On the Simplification of Tensor Expressions
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Tensors and Differential Forms on Vector Spaces
APPENDIX A TENSORS AND DIFFERENTIAL FORMS ON VECTOR SPACES Since only so much of the vast and growing field of differential forms and differentiable manifolds will be actually used in this survey, we shall attempt to briefly review how the calculus of exterior differential forms on vector spaces can serve as a replacement for the more conventional vector calculus and then introduce only the most elementary notions regarding more topologically general differentiable manifolds, which will mostly be used as the basis for the discussion of Lie groups, in the following appendix. Since exterior differential forms are special kinds of tensor fields – namely, completely-antisymmetric covariant ones – and tensors are important to physics, in their own right, we shall first review the basic notions concerning tensors and multilinear algebra. Presumably, the reader is familiar with linear algebra as it is usually taught to physicists, but for the “basis-free” approach to linear and multilinear algebra (which we shall not always adhere to fanatically), it would also help to have some familiarity with the more “abstract-algebraic” approach to linear algebra, such as one might learn from Hoffman and Kunze [ 1], for instance. 1. Tensor algebra. – A tensor algebra is a type of algebra in which multiplication takes the form of the tensor product. a. Tensor product. – Although the tensor product of vector spaces can be given a rigorous definition in a more abstract-algebraic context (See Greub [ 2], for instance), for the purposes of actual calculations with tensors and tensor fields, it is usually sufficient to say that if V and W are vector spaces of dimensions n and m, respectively, then the tensor product V ⊗ W will be a vector space of dimension nm whose elements are finite linear combinations of elements of the form v ⊗ w, where v is a vector in V and w is a vector in W. -
Tensors Notation • Contravariant Denoted by Superscript Ai Took a Vector and Gave for Vector Calculus Us a Vector
Tensors Notation • contravariant denoted by superscript Ai took a vector and gave For vector calculus us a vector • covariant denoted by subscript Ai took a scaler and gave us a vector Review To avoid confusion in cartesian coordinates both types are the same so • Vectors we just opt for the subscript. Thus a vector x would be x1,x2,x3 R3 • Summation representation of an n by n array As it turns out in an cartesian space and other rectilinear coordi- nate systems there is no difference between contravariant and covariant • Gradient, Divergence and Curl vectors. This will not be the case for other coordinate systems such a • Spherical Harmonics (maybe) curvilinear coordinate systems or in 4 dimensions. These definitions are closely related to the Jacobian. Motivation If you tape a book shut and try to spin it in the air on each indepen- Definitions for Tensors of Rank 2 dent axis you will notice that it spins fine on two axes but not on the Rank 2 tensors can be written as a square array. They have con- third. That’s the inertia tensor in your hands. Similar are the polar- travariant, mixed, and covariant forms. As we might expect in cartesian izations tensor, index of refraction tensor and stress tensor. But tensors coordinates these are the same. also show up in all sorts of places that don’t connect to an anisotropic material property, in fact even spherical harmonics are tensors. What are the similarities and differences between such a plethora of tensors? Vector Calculus and Identifers The mathematics of tensors is particularly useful for de- Tensor analysis extends deep into coordinate transformations of all scribing properties of substances which vary in direction– kinds of spaces and coordinate systems. -
Arxiv:0911.0334V2 [Gr-Qc] 4 Jul 2020
Classical Physics: Spacetime and Fields Nikodem Poplawski Department of Mathematics and Physics, University of New Haven, CT, USA Preface We present a self-contained introduction to the classical theory of spacetime and fields. This expo- sition is based on the most general principles: the principle of general covariance (relativity) and the principle of least action. The order of the exposition is: 1. Spacetime (principle of general covariance and tensors, affine connection, curvature, metric, tetrad and spin connection, Lorentz group, spinors); 2. Fields (principle of least action, action for gravitational field, matter, symmetries and conservation laws, gravitational field equations, spinor fields, electromagnetic field, action for particles). In this order, a particle is a special case of a field existing in spacetime, and classical mechanics can be derived from field theory. I dedicate this book to my Parents: Bo_zennaPop lawska and Janusz Pop lawski. I am also grateful to Chris Cox for inspiring this book. The Laws of Physics are simple, beautiful, and universal. arXiv:0911.0334v2 [gr-qc] 4 Jul 2020 1 Contents 1 Spacetime 5 1.1 Principle of general covariance and tensors . 5 1.1.1 Vectors . 5 1.1.2 Tensors . 6 1.1.3 Densities . 7 1.1.4 Contraction . 7 1.1.5 Kronecker and Levi-Civita symbols . 8 1.1.6 Dual densities . 8 1.1.7 Covariant integrals . 9 1.1.8 Antisymmetric derivatives . 9 1.2 Affine connection . 10 1.2.1 Covariant differentiation of tensors . 10 1.2.2 Parallel transport . 11 1.2.3 Torsion tensor . 11 1.2.4 Covariant differentiation of densities . -
On the Representation of Symmetric and Antisymmetric Tensors
Max-Planck-Institut fur¨ Mathematik in den Naturwissenschaften Leipzig On the Representation of Symmetric and Antisymmetric Tensors (revised version: April 2017) by Wolfgang Hackbusch Preprint no.: 72 2016 On the Representation of Symmetric and Antisymmetric Tensors Wolfgang Hackbusch Max-Planck-Institut Mathematik in den Naturwissenschaften Inselstr. 22–26, D-04103 Leipzig, Germany [email protected] Abstract Various tensor formats are used for the data-sparse representation of large-scale tensors. Here we investigate how symmetric or antiymmetric tensors can be represented. The analysis leads to several open questions. Mathematics Subject Classi…cation: 15A69, 65F99 Keywords: tensor representation, symmetric tensors, antisymmetric tensors, hierarchical tensor format 1 Introduction We consider tensor spaces of huge dimension exceeding the capacity of computers. Therefore the numerical treatment of such tensors requires a special representation technique which characterises the tensor by data of moderate size. These representations (or formats) should also support operations with tensors. Examples of operations are the addition, the scalar product, the componentwise product (Hadamard product), and the matrix-vector multiplication. In the latter case, the ‘matrix’belongs to the tensor space of Kronecker matrices, while the ‘vector’is a usual tensor. In certain applications the subspaces of symmetric or antisymmetric tensors are of interest. For instance, fermionic states in quantum chemistry require antisymmetry, whereas bosonic systems are described by symmetric tensors. The appropriate representation of (anti)symmetric tensors is seldom discussed in the literature. Of course, all formats are able to represent these tensors since they are particular examples of general tensors. However, the special (anti)symmetric format should exclusively produce (anti)symmetric tensors. -
Multilinear Algebra
Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R. -
Tensor Calculus and Differential Geometry
Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane . -
The Language of Differential Forms
Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects. -
Semi-Classical Scalar Propagators in Curved Backgrounds: Formalism and Ambiguities
Semi-classical scalar propagators in curved backgrounds : formalism and ambiguities J. Grain1,2 & A. Barrau1 1Laboratory for Subatomic Physics and Cosmology, Grenoble Universit´es, CNRS, IN2P3 53, avenue de Martyrs, 38026 Grenoble cedex, France 2AstroParticle & Cosmology, Universit´eParis 7, CNRS, IN2P3 10, rue Alice Domon et L´eonie Duquet, 75205 Paris cedex 13, France Abstract The phenomenology of quantum systems in curved space-times is among the most fascinating fields of physics, allowing –often at the gedankenexperiment level– constraints on tentative the- ories of quantum gravity. Determining the dynamics of fields in curved backgrounds remains however a complicated task because of the highly intricate partial differential equations in- volved, especially when the space metric exhibits no symmetry. In this article, we provide –in a pedagogical way– a general formalism to determine this dynamics at the semi-classical order. To this purpose, a generic expression for the semi-classical propagator is computed and the equation of motion for the probability four-current is derived. Those results underline a direct analogy between the computation of the propagator in general relativistic quantum mechanics and the computation of the propagator for stationary systems in non-relativistic quantum me- chanics. PACS numbers: 04.62.+v, 11.15.Kc arXiv:0705.4393v1 [hep-th] 30 May 2007 1 Introduction The dynamics of a scalar field propagating in a curved background is governed by partial differ- ential equations which, in most cases, have no analytical solution. Investigating the behavior of those fields in the semi-classical approximation is a promising alternative to numerical studies, allowing accurate predictions for many phenomena including the Hawking radiation process [1, 2, 3, 4] and the primordial power spectrum [5, 6]. -
Constructing $ P, N $-Forms from $ P $-Forms Via the Hodge Star Operator and the Exterior Derivative
Constructing p, n-forms from p-forms via the Hodge star operator and the exterior derivative Jun-Jin Peng1,2∗ 1School of Physics and Electronic Science, Guizhou Normal University, Guiyang, Guizhou 550001, People’s Republic of China; 2Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang, Guizhou 550001, People’s Republic of China Abstract In this paper, we aim to explore the properties and applications on the operators consisting of the Hodge star operator together with the exterior derivative, whose action on an arbi- trary p-form field in n-dimensional spacetimes makes its form degree remain invariant. Such operations are able to generate a variety of p-forms with the even-order derivatives of the p- form. To do this, we first investigate the properties of the operators, such as the Laplace-de Rham operator, the codifferential and their combinations, as well as the applications of the arXiv:1909.09921v2 [gr-qc] 23 Mar 2020 operators in the construction of conserved currents. On basis of two general p-forms, then we construct a general n-form with higher-order derivatives. Finally, we propose that such an n-form could be applied to define a generalized Lagrangian with respect to a p-form field according to the fact that it incudes the ordinary Lagrangians for the p-form and scalar fields as special cases. Keywords: p-form; Hodge star; Laplace-de Rham operator; Lagrangian for p-form. ∗ [email protected] 1 1 Introduction Differential forms are a powerful tool developed to deal with the calculus in differential geom- etry and tensor analysis. -
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362 INDEX A Cauchy stress law 216 Absolute differentiation 120 Cauchy-Riemann equations 293,321 Absolute scalar field 43 Charge density 323 Absolute tensor 45,46,47,48 Christoffel symbols 108,110,111 Acceleration 121, 190, 192 Circulation 293 Action integral 198 Codazzi equations 139 Addition of systems 6, 51 Coefficient of viscosity 285 Addition of tensors 6, 51 Cofactors 25, 26, 32 Adherence boundary condition 294 Compatibility equations 259, 260, 262 Aelotropic material 245 Completely skew symmetric system 31 Affine transformation 86, 107 Compound pendulum 195,209 Airy stress function 264 Compressible material 231 Almansi strain tensor 229 Conic sections 151 Alternating tensor 6,7 Conical coordinates 74 Ampere’s law 176,301,337,341 Conjugate dyad 49 Angle between vectors 80, 82 Conjugate metric tensor 36, 77 Angular momentum 218, 287 Conservation of angular momentum 218, 295 Angular velocity 86,87,201,203 Conservation of energy 295 Arc length 60, 67, 133 Conservation of linear momentum 217, 295 Associated tensors 79 Conservation of mass 233, 295 Auxiliary Magnetic field 338 Conservative system 191, 298 Axis of symmetry 247 Conservative electric field 323 B Constitutive equations 242, 251,281, 287 Continuity equation 106,234, 287, 335 Basic equations elasticity 236, 253, 270 Contraction 6, 52 Basic equations for a continuum 236 Contravariant components 36, 44 Basic equations of fluids 281, 287 Contravariant tensor 45 Basis vectors 1,2,37,48 Coordinate curves 37, 67 Beltrami 262 Coordinate surfaces 37, 67 Bernoulli’s Theorem 292 Coordinate transformations -
A Mathematica Package for Doing Tensor Calculations in Differential Geometry User's Manual
Ricci A Mathematica package for doing tensor calculations in differential geometry User’s Manual Version 1.32 By John M. Lee assisted by Dale Lear, John Roth, Jay Coskey, and Lee Nave 2 Ricci A Mathematica package for doing tensor calculations in differential geometry User’s Manual Version 1.32 By John M. Lee assisted by Dale Lear, John Roth, Jay Coskey, and Lee Nave Copyright c 1992–1998 John M. Lee All rights reserved Development of this software was supported in part by NSF grants DMS-9101832, DMS-9404107 Mathematica is a registered trademark of Wolfram Research, Inc. This software package and its accompanying documentation are provided as is, without guarantee of support or maintenance. The copyright holder makes no express or implied warranty of any kind with respect to this software, including implied warranties of merchantability or fitness for a particular purpose, and is not liable for any damages resulting in any way from its use. Everyone is granted permission to copy, modify and redistribute this software package and its accompanying documentation, provided that: 1. All copies contain this notice in the main program file and in the supporting documentation. 2. All modified copies carry a prominent notice stating who made the last modifi- cation and the date of such modification. 3. No charge is made for this software or works derived from it, with the exception of a distribution fee to cover the cost of materials and/or transmission. John M. Lee Department of Mathematics Box 354350 University of Washington Seattle, WA 98195-4350 E-mail: [email protected] Web: http://www.math.washington.edu/~lee/ CONTENTS 3 Contents 1 Introduction 6 1.1Overview.............................. -
Appendix a Relations Between Covariant and Contravariant Bases
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 or or a gk g  g ¼  ðA:1Þ i j oui ou j where a is the scalar factor; gk is the contravariant basis of the curvilinear coordinate of uk. Multiplying Eq. (A.1) by the covariant basis gk, the scalar factor a results in k k ðgi  gjÞ: gk ¼ aðg : gkÞ¼ad ¼ a ÂÃk ðA:2Þ ) a ¼ðgi  gjÞ : gk gi; gj; gk The scalar triple product of the covariant bases can be written as pffiffiffi a ¼ ½¼ðg1; g2; g3 g1  g2Þ : 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. Therefore, the Levi-Civita permutation symbols (pseudo-tensor components) can be used in expression of the contravariant basis. ffiffiffi p k k g g ¼ J g ¼ðgi  gjÞ¼Àðgj  giÞ eijkðgi  gjÞ eijkðgi  gjÞ ðA:4Þ ) gk ¼ pffiffiffi ¼ g J where the Levi-Civita permutation symbols are defined by 8 <> þ1ifði; j; kÞ is an even permutation; eijk ¼ > À1ifði; j; kÞ is an odd permutation; : A:5 0ifi ¼ j; or i ¼ k; or j ¼ k ð Þ 1 , e ¼ ði À jÞÁðj À kÞÁðk À iÞ for i; j; k ¼ 1; 2; 3 ijk 2 H.