange identity [2]. Appendix E: Euclidean and Riemannian Manifolds 355

u2 a2 a du2 2 du1 1 2 P(u ,u ) dS a1 1 a1du r(u1,u2) du2

0 u1

Fig. E.11 Surface area in the curvilinear coordinates

ZZ ∂r ∂r S ¼ Â duiduj ∂ i ∂ j ZZ u u i j ¼ ai Â aj du du ZZ : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ ðE 47Þ 2 2 2 i j ¼ jjai Á aj À ai Á aj du du ZZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ 2 i j ¼ aðÞii Á aðÞjj À aij du du

Therefore, ZZ 1 2 S ¼ jja1 Â a2 du du for i ¼ 1; j ¼ 2 ZZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðE:48Þ 2 1 2 ¼ a11 Á a22 À ðÞa12 du du

In Eq. (E.47), the vector length squared can be calculated as 2 jjai ¼ ai Á ai ¼ aðÞii , no summation over (ii)

E.3 Kronecker Delta

The Kronecker delta is very useful in tensor analysis and is deﬁned as 356 Appendix E: Euclidean and Riemannian Manifolds & ∂uj 0fori 6¼ j δ j ¼ ðE:49Þ i ∂ui 1fori ¼ j where ui and uj are in the same coordinate system and independent of each other. Some properties of the Kronecker delta (Kronecker tensor) are considered [2, 5]. We summarize a few properties of the Kronecker delta: Property 1 The chain rule of differentiation of the Kronecker delta using the contraction rule (cf. Appendix A)

∂uj ∂uj ∂uk δ j ¼ ¼ ¼ δ jδ k ðE:50Þ i ∂ui ∂uk ∂ui k i

Property 2 Kronecker delta in Einstein summation convention

δ i jk δ i 1k δ i ik δ i Nk j a ¼ 1a þÁÁÁþ i a þÁÁÁþ Na ¼ 0 þÁÁÁþ1 Á aik þÁÁÁþ0 ðE:51Þ ¼ aik

Property 3 Product of Kronecker deltas

δ iδ j δ i δ1 δ iδ i δ i δ N j k ¼ 1 k þÁÁÁþ i k þÁÁÁþ N k δ i : ¼ 0 þÁÁÁþ1 Á k þÁÁÁþ0 ðE 52Þ δ i ¼ k

Note that δðÞi δ1 δ2 δ N ðÞi 1 ¼ 2 ¼ÁÁÁ¼ N ¼ 1 (no summation over the free index i); However, δ i δ1 δ2 δ N i 1 þ 2 þÁÁÁþ N ¼ N (summation over the dummy index i).

E.4 Levi-Civita Permutation Symbols

Levi-Civita permutation symbols in a three-dimensional space are third-order pseudo-tensors. They are a useful tool to simplify the mathematical expressions and computations [1–3]. The Levi-Civita permutation symbols can simply be deﬁned as Appendix E: Euclidean and Riemannian Manifolds 357

j = 1,2,3

⎛ ⎞ ⎜ 0 1 0⎟ i = 1,2,3 ⎜−1 0 0⎟ ⎜ ⎟ ⎝ 0 0 0⎠ = ⎛ − ⎞ εijk ⎜0 0 1⎟ ⎜0 0 0 ⎟ ⎜ ⎟ ⎝1 0 0 ⎠ ⎛ ⎞ ⎜0 0 0⎟ ⎜0 0 1⎟ ⎜ ⎟ ⎝0 −1 0⎠

Fig. E.12 27 Levi-Civita permutation symbols

8 < þ1ifðÞi, j, k is an even permutation; ε ¼ À1ifðÞi, j, k is an odd permutation; ijk : : 0ifi ¼ j, or i ¼ k; or j ¼ k ðE 53Þ 1 , ε ¼ ðÞÁi À j ðÞÁj À k ðÞk À i for i, j, k ¼ 1, 2, 3 ijk 2

Here, we abstain from giving an exact deﬁnition of even and odd permutation because this would go beyond the scope of this book. The reader is referred to the literature [8, 9]. According to Eq. (E.53), the Levi-Civita permutation symbols can be expressed as & εijk ¼ εjki ¼ εkij ðÞeven permutation ; εijk ¼ ðE:54Þ Àεikj ¼Àεkji ¼ÀεjikðÞodd permutation

The 27 Levi-Civita permutation symbols for a three-dimension coordinate system are graphically displayed in Fig. E.12.

References

1. Klingbeil, E.: Tensorrechnung fur€ Ingenieure (in German). B.I.-Wissenschafts-verlag, Mann- heim (1966) 2. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 3. Kay, D.C.: Tensor Calculus. Schaum’s Outline Series. McGraw-Hill, New York, NY (2011) 4. Grifﬁths, D.J.: Introduction to Quantum Mechanics, 2nd edn. Pearson Prentice Hall, Upper Saddle River, NJ (2005) 5. Oeijord, N.K.: The Very Basics of Tensors. IUniverse, New York, NY (2005) 6. Springer, C.E.: Tensor and Vector Analysis. Dover Publications, Mineola, NY (2012) 7. Lang, S.: Fundamentals of Differential Geometry. Springer, Berlin (1999) 358 Appendix E: Euclidean and Riemannian Manifolds

8. Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2000) 9. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 10. Riemann, B.: U¨ ber die Hypothesen, welche der Geometrie zu Grunde liegen. Springer Spektrum, Berlin (2013) Appendix F: Probability Function for the Quantum Interference

In case of the interference pattern, the probability function of the electrons for ﬁnding the electron at position z in a long observing time is the squared amplitude of the sum of the wave functions ψA(z) and ψB(z). The complex wave functions ψA(z) and ψB(z) are displayed in a complex plane, as shown in Fig. F.1.

ψ B

ψ AB

θ ϕ B ψ α A ϕ A 0 Re

Fig. F.1 The sum function ψ AB of two complex wave functions

© Springer-Verlag Berlin Heidelberg 2017 359 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 360 Appendix F: Probability Function for the Quantum Interference

Using the law of cosines for a triangle, the probability function PAB(z) at position z is calculated as ψ 2 ψ ψ 2 PABðÞz kkAB ¼ kkA þ B ¼ kkψ 2 þ kkψ 2 À 2kkψ Á kkψ cos θ A B A B ðF:1Þ ψ 2 ψ 2 ψ ψ α ¼ kkA þ kkB þ 2kkA Á kkB cos ψ ψ α ¼ PAðÞþz PBðÞþz 2kkA Á kkB cos where PA(z) and PB(z) are the probability functions at position z of the wave functions ψA(z) and ψB(z). The wave functions are expressed in the complex form as

φ ψ ψ φ φ ψ i A ; A ¼ kkÁA ðÞ¼cos A þ i sin A kkA e φ ðF:2Þ ψ ψ φ φ ψ i B B ¼ kkB Á ðÞ¼cos B þ i sin B kkB e

The complex conjugate of the wave function of ψB results as

ψ * ¼ kkψ Á ðÞcos φ À i sin φ B B B B : φ ðF 3Þ ψ φ φ ψ Ài B ¼ kkÁB ½¼cos ðÞþÀ B i sin ðÞÀ B kkB e

The product of two complex wave functions is calculated as

φ φ ψ ψ * ¼ kkψ Á kkψ eiðÞAÀ B A B A B ðF:4Þ ψ ψ φ φ φ φ ¼ kkA Á kkB Á ½cos ðÞþA À B i sin ðÞA À B

The difference angle between the phase angles of the wave functions results as φ φ α φ φ α : B À A ¼ ) A À B ¼À ðF 5Þ

Substituting Eqs. (F.4) and (F.5), one calculates the real part of Eq. (F.4) ÀÁ ψ ψ * ψ ψ φ φ Re A B ¼ kkA Á kkB cos ðÞA À B ψ ψ α ¼ kkA Á kkB cos ðÞÀ ðF:6Þ ¼ kkψ Á kkψ cos α ÀÁA B ψ * ψ ¼ Re A B

Substituting Eq. (F.6) into Eq. (F.1), one obtains the probability function PAB(z)at position z ÀÁ P ðÞ¼z kkψ 2 þ kkψ 2 þ 2Re ψ ψ * AB A B ÀÁA B ¼ P ðÞþz P ðÞþz 2Re ψ ψ * ðF:7Þ A B ÀÁA B ψ * ψ ¼ PAðÞþz PBðÞþz 2Re A B

The third term on the RHS of Eq. (F.7) represents the quantum interference in the probability function. Appendix G: Lorentz and Minkowski Transformations in Spacetime

The Lorentz and Minkowski transformations deal with the special relativity theory (SRT) in a four-dimensional spacetime. Let S be an inertial coordinate system; S0 be a moving coordinate system with a velocity v relative to S (s. Fig. G.1). In the case that the velocity v is parallel to the coordinate x, the Lorentz transformation between two coordinate systems is written as

0 x À vt x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffi γ Á ðÞx À vt v2 1 À c2 y0 ¼ y 0 : z ¼ z ðG 1Þ vx t À 0 2 vx t ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffic γ Á t À v2 c2 1 À c2

z z′

r = ct y y′ r′ = ct′

v < c x x′ S S′

Fig. G.1 Inertial and moving coordinate systems

© Springer-Verlag Berlin Heidelberg 2017 361 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 362 Appendix G: Lorentz and Minkowski Transformations in Spacetime

The transformed coordinates of the moving system S0 are written in the transformation matrix L as 2 3 2 3 0 12 3 0 γ γ x x 00À v x 6 0 7 6 7 B 0100C6 7 6 y 7 6 y 7 B C6 y 7 : 4 0 5 ¼ L4 5 ¼ @ 0010A4 5 ðG 2Þ z z γv z 0 À 00 γ t t c2 t where c is the constant light speed, t and t0 are the time coordinates in the systems S and S0, respectively; γ is called the Lorentz factor, which is deﬁned as

1 γ qffiffiffiffiffiffiffiffiffiffiffiffi 1 v2 1 À c2

The Lorentz backtransformation results from Eq. (G.2)as 2 3 2 3 0 12 3 0 γ γ 0 x x 00 v x 6 7 6 0 7 B 0100C6 0 7 6 y 7 À16 y 7 B C6 y 7 : 4 5 ¼ L 4 0 5 ¼ @ 0010A4 0 5 ðG 3Þ z z γv z 0 00 γ 0 t t c2 t

Thus, the coordinates of the inertial system S are written as

0 0 x þ vt 0 0 x ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ γ Á x þ vt v2 1 À c2 y ¼ y0 0 : z ¼ z ðG 4Þ 0 0 vx t þ 0 2 0 vx t ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffic ¼ γ Á t þ v2 c2 1 À c2

Note that the larger the moving speed v, the shorter the length of an object in the moving coordinate system S0 becomes. This effect is called the length contraction. rffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 l v l ¼ ¼ l Á 1 À l ðG:5Þ γ c2

On the contrary, the larger the moving speed v, the longer the time interval between two events encounters compared to the standing coordinate system S. This effect is deﬁned as the time dilation. This result indicates that the clock runs more slowly in the moving coordinate system. If the moving speed v reaches the light speed c, the Appendix G: Lorentz and Minkowski Transformations in Spacetime 363 clock would stop running since the time interval between two events will go to inﬁnity (t0 !1) in the moving coordinate system.

0 t t ¼ γt ¼ qffiffiffiffiffiffiffiffiffiffiffiffi t ðG:6Þ v2 1 À c2

However, if the velocity v is in an arbitrary direction in the coordinate system S, the propagating distances of a light signal transmitting with a constant light speed c at the times t and t0 result in the coordinate systems S and S0 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ qxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 þ y2 þ z2 ¼ ct; ðG:7Þ r0 ¼ x0 2 þ y0 2 þ z0 2 ¼ ct0

Having squared both sides of Eq. (G.7), one obtains

2 2 2 2 2 ; x þ y þ z À c t ¼ 0 : 0 2 0 2 0 2 0 2 ðG 8Þ x þ y þ z À c2t ¼ 0:

The generalized Lorentz transformation between two arbitrary coordinate systems results from Eq. (G.8) for any proportional parameter σ as ÀÁ 0 2 0 2 0 2 0 2 x þ y þ z À c2t ¼ σ Á x2 þ y2 þ z2 À c2t2 ¼ 0 ðG:9Þ

Equation (G.9) is also valid for σ ¼ 1; therefore, the generalized Lorentz transformation becomes

0 2 0 2 0 2 0 2 x þ y þ z À c2t ¼ x2 þ y2 þ z2 À c2t2 ðG:10Þ

The Minkowski transformation in the field-free spacetime with four independent dimensions of x1, x2, x3, and x4 is written as 0 1 2 3 0 12 3 x1 ¼ x x1 1000 x B C 6 7 B C6 7 B x2 ¼ y C 6 x2 7 B 0100C6 y 7 @ A , 4 5 ¼ @ A4 5 ðG:11Þ x3 ¼ pz ffiffiffiffiffiffiffi x3 0010 z x4 ¼ À1ct ¼ jct x4 000jc t

Therefore,

2 2 2 2 2 2 2 2 2 : x1 þ x2 þ x3 þ x4 ¼ x þ y þ z À c t ðG 12Þ

Thus, the Minkowski transformation satisﬁes the generalized Lorentz transformation of Eq. (G.10) using in the special relativity theory, cf. Fig. 5.11. Both transformations are used for spacetimes without gravitational ﬁeld. 364 Appendix G: Lorentz and Minkowski Transformations in Spacetime

The general relativity theory (GRT) is the field theory of gravitation, in which the spacetime is curved with nonzero Riemannian curvature at any point on its surface. On the contrary, the SRT is only valid in a flat spacetime in which gravity does not exist. Hence, the GRT is a geometric theory of gravity that determines the structure of spacetime by means of curvatures at each point on the spacetime surface. The principle of equivalence indicates that the effect of an accelerating coordinate system has the same effect of gravity, in which the spacetime is affected by a gravitational field. As a result, the geometric structure of an accelerated spacetime must be curved. If a matter is under a gravitational or electromagnetic field in a spacetime, it is moved in the spacetime. In turn, the matter curves the spacetime by means of its gravitational or electromagnetic force acting upon the spacetime. Therefore, the four-dimensional Minkowski transformation is a pseudo-Euclidean flat spacetime is only obtained in the absence of gravity. In the GRT, a metric tensor gμv is necessary for the curved spacetime under a gravitational field, in which the geodesic distance ds (i.e. the shortest distance between two arbitrary points) is written in a four-dimensional spacetime as

2 ; μ : ds ¼ gμvdxμdxv 8 , 8v ¼ 1, 2, 3, 4 ðG 13Þ

Using the Schwarzschild metric tensor for a gravitational ﬁeld, the geodesic in a curved spacetime is expressed in the rectangular coordinates as

ds2 ¼ g dx2 þ g dx2 þ g dx2 þ g dx2 11 1 22 2 33 3 44 4 G:14 2 2 2 2 ð Þ ¼ g11dx þ g22dy þ g33dz þ g44dt

The Schwarzschild metric tensor in Eq. (G.14) is written as 0 1 g 000 B 11 C B 0 g22 00C : gμv ¼ @ A ðG 15Þ 00g33 0 000g44

The principle components of the Schwarzschild metric tensor are deﬁned as

1 g ¼ g ¼ g ; 11 22 33 2MG 1 À c2r ðG:16Þ 2MG g Àc2 1 À 44 c2r where G is the gravitational constant (G ¼ 6.673 Â 10À11 m3 kgÀ1 sÀ2); M is the mass of the object; r is the radial coordinate of the spherical spacetime. Appendix H: The Law of Large Numbers in Statistical Mechanics

Let X1, X2, ..., XN be N independent variables that have the probability densities ρ1(X1), ρ2(X2),..., ρN(XN), respectively. The probability densities satisfy the condition

þ1ð ρ ... : iðÞXi dXi ¼ 1 fori ¼ 1, 2, , N ðH 1Þ À1

The joint probability density of N independent variables gives the relation

ρ ; ...; ... ρ ...ρ :::: : ðÞX1 XN dX1 dXN ¼ 1ðÞX1 NðÞXN dX1 dXN ðH 2Þ

Integrating the joint probability density over the entire space, one obtains using Eq. (H.1)

þ1ð þ1ð þ1ð þ1ð ... ρðÞX ; ...; X dX ...dX ¼ ρ ðÞX dX ... ρ ðÞX dX 1 N 1 N 1 1 1 N N N ðH:3Þ À1 À1 À1 À1 ¼ 1

Let ΔX1,...,ΔXN be the differences of the independent variables X1, ...,XN to their average value, respectively. They can be expressed as

ΔXi ¼ Xi À hiXi fori ¼ 1, 2, ..., N ðH:4Þ where the expectation (average value) of the variable Xi is deﬁned as

© Springer-Verlag Berlin Heidelberg 2017 365 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 366 Appendix H: The Law of Large Numbers in Statistical Mechanics

þ1ð ρ : hiXi Xi iðÞXi dXi ðH 5Þ À1

Similarly, the variance of variable Xi is written as DEDE 2 2 VarðÞ Xi ðÞΔXi ¼ ðÞXi À hiXi þ1ð ðH:6Þ 2ρ ¼ ðÞXi À hiXi iðÞXi dXi À1

The mean value Xm of the N independent variables is deﬁned as

1 XN Xm ¼ Xj ðH:7Þ N j¼1

Using Eqs. (H.5) and (H.7), the expectation of the mean value Xm is calculated as

þ1ð

hiXm XmρðÞX1; ...; XN dX1 ...dXN À1 þ1ð ρ ...ρ ... : ¼ Xm 1ðÞX1 NðÞXN dX1 dXN ðH 8Þ À1 þ1ð XN XN 1 ρ 1 ¼ Xi iðÞXi dXi hiXi N i¼1 N i¼1 À1

The variance of the mean value Xm is written as DEDE 2 2 VarðÞ Xm ðÞΔXm ¼ ðÞXm À hiXm þ1ð ðH:9Þ 2 ¼ ðÞXm À hiXm ρðÞX1; ...; XN dX1 ...dXN À1

Substituting Eqs. (H.7) and (H.8) into Eq. (H.9), one obtains the variance of Xm Appendix H: The Law of Large Numbers in Statistical Mechanics 367

þ1ð DEXN Δ 2 1 2ρ ðÞXm ¼ 2 ðÞXi À hiXi iðÞXi dXi N i¼1 À1 ðH:10Þ XN DE 1 Δ 2 σ2 ¼ 2 ðÞXi m N i¼1

Thus, the standard deviation of the mean value Xm results as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u DEuXN DE 1 t σ ¼ ðÞΔX 2 ¼ ðÞΔX 2 ðH:11Þ m m N i i¼1

In the case of equal probabilities for all N independent variables, the standard deviation of the mean value Xm results as DE DEΔ 2 N ðÞXi Var X σ2 Δ 2 ðÞi m ¼ ðÞXm ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N2 N ðH:12Þ VarðÞ Xi σ ) σm ¼ pffiffiffiffi pffiffiffiffi N N where σ is the standard deviation of variable Xi. Equation (H.12)isthe law of large numbers in statistical mechanics. It denotes that the mean value Xm of N independent variables Xi for i ¼ 1, 2,...,N has a 1/2 standard deviation that is only 1/N of the standard deviation of variable Xi in case of equal probabilities for N independent variables. In statistics, there are two laws of large numbers, the weak and strong laws. The weak law states that the mean value of N independent variables converges in probability towards the expected value when the number of variables is very large.

XN 1 P lim Xm lim Xi ¼ hiXm ðH:13Þ N!1 N!1 N i¼1

In this case, the probability density function for the weak law is written for any positive real number ε as

lim ρðÞ¼jjXm À hiXm > ε 0 ðH:14Þ N!1

The strong law denotes that the mean value of N independent variables always converges to the expected value when the number of variables is very large. 368 Appendix H: The Law of Large Numbers in Statistical Mechanics

1 XN lim Xm lim Xi ¼ hiXm ðH:15Þ N!1 N!1 N i¼1

In this case, the probability density function for the strong law results as

ρ lim Xm ¼ hiXm ¼ 1: ðH:16Þ N!1 Mathematical Symbols in This Book

• First partial derivative of a second-order tensor with respect to uk

∂Tij Tij ð1Þ ,k ∂uk

Do not confuse Eq. (1) with the symbol used in some books:

∂Tij Tij þ Γ i Tmj þ Γ j Tim ,k ∂uk km km

This symbol is equivalent to Eq. (2) used in this book. • Covariant derivative of a second-order tensor with respect to uk

ij ij Γ i mj Γ j im T jk T, k þ kmT þ kmT ð2Þ

• Second partial derivative of a ﬁrst-order tensor with respect to uj and uk

∂2T T i ð3Þ i,jk ∂uj∂uk

Do not confuse Eq. (3) with the symbol used in some books:

∂2T ∂T ∂T T i À Γ m T À Γ m m À Γ m m i,jk ∂uj∂uk ik, j m ik ∂uj ij ∂uk ∂T þΓ mΓ n T À Γ m i þ Γ mΓ n T ij mk n jk ∂um jk im n

This symbol is equivalent to Eq. (4) used in this book. • Second covariant derivative of a ﬁrst-order tensor with respect to uj and uk

© Springer-Verlag Berlin Heidelberg 2017 369 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 370 Mathematical Symbols in This Book

Γ m Γ m Γ m Tikj Ti, jk À ik, jTm À ik Tm, j À ij Tm, k Γ mΓ n Γ m Γ mΓ n ð4Þ þ ij mkTn À jk Ti,m þ jk imTn

• Christoffel symbols of ﬁrst kind Γijk instead of ½ij, k used in some books • Christoffel symbols&' of second kind k Γk instead of used in some books. ij ij Further Reading

1. Aris, R.: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover Publications, New York, NY (1989) 2. Ba¨r, C.: Elementare Differentialgeometrie (in German), Zweiteth edn. De Gruyter, Berlin (2001) 3. Cahill, K.: Physical Mathematics. Cambridge University Press, Cambridge (2013) 4. Chen, N.: Aerothermodynamics of Turbomachinery, Analysis and Design. Wiley, Singapore (2010) 5. De, U.C., Shaikh, A., Sengupta, J.: Tensor Calculus, 2nd edn. Alpha Science, Oxford (2012) 6. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford Science, Oxford (1958) 7. Grifﬁths, D.J.: Introduction to Quantum Mechanics, 2nd edn. Pearson Prentice Hall, Upper Saddle River, NJ (2005) 8. Grinfeld, P.: Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer Science + Business Media, New York, NY (2013) 9. Itskov, M.: Tensor Algebra and Tensor Analysis for Engineers – With Applications to Continuum Mechanics, 2nd edn. Springer, Berlin (2010) 10. Kay, D.C.: Tensor Calculus. Schaum’s Outline Series. McGraw-Hill, New York, NY (2011) 11. Klingbeil, E.: Tensorrechnung fur€ Ingenieure (in German). B.I.-Wissenschafts-verlag, Mann- heim (1966) 12. Kuhnel,€ W.: Differentialgeometrie – Kurven, Fla¨chen, Mannigfaltigkeit (in German), 6th edn. Springer-Spektrum, Wiesbaden (2013) 13. Landau, D., Lifshitz, E.M.: The Classical Theory of Fields. Addison-Wesley, Reading, MA (1962) 14. Lang, S.: Fundamentals of Differential Geometry, 2nd edn. Springer, New York, NY (2001) 15. Lawden, D.F.: Introduction to Tensor Calculus, Relativity and Cosmology, 3rd edn. Dover Publications, Mineola, NY (2002) 16. Longair, M.: Quantum Concepts in Physics. Cambridge University Press, Cambridge (2013) 17. Lovelock, D., Rund, H.: Tensors, Differential Forms, and Variational Principles. Dover Publications, New York, NY (1989) 18. McConnell, A.J.: Applications of Tensor Analysis. Dover Publications, New York, NY (1960) 19. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 20. Oeijord, N.K.: The Very Basics of Tensors. IUniverse, New York, NY (2005) 21. Penrose, R.: The Road to Reality. Alfred A. Knopf, New York, NY (2005)

© Springer-Verlag Berlin Heidelberg 2017 371 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 372 Further Reading

22. Schobeiri, M.: Turbomachinery Flow Physics and Dynamic Performance, 2nd edn. Springer, Berlin (2012) 23. Shankar, R.: Principles of Quantum Mechanics, 2nd edn. Springer, Berlin (1980) 24. Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York, NY (1982) 25. Springer, C.E.: Tensor and Vector Analysis. Dover Publications, Mineola, NY (2012) 26. Susskind, L., Lindesay, J.: An Introduction to Black Holes, Information and the String Theory Revolution. World Scientiﬁc Publishing, Singapore (2005) 27. Synge, J.L., Schild, A.: Tensor Calculus. Dover Publications, New York, NY (1978) 28. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 29. Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2000) 30. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. AMS, New York, NY (1978) 31. Messiah, A.: Quantum Mechanics – Two Volumes. Dover Publications, New York, NY (2014) 32. Cohen-Tannoudji, C., Diu, B., Laloe, F.: Quantum Mechanics – Two Volumes. Wiley, New York, NY (1977) Index

A Compton wavelength, 291 Absolute probability, 259 Computational fluid dynamics (CFD), 204 Abstract index notation, 97, 239 Congruence, 126 Adjoint, 17 Constitutive equations, 213 Ambient coordinate, 140, 141 Continuity equation, 204 Ampere-Maxwell law, 226 Contravariant basis, 337 Angle between two vectors, 345 Contravariant metric tensor components, 69 Anti-symmetric, 21, 64 Coordinate velocity, 140, 141 Arc length, 340, 341, 350 Coriolis acceleration, 207 Area differential, 105 Cosmological constant, 240 Auto-parallel, 150 Cotangent bundle, 146 Cotangent space, 146 Covariant and contravariant bases, 42 B Covariant basis, 337 Basis tensors, 51 Covariant derivative, 90 Bianchi first identity, 92 Covariant Einstein tensor, 99 Bianchi second identity, 100 Covariant first-order tensor, 88 Black hole, 243 Covariant metric tensor components, 69 Block symmetry, 92 Covariant partial derivative, 88 Bra, 17 Covariant Riemann curvature tensor, 92 Bra and ket, 15 Cross product, 56 Curl (rotation), 188 Curl identity, 191 C Curvature vector, 108 Cartan’s formula, 136 Curved spacetime, 364 Cauchy’s strain tensor, 220, 222 Curved surfaces, 349 Cauchy’s stress tensor, 215 Cyclic property, 92 Cayley-Hamilton theorem, 219 Cylindrical coordinates, 5 CFD. See Computational fluid dynamics (CFD) Characteristic equation, 13, 32, 219 Christoffel symbol, 84 D Codazzi’s equation, 125 Density operator, 273 Commutator, 282 Derivative of the contravariant basis, 84 Complex conjugate, 257 Derivative of the covariant basis, 76 Compton effect, 289 Derivative of the product, 84

© Springer-Verlag Berlin Heidelberg 2017 373 H. Nguyen-Scha¨fer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI 10.1007/978-3-662-48497-5 374 Index

Differential forms, 157 Frenet orthonormal frame, 120 Dirac delta function, 262 Friction stress contravariant tensor, 210 Dirac equation, 310 Divergence, 184, 186 Divergence theorem, 197 G Dual bases, 42 Gauss-Bonnet theorem, 120 Dual space, 146 Gauss-Codazzi equations, 124 Dual vector spaces, 36 Gauss derivative equations, 122 Gauss divergence theorem, 198 Gauss equation, 126 E Gaussian curvature, 113, 117 Eigenfrequency, 13 Gaussian surface parameters, 350 Eigenkets, 30 Gauss’s law for electric fields, 225 Eigenvalue, 11 Gauss’s law for magnetic fields, 225 Eigenvector, 13 Gauss’s Theorema Egregium, 114 Einstein field equations, 239 Gauss theorem, 197 Einstein-Maxwell equations, 240 Geodesic curvature, 109 Einstein tensor, 99 Geodesic equation, 150 Elasticity tensor, 223 Gradient, 182 Electric charge density, 226 Gradient of a contravariant vector, 183 Electric displacement, 224 Gradient of a covariant vector, 183 Electric field strength, 224, 226, 241 Gradient of an invariant, 182–183 Electrodynamics, 224 Gram-Schmidt scheme, 343, 344 Electromagnetic stress-energy tensor, 240 Gravitational constant, 240, 242, 364 Electromagnetic waves, 227 Green’s identities, 199 Elementary k-form, 158 Elliptic point, 111 Energy-momentum tensor, 240 H Energy or rothalpy equation, 210 Entanglement, 249 Heisenberg picture, 300 EPR paradox, 254 Hermitian matrix, 31 Euclidean N-space, 335 Hermitian operator, 29 Euclidean space, 348 Hessian tensor, 113, 116 Euler’s characteristic, 120 Hilbert space, 249 Exterior algebra, 155 Hodge dual, 172 Exterior derivative, 147, 163 Hodge * operator, 171 Exterior product, 158 Homeomorphic, 155 Hooke’s law, 223 Hyperbolic point, 111 F Faraday’s law, 225 First fundamental form, 109 I First-kind Christoffel symbol, 79 Identity matrix, 22 First-kind Ricci tensor, 97, 98 Interior derivative, 166 Flat space, 93 Interior product, 136, 166 Flat surface, 348 Intrinsic geometry, 351 Four-current density vector, 229 Intrinsic value, 56 Four-dimensional manifold, 227 Invariant time derivative, 141 Index 375

J Nonlocality, 250 Jacobian, 5, 41 N-order tensor, 35 Jacobi identity, 129 Normal curvature, 109 Normal vector, 353

K Ket orthonormal bases, 18 O Ket transformation, 28 Observables, 270 Ket vector, 16 Orthonormal coordinate, 338 Killing vector, 139 Orthonormalization, 18, 343 Kinetic energy-momentum tensor, 240 Outer product, 22 Klein-Gordon equation, 306 Koszul formula, 152 Kronecker delta, 342, 355 P Parabolic point, 111 Parameterized curve, 103 L Parameterized curves, 352 Lagrange identity, 354 Partial derivatives of the Christoffel symbols, Laplacian of a contravariant vector, 191 92 Laplacian of an invariant, 190 Photon sphere, 245 Levi-Civita connection, 148 Physical tensor component, 74 Levi-Civita permutation, 356 Physical vector components, 203 Levi-Civita permutation symbol, 57 Poincare´ group, 227 Lie derivative, 131 Poincare´ Lemma, 164 Lie dragged, 130 Poincare´ transformation, 227 Light speed, 226 Polar materials, 216 Linear adjoint operator, 21 Positive definite, 21 Lorentz transformation, 227 Principal curvature planes, 113 Principal normal curvatures, 113 Principal strains, 222 M Principal stresses, 218 Magnetic field density, 224 Principle of equivalence, 364 Magnetic field strength, 224 Probability density, 259, 260 Many-worlds interpretation (MWI), 297 Projection operator, 22 Maxwell’s equations, 224 Propagating speed, 227 Mean curvature, 113 Pullback, 168 Metric tensor, 151, 342 Pushforward, 171 Minkowski space, 227 Mixed components, 55 Mixed metric tensor components, 70 Q Moving surface, 140 Qubit, 251 Multi-fold-N-dimensional tensor space, 65 Qubit state, 251 Multilinear functional, 2, 36 MWI. See Many-worlds interpretation (MWI) R Reduced Compton wavelength, 291 N Relative probability, 257 Nabla operator, 181 Ricci curvature, 99 Navier-Stokes equations, 204 Ricci’s lemma, 95 Newton’s constant, 242, 364 Ricci tensor, 97, 239 376 Index

Riemann-Christoffel tensor, 92, 93, 97 Surface curvature tensor, 124–126 Riemann curvature, 117, 118 Surface triangulation, 120 Riemann curvature tensor, 92 Riemannian manifold, 347 Riemann Levi-Civita (RLC) connection, 148 T Tangent bundle, 127, 146 Tangential coordinate velocity, 140, 141 S Tangential surface, 352 Schrodinger€ picture, 300 Tangent space, 127, 146 Schrodinger€ ’s cat, 256 Tangent surface, 127 Schwarzschild metric, 241 Tensor, 35 Schwarzschild space, 242 Tensor product, 51 Schwarzschild’s solution, 241 Time evolution, 298 Second covariant derivative, 90 Transformed ket basis, 25 Second fundamental form, 110 Transpose conjugate, 17 Second-kind Christoffel symbol, 77, 78, 122 Second-kind Ricci tensor, 98, 331 Second-order tensor, 51 U Second partial derivative, 91 Uncertainty principle, 251 Shear modulus, 224 Unit normal vector, 107, 354 Shift tensor, 40 Unit tangent vector, 107 Skew-symmetric, 21 Spacetime, 227 Spacetime dimensions, 224 V Spacetime equations, 224 Volume differential, 202 Spherical coordinates, 8 Star operator, 171 Stokes theorem, 199 W Stress and strain tensors, 212 Wave-particle duality, 286 Surface area, 354 Wedge product, 158 Surface coordinates, 140 Weingarten’s equations, 123