Emil De Souza Sбnchez Filho

Emil de Souza Sánchez Filho

Tensor Calculus for Engineers and Physicists Tensor Calculus for Engineers and Physicists

Emil de Souza Sa´nchez Filho

Tensor Calculus for Engineers and Physicists Emil de Souza Sa´nchez Filho Fluminense Federal University Rio de Janeiro, Rio de Janeiro Brazil

ISBN 978-3-319-31519-5 ISBN 978-3-319-31520-1 (eBook) DOI 10.1007/978-3-319-31520-1

Library of Congress Control Number: 2016938417

© Springer International Publishing Switzerland 2016 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 International Publishing AG Switzerland To Sandra, Yuri, Natalia and Lara

Preface

The Tensor Calculus for Engineers and Physicist provides a rigorous approach to tensor manifolds and their role in several issues of these professions. With a thorough, complete, and unified presentation, this book affords insights into several topics of tensor analysis, which covers all aspects of N-dimensional spaces. Although no emphasis is placed on special and particular problems of Engineer- ing or Physics, the text covers the fundamental and complete study of the aim of these fields of the science. The book makes a brief introduction to the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having a dominium over the subsequent themes, without needing to resort to other bibliographical sources on tensors. This book did not have the framework of a math book, which is a work that seeks, above all else, to organize ideas and concepts in a didactic manner so as to allow the familiarity with the tensorial approach and its application of the practical cases of Physics and the areas of Engineering. The development of the various chapters does not cling to any particular field of knowledge, and the concepts and the deductions of the equations are presented so as to permit engineers and physicists to read the text without being experts in any branch of science to which a specific topic applies. The chapters treat the various themes in a sequential manner and the deductions are performed without omission of the intermediary steps, the subjects being treated in a didactic manner and supplemented with various examples in the form of solved exercises with the exception of Chap. 3 that broaches review topics. A few problems with answers are presented at the end of each chapter, seeking to allow the reader to improve his practice in solving exercises on the themes that were broached. Chapter 1 is a brief introduction to the basic concepts of tensorial formalism so as to permit the reader to make a quick and easy review of the essential topics that make possible the knowledge of the subsequent themes that come later, without needing to resort to other bibliographic sources on tensors.

vii viii Preface

The concepts of covariant, absolute, and contravariant derivatives, with the detailed development of all the expressions concerning these parameters, as well as the deductions of the Christoffel symbols of the first and second kind, are the essence of Chap. 2. Chapter 3 presents the Green, Stokes, and Gauss–Ostrogradsky theorems using a vectorial formulation. The expansion of the concepts of the differential operators studied in Differential Calculus is performed in Chap. 4. The scalar, vectorial, and tensorial fields are defined, and the concepts and expressions for gradient, divergence, and curl are formulated. With the definition of the nabla operator, successive applications of this linear differential operator are carried out and various fundamental relations between the differential operators are deducted, defining the Laplace operator. All the formulas are deducted by means of tensorial approach. The definition of metric spaces with several dimensions, with the introduction of Riemann curvature concept, and the Ricci tensor formulations, the scalar curvature, and the Einstein tensor are the subjects studied in detail in Chap. 5. Various particular cases of Riemann spaces are analyzed, such as the bidimensional spaces, the spaces with constant curvature, the Minkowski space, and the conformal spaces, with the definition of the Weyl tensor. Chapter 6 broaches metric spaces provided with curvature with the introduction of the concepts of the geodesics and the geodesics and Riemann coordinate systems. The geodesics deviation and the parallelism of vectors in curved spaces are studied, with the definition of the torsion tensor concept. The purpose of this book is to give a simple, correct, and comprehensive mathematical explanation of Tensor Calculus, and it is self-contained. Postgraduate and advanced undergraduate students and professionals will find clarity and insight into the subject of this textbook. The preparation of a book is a hard and long work that requires the participation of other people besides the author, which are of fundamental importance in the preparation of the originals and in the tiresome task of reviewing the typing, chiefly in a text such as the one in this book. So, our sincere thanks to all those who helped in the preparation and editing of these pages. In relation to the errors in this text which were not corrected by a more diligent review, it is stressed that they are the author’s responsibility and the author apologizes for them.

Rio de Janeiro, Brazil Emil de Souza Sa´nchez Filho December 17, 2015 Historical Introduction

This brief history of Tensor Calculus broaches the development of the idea of vector and the advent of the concept of tensor in a synthetic way. The following paragraphs aim to show the history of the development of these themes in the course of time, highlighting the main stages that took place in this evolution of the mathematical knowledge. A few items of bibliographic data of the mathematicians and scientists who participated on this epic journey in a more striking manner are described. The perception of Nature under a purely philosophical focus led Plato in 360 BC to the study of geometry. This philosopher classified the geometric figures into triangles, rectangles, and circles, and with this system, he grounded the basic concepts of geometry. Later Euclid systemized geometry in axiomatic form, starting from the fundamental concepts of points and lines. The wise men of ancient Greece also concerned themselves with the study of the movement of bodies by means of geometric concepts. The texts of Aristotle (384– 322 BC) in Mechanics show that he had the notion of composition of movements. In this work, Aristotle enounced in an axiomatic form that the force that moves a body is collinear with the direction of the body’s movement. In a segment of Mechanics, he describes the velocity of two bodies in linear movement with constant proportions between each other, explaining that “When a body moves with a certain proportion, the body needs to move in a straight line, and this is the diameter of the figure formed with the straight lines which have known proportions.” This statement deals with the displacements of two bodies—the Greek sage acknowleding that the resultant of these displacements would be the diagonal of the rectangle (the text talks about the diameter) from the composition of the speeds. In the Renaissance, the prominent figure of Leonardo da Vinci (1452–1519) also stood out in the field of sciences. In his writings, he reports that “Mechanics is the paradise of mathematical science, because all the fruits of mathematics are picked here.” Da Vinci conceived concepts on the composition of forces for maintaining the balance of the simple structures, but enunciated them in an erroneous and contradictory manner in view of the present-day knowledge.

ix x Historical Introduction

The awakening of a new manner of facing the uniform was already blossoming in the 1600s. The ideas about the conception and study of the world were no longer conceived from the scholastic point of view, for reason more than faith had become the way to new discoveries and interpretations of the outside world. In the Nether- lands, where liberal ideas were admitted and free thought could be exercised in full, the Dutch mathematician Simon Stevin (1548–1620), or Stevinus in a Latinized spelling, was the one who demonstrated in a clear manner the rule for the composition of forces, when analyzing the balance of a body located in an inclined plane and supported by weights, one hanging at the end of a lever, and the other hanging from a pulley attached to the vertical cathetus of the inclined plane. This rule is a part of the writings of Galileo Galilei (1564–1642) on the balance of bodies in a tilted plane. However, it became necessary to conceive mathematical formalism that translated these experimental veriﬁcations. The start of the concept of vector came about in an empirical mode with the formulation of the parallelogram rule, for Stevinus, in a paper published in 1586 on applied mechanics, set forth this principle of Classic Mechanics, formalizing by means of the balance of a force system the concept of a variety depending on the direction and orientation of its action, enabling in the future the theoretical preparation of the concept of vector. The creation of the Analytical Geometry by Rene´ Du Perron Descartes (1596– 1650) brought together Euclid’s geometry and algebra, establishing a univocal correspondence between the points of a straight line and the real numbers. The introduction of the orthogonal coordinates system, also called Cartesian coordinates, allowed the calculation of the distance ds between two points in the Euclid- ean space by algebraic means, given by ds2 ¼ dx2 þ dy2 þ dz2, where dx, dy, dz are the coordinates of the point.

The movement of the bodies was a focus of attention of the mathematicians and scientists, and a more elaborate mathematical approach was necessary when it was studied. This was taken care of by Leonhard Paul Euler (1707–1783), who conceived the concept of inertia tensor. This concept is present in his book Theoria Motus Corporum Solidorum seu Rigidorum (Theory of the Movement of the Solid and Rigid Bodies) published in 1760. In this paper, Euler studies the curvature lines, initiating the study of Differential Geometry. He was the most published Historical Introduction xi mathematician of the all time, 860 works are known from him, and it is known that he published 560 papers during his lifetime, among books, articles, and letters.

In the early 1800s, Germany was becoming the world’s largest center in mathematics. Among many of its brilliant minds, it counted Johann Karl Friedrich Gauß (1777–1855). On occupying himself with the studies of curves and surfaces, Gauß coined the term non-Euclidean geometries; in 1816, he’d already conceived concepts relative to these geometries. He prepared a theory of surfaces using curvilinear coordinates in the paper Disquisitones Generales circa Superfı´cies Curvas, published in 1827. Gauß argued that the space geometry has a physical aspect to be discovered by experimentation. These ideas went against the philosophical concepts of Immanuel Kant (1724–1804), who preconized that the conception of the space is a priori Euclidian. Gauß conceived a system of local coordinates system u, v, w located on a surface, which allowed him to calculate the distance between two points on this surface, given by the quadratic expression ds2 ¼ Adu2þ Bdv2 þ Cdw2 þ 2Edu Á dv þ 2Fdv Á dw þ 2Gdu Á dw, where A, B, C, F, G are functions of the coordinates u, v, w. The idea of force associated with a direction could be better developed analyt- ically after the creation of the Analytical Geometry by Descartes. The representation of the complex numbers by means of two orthogonal axes, one axis representing the real numbers and the other axis representing the imaginary number, was developed by the Englishman John Wallis (1616–1703). This representation allowed the Frenchman Jean Robert Argand (1768–1822) to develop in 1778, in a manner independent from the Dane Gaspar Wessel (1745–1818), the mathematical operations between the complex numbers. These operations served as a framework for the Irish mathematician William Rowan Hamilton (1805–1865) to develop a more encompassing study in three dimensions, in which the complex numbers are contained in a new variety: the Quaternions. xii Historical Introduction

This development came about by means of the works of Hamilton, who had the beginning of his career marked by the discovery of an error in the book Me´canique Celeste authored by Pierre Simon-Laplace (1749–1827), which gave him prestige in the intellectual environment. In his time, there was a great discrepancy between the mathematical production from the European continent and from Great Britain, for the golden times of Isaac Newton (1642–1727) had already passed. Hamilton studied the last advances of the continental mathematics, and between 1834 and 1835, he published the books General Methods in Dynamics. In 1843, he published the Quaternions Theory, printed in two volumes, the ﬁrst one in 1853 and the second one in 1866, in which a theory similar to the vector theory was outlined, stressing, however, that these two theories differ in their grounds.

In the ﬁrst half of the nineteenth century, the German Hermann Gunther€ Graßmann (1809–1877), a secondary school teacher of the city of Stettin located in the region that belongs to Pomerania and that is currently a part of Poland, published the book Die Lineale Ausdehnunsgleher ein neuer Zweig der Mathematik (Extension Theory), in which he studies a geometry of more than three dimensions, treating N dimensions, and formulating a generalization of the classic geometry. To outline this theory, he used the concepts of invariants (vectors and tensors), which later enabled other scholars to develop calculus and vector analysis. Historical Introduction xiii

The great mathematical contribution of the nineteenth century, which definitely marked the development of Physics, is due to Georg Friedrich Bernhard Riemann (1826–1866). Riemann studied in Gottingen,€ where he was a pupil of Gauß, and afterward in Berlin, where he was a pupil of Peter Gustav Lejeune Dirichlet (1805– 1859), and showed an exceptional capacity for mathematics when he was still young. His most striking contribution was when he submitted in December 1853 his Habilititationsschrift (thesis) to compete for the position of Privatdozent at the University of Gottingen.€ This thesis titled Uber€ die Hypothesen welche der Geometrie zu Grunde liegen enabled a genial revolution in the structure of Physics in the beginning of the twentieth century, providing Albert Einstein (1879–1955) with the mathematical background necessary for formulating his Theory of Rela- tivity. The exhibition of this work in a defense of thesis carried out in June 10, 1854, sought to show his capacity to teach. Gauß was a member of examination board and praised the exhibition of Riemann’s new concepts. His excitement for the new formulations was expressed in words: “... the depth of the ideas that were presented....” This work was published 14 years later, in 1868, two years after the death of its author. Riemann generalized the geometric concepts of Gauß, conceiving a system of more general coordinates spelled as dxi, and established a fundamental relation for the space of N dimensions, where the distance between 2 i j two points ds is given by the quadratic form ds ¼ gijdx dx , having gij a symmetrical function, positive and defined, which characterizes the space in a unique manner. Riemann developed a non-Euclidean, elliptical geometry, different from the geometries of Jańos Bolyai and Nikolai Ivanovich Lobachevsky. The Riemann Geometry unified these three types of geometry and generalized the concepts of curves and surfaces for hyperspaces. The broaching of the Euclidean space in terms of generic coordinates was carried out for the first time by Gabriel Lame´ (1795–1870) in his work Leçons sur les Fonctions Inverses des Transcedentes et les Surfaces Isothermes, published in Paris in 1857, and in another work Leçons sur les Coordoneés Curvilignes, published in Paris in 1859. xiv Historical Introduction

The new experimental discoveries in the ﬁelds of electricity and magnetism made the development of an adequate mathematical language necessary to translate them in an effective way. These practical needs led the North American Josiah Willard Gibbs (1839–1903) and the Englishman Olivier Heaviside (1850–1925), in an independent manner, to reformulate the conceptions of Graßmann and Hamilton, creating the vector calculus. Heaviside had thoughts turned toward the practical cases and sought applications for the vectors and used vector calculus in electromagnetism problems in the industrial areas.

With these practical applications, the vectorial formalism became a tool to be used in problems of engineering and physics, and Edwin Bidwell Wilson, a pupil of Gibbs, developed his master’s idea in the book Vector Analysis: A Text Book for the Use for Students of Mathematics and Physics Founded upon Lectures of Josiah Willard Gibbs, published in 1901 where he disclosed this mathematical apparatus, making it popular. This was the ﬁrst book to present the modern system of vectorial analysis and became a landmark in broadcasting the concepts of calculus and vectorial analysis. Historical Introduction xv

The German mathematician and prominent professor Elwin Bruno Christoffel (1829–1900) developed researches on the Invariant Theory, writing six articles about this subject. In the article Uber€ die Transformation der Homogenel Differentialausdrucke€ zweiten Grade, published in the Journal fur€ Mathematik, 70, 1869, he studied the differentiation of the symmetric tensor gij and introduced two functions formed by combinations of partial derivatives of this tensor, conceiving two differential operators called Christoffel symbols of the ﬁrst and second kind, which are fundamental in Tensorial Analysis. With this, he contributed in a fundamental way to the arrival of Tensor Calculus later developed by Gregorio Ricci-Curbastro and Tullio Levi Civita. The metrics of the Riemann spaces and the Christoffel symbols are the fundaments of Tensor Calculus. The importance of tensors in problems of Physics is due to the fact that physical phenomena are analyzed by means of models which include these varieties, which are described in terms of reference systems. However, the coordinates which are described in terms of the reference systems are not a part of the phenomena, only a tool used to represent them mathematically. As no privileged reference systems exist, it becomes necessary to establish relations which transform the coordinates from one referential system to another, so as to relate the tensors’ components. These components in a coordinate system are linear and homogeneous functions of the components in another reference system.

The technological development at the end of the nineteenth century and the great advances in the theory of electromagnetism and in theoretical physics made the conception of a new mathematical tool which enabled expressing new concepts and xvi Historical Introduction laws imperious. The vectorial formalism did not fulfill the broad field and the variety of new knowledge that needed to be studied more and interpreted better. This tool began to be created by the Italian mathematician Gregorio Ricci- Curbastro (1853–1925), who initiated the conception of Absolute Differential Calculus in 1884. Ricci-Curbastro was a mathematical physicist par excellence. He was a pupil of the imminent Italian professors Enrico Betti (1823–1892) and Eugenio Beltrami (1835–1900). He occupied himself mainly with the Riemann geometry and the study of the quadratic differential form and was influenced by Christoffel’s idea of covariant differentiation which allowed achieving great advances in geometry. He created a research group in which Tullio Levi-Civita participated and worked for 10 years (1887–1896) in the exploration of the new concepts and of an elegant and synthetic notation easily applicable to a variety of problems of mathematical analysis, geometry, and physics. In his article, Me´thodes de Calcul Differeńtiel Absolu et leurs Applications, published in 1900 in vol. 54 of the Mathematische Annalen, in conjunction with his pupil Levi-Civita, the applications of the differential invariants were broached, subject of the Elasticity Theory, of the Classic Mechanics and the Differential Geometry. This article is considered as the beginning of the creation of Tensor Calculus. He published the first explanation of his method in the Volume XVI of the Bulletin des Sciences Mathe´matiques (1892), applying it to problems from Differential Geometry to Mathematical Physics. The transformation law of a function system is due to Ricci-Curbastro, who published it in an article in 1887, and which is also present in another article published 1889, in which he introduces the use of upper and lower indexes, showing the differences between the contravariant and covariant transformation laws. In these papers, he exhibits the framework of Tensor Calculus.

The pupil and collaborator of Ricci-Curbastro, Tullio Levi-Civita (1873–1941) published in 1917 in the Rediconti del Circolo Matematico di Palermo, XLII (pp. 173–215) the article Nozione di Parallelismo in una Varieta Qualunque e Conseguente Speciﬁcazione Geometrica della Curvatura Riemanniana, contribut- ing in a considerable way to the development of Tensor Calculus. In this work, he describes the parallelism in curved spaces. This study was presented in lectures addressed in two courses given at the University of Rome in the period of Historical Introduction xvii

1920–1921 and 1922–1923. He corresponded with Einstein, who showed great interest in the new mathematical tool. In 1925, he published the book Lezione di Calcolo Differenziale Absoluto which is a classic in the mathematical literature.

It was the German Albert Einstein in 1916 who called the Absolute Differential Calculus of Ricci-Curbastro and Levi-Civita Tensor Calculus, but the term tensor, such as it is understood today, had been introduced in the literature in 1908 by the physicist and crystallographer Gottingen,€ Waldemar Voigt (1850–1919). The development of the theoretical works of Einstein was only possible after he became aware of by means of his colleague from Zurich, Marcel Grossmann (1878–1936), head professor of descriptive geometry at the Eidgenossische€ Technische Hochschule, the article Me´thodes de Calcul Differe´ntiel Absolut, which provided him the mathematical tool necessary to conceive his theory, publishing in 1916 in the Annalen der Physik the article Die Grundlagen der algemeinnen Relativitatstheorie. His contribution Tensor Calculus also came about with the conception of the summation rule incorporated to the index notation. The term tensor became popular mainly due to the Theory of Relativity, in which Einstein used this denomination. His researches on the gravitational ﬁeld also had the help of Grossmann, Tulio Levi-Civita, and Gregorio Ricci-Curbastro, conceiving the Gen- eral Relativity Theory. On the use of the Tensor Calculus in his Gravitation Theory, Einstein wrote: “Sie bedeutet einen wahren Triumph der durch Gauss, Riemann, Christoffel, Ricci ...begrundeten€ Methoden des allgemeinen Differentialkalculus.” xviii Historical Introduction

Other notable mathematics contributed to the development of the study of tensors. The Dutch Jan Arnoldus Schouten (1873–1941), professor of the T. U. Delft, discovered independently of Levi-Civita the parallelism and systematized the Tensor Calculus. Schouten published in 1924 the book Ricci-Kalkul€ which became a reference work on the subject, where he innovates the tensorial notation, placing the tensor indexes in brackets to indicate that it was an antisymmetric tensor.

The Englishman Arthur Stanley Eddington (1882–1944) conceived new in Tensor Calculus and was major promoter of the Theory of Relativity to the lay public.

The German Hermann Klaus Hugo Weyl (1885–1955) published in 1913 Die Idee der Riemannschen Flache,€ which gave a uniﬁed treatment of Riemann Historical Introduction xix surfaces. He contributed to the development and disclosure of Tensor Calculus, publishing in 1918 the book Raum-Zeit-Materie a classic on the Theory of Rela- tivity. Weyl was one of the greatest and most inﬂuential mathematicians of the twentieth century, with broad dominium of themes with knowledge nearing the “universalism.”

The American Luther Pfahler Eisenhart (1876–1965) who contributed greatly to semi-Riemannian geometry wrote several fundamental books with tensorial approach. The work of French mathematician E´ lie Joseph Cartan (1869–1951) in differential forms, one of the basic kinds of tensors used in mathematics, is principal reference in this theme. He published the famous book Leçons sur la Geóme´trie des Espaces de Riemann (first edition in 1928 and second edition in 1946).

Contents

1 Review of Fundamental Topics About Tensors ...... 1 1.1 Preview ...... 1 1.1.1 Index Notation and Transformation of Coordinates . . . . . 1 1.2 Space of N Dimensions ...... 2 1.3 Tensors ...... 2 1.3.1 Vectors ...... 2 1.3.2 Kronecker Delta and Permutation Symbol ...... 3 1.3.3 Dual (or Reciprocal) Basis ...... 3 1.3.4 Multilinear Forms ...... 10 1.4 Homogeneous Spaces and Isotropic Spaces ...... 16 1.5 Metric Tensor ...... 16 1.5.1 Conjugated Tensor ...... 22 1.5.2 Dot Product in Metric Spaces ...... 30 1.6 Angle Between Curves ...... 39 1.6.1 Symmetrical and Antisymmetrical Tensors ...... 43 1.7 Relative Tensors ...... 52 1.7.1 Multiplication by a Scalar ...... 54 1.8 Physical Components of a Tensor ...... 62 1.8.1 Physical Components of a Vector ...... 62 1.9 Tests of the Tensorial Characteristics of a Variety ...... 66 2 Covariant, Absolute, and Contravariant Derivatives ...... 73 2.1 Initial Notes ...... 73 2.2 Cartesian Tensor Derivative ...... 74 2.2.1 Vectors ...... 75 2.2.2 Cartesian Tensor of the Second Order ...... 77 2.3 Derivatives of the Basis Vectors ...... 78 2.3.1 Christoffel Symbols ...... 81 2.3.2 Relation Between the Christoffel Symbols ...... 83 2.3.3 Symmetry ...... 84 2.3.4 Cartesian Coordinate System ...... 84

xxi xxii Contents

2.3.5 Notation ...... 85 2.3.6 Number of Different Terms ...... 85 2.3.7 Transformation of the Christoffel Symbol of First Kind . . 86 2.3.8 Transformation of the Christoffel Symbol of Second Kind 87 2.3.9 Linear Transformations ...... 88 2.3.10 Orthogonal Coordinate Systems ...... 88 2.3.11 Contraction ...... 89 2.3.12 Christoffel Relations ...... 91 2.3.13 Ricci Identity ...... 92 2.3.14 Fundamental Relations ...... 93 2.4 Covariant Derivative ...... 100 2.4.1 Contravariant Tensor ...... 101 2.4.2 Contravariant Tensor of the Second-Order ...... 104 2.4.3 Covariant Tensor ...... 109 2.4.4 Mixed Tensor ...... 113 2.4.5 Covariant Derivative of the Addition, Subtraction, and Product of Tensors ...... 116 ij δi 2.4.6 Covariant Derivative of Tensors gij, g , j ...... 117 2.4.7 Particularities of the Covariant Derivative ...... 121 2.5 Covariant Derivative of Relative Tensors ...... 123 2.5.1 Covariant Derivative of the Ricci Pseudotensor ...... 125 2.6 Intrinsic or Absolute Derivative ...... 128 2.6.1 Uniqueness of the Absolute Derivative ...... 131 2.7 Contravariant Derivative ...... 133 3 Integral Theorems ...... 137 3.1 Basic Concepts ...... 137 3.1.1 Smooth Surface ...... 137 3.1.2 Simply Connected Domain ...... 137 3.1.3 Multiply Connected Domain ...... 138 3.1.4 Oriented Curve ...... 138 3.1.5 Surface Integral ...... 138 3.1.6 Flow ...... 139 3.2 Oriented Surface ...... 140 3.2.1 Volume Integral ...... 141 3.3 Green’s Theorem ...... 142 3.4 Stokes’ Theorem ...... 147 3.5 Gauß–Ostrogradsky Theorem ...... 150 4 Differential Operators ...... 155 4.1 Scalar, Vectorial, and Tensorial Fields ...... 155 4.1.1 Initial Notes ...... 155 4.1.2 Scalar Field ...... 156 Contents xxiii

4.1.3 Pseudoscalar Field ...... 156 4.1.4 Vectorial Field ...... 156 4.1.5 Tensorial Field ...... 158 4.1.6 Circulation ...... 159 4.2 Gradient ...... 160 4.2.1 Norm of the Gradient ...... 164 4.2.2 Orthogonal Coordinate Systems ...... 165 4.2.3 Directional Derivative of the Gradient ...... 166 4.2.4 Dyadic Product ...... 167 4.2.5 Gradient of a Second-Order Tensor ...... 169 4.2.6 Gradient Properties ...... 170 4.3 Divergence ...... 174 4.3.1 Divergence Theorem ...... 177 4.3.2 Contravariant and Covariant Components ...... 179 4.3.3 Orthogonal Coordinate Systems ...... 181 4.3.4 Physical Components ...... 183 4.3.5 Properties ...... 183 4.3.6 Divergence of a Second-Order Tensor ...... 183 4.4 Curl ...... 194 4.4.1 Stokes Theorem ...... 196 4.4.2 Orthogonal Curvilinear Coordinate Systems ...... 201 4.4.3 Properties ...... 202 4.4.4 Curl of a Tensor ...... 202 4.5 Successive Applications of the Nabla Operator ...... 207 4.5.1 Basic Relations ...... 207 4.5.2 Laplace Operator ...... 214 4.5.3 Properties ...... 216 4.5.4 Orthogonal Coordinate Systems ...... 218 4.5.5 Laplacian of a Vector ...... 218 4.5.6 Curl of the Laplacian of a Vector ...... 219 4.5.7 Laplacian of a Second-Order Tensor ...... 220 4.6 Other Differential Operators ...... 224 4.6.1 Hesse Operator ...... 224 4.6.2 D’Alembert Operator ...... 225 5 Riemann Spaces ...... 227 5.1 Preview ...... 227 5.2 The Curvature Tensor ...... 227 5.2.1 Formulation ...... 228 5.2.2 Differentiation Commutativity ...... 231 i 5.2.3 Antisymmetry of Tensor R‘jk ...... 233 i 5.2.4 Notations for Tensor R‘jk ...... 233 ‘ 5.2.5 Uniqueness of Tensor Rijk ...... 234 5.2.6 First Bianchi Identity ...... 234 xxiv Contents

5.2.7 Second Bianchi Identity ...... 235 5.2.8 Curvature Tensor of Variance (0, 4) ...... 238 5.2.9 Properties of Tensor Rpijk ...... 240 5.2.10 Distinct Algebraic Components of Tensor Rpijk ...... 241 5.2.11 Classiﬁcation of Spaces ...... 245 5.3 Riemann Curvature ...... 246 5.3.1 Deﬁnition ...... 246 5.3.2 Invariance ...... 247 5.3.3 Normalized Form ...... 248 5.4 Ricci Tensor and Scalar Curvature ...... 250 5.4.1 Ricci Tensor with Variance (0, 2) ...... 251 5.4.2 Divergence of the Ricci Tensor with Variance Ricci (0, 2) ...... 253 5.4.3 Bianchi Identity for the Ricci Tensor with Variance (0, 2) ...... 253 5.4.4 Scalar Curvature ...... 254 5.4.5 Geometric Interpretation of the Ricci Tensor with Variance (0, 2) ...... 254 5.4.6 Eigenvectors of the Ricci Tensor with Variance (0, 2) . . . 256 5.4.7 Ricci Tensor with Variance (1, 1) ...... 257 5.4.8 Notations ...... 259 5.5 Einstein Tensor ...... 262 5.6 Particular Cases of Riemann Spaces ...... 264 5.6.1 Riemann Space E2 ...... 265 5.6.2 Gauß Curvature ...... 267 5.6.3 Component R1212 in Orthogonal Coordinate Systems .... 269 5.6.4 Einstein Tensor ...... 271 5.6.5 Riemann Space with Constant Curvature ...... 273 5.6.6 Isotropy ...... 274 5.6.7 Minkowski Space ...... 280 5.6.8 Conformal Spaces ...... 281 5.7 Dimensional Analysis ...... 291 6 Geodesics and Parallelism of Vectors ...... 295 6.1 Introduction ...... 295 6.2 Geodesics ...... 295 6.2.1 Representation by Means of Curves in the Surfaces ..... 299 6.2.2 Constant Direction ...... 299 6.2.3 Representation by Means of the Unit Tangent Vector . . . 301 6.2.4 Representation by Means of an Arbitrary Parameter .... 302 6.3 Geodesics with Null Length ...... 307 6.4 Coordinate Systems ...... 309 6.4.1 Geodesic Coordinates ...... 309 6.4.2 Riemann Coordinates ...... 311 6.5 Geodesic Deviation ...... 313 Contents xxv

6.6 Parallelism of Vectors ...... 319 6.6.1 Initial Notes ...... 319 6.6.2 Parallel Transport of Vectors ...... 321 6.6.3 Torsion ...... 332 Bibliography ...... 337

Index ...... 341

Notations

ℜ Set of the real numbers Z Set of the complex numbers jjÁÁÁ Determinant kkÁÁÁ Modulus, absolute value · Dot product, scalar product, inner product Â Cross product, vectorial product Tensorial product Two contractions of the tensorial product δ δij δj Kronecker delta ij, , i ij ...m δij ...m, δ , Generalized Kronecker delta δj:::: n i ... m ijk eijk, e Permutation symbol ijk ...m eijk ...m, e Generalized permutation symbol ijk εijk, ε Ricci pseudotensor ε εi1i2i3ÁÁÁin i1i2i3ÁÁÁin , Ricci pseudotensor for the space EN E3 Euclidian space J Jacobian EN Vectorial space or tensorial space with N dimension ‘nÁÁÁ Natural logarithm ijk ...m εijk ...m, ε Ricci pseudotensor for the space EN d ... Differentiation with respect to variable xk dxk ϕ, i Comma notation for differentiation x_: Differentiation with respect to time ∂ ... Partial differentiation with respect to variable xk ∂xk ∂kÁÁÁ Covariant derivative δÁÁÁ Intrinsic or absolute derivative δt ∇ÁÁÁ Nabla operator

xxvii xxviii Notations

∇2ÁÁÁ Laplace operator, Laplacian HÁÁÁ Hesse operator, Hessian □... D’Alembert operator, D’alembertian divÁÁÁ Divergent gradÁÁÁ Gradient lapÁÁÁ Laplacian rotÁÁÁ Rotational, curl ij j Metric tensor gij, g , gi Γij,k Christoffel symbol of ﬁrst kind Γm ip Christoffel symbol of the second kind ij Gij, G Einstein tensor k Gm Einstein tensor with variance (1,1) K Riemann curvature R Scalar curvature Rij Ricci tensor of the variance (0,2) i Rj Ricci tensor of the variance (1,1) ‘ Rijk Riemann–Christoffel curvature tensor, Riemann–Christoffel mixed tensor, Riemann–Christoffel tensor of the second kind, curvature tensor Rpijk Curvature tensor of variance (0, 4) trÁÁÁ Trace of the matrix Wijk‘ Weyl curvature tensor

Greek Alphabets Sound Letter Alpha α, Α Beta β, Β Gamma γ, Γ Delta δ, Δ Epsilon ε, Ε Zeta ζ, Ζ Eta η, Η Theta θ, Θ Iota ι, I Kappa κ, K Lambda λ, Λ Mu€ μ, M Nu€ ν, Ν Ksi ξ, Ξ Omicron o, Ο Pi π, Π Rho ρ, Ρ Notations xxix

Sigma σ, Σ Tau τ, Τ U¨ psı´lon υ, Υ Phi φ, ϕ, Φ Khi χ, Χ Psi ψ, Ψ Omega ω, Ω Chapter 1 Review of Fundamental Topics About Tensors

1.1 Preview

This chapter presents a brief review of the fundamental concepts required for the consistent development of the later chapters. Various subjects are admitted as being previously known, which allows avoiding demonstrations that overload the text. It is assumed that the reader has full knowledge of Differential and Integral Calculus, Vectorial Calculus, Linear Algebra, and the fundamental concepts about tensors and dominium of the tensorial formalism. However, are presented succinctly the essential topics for understanding the themes that are developed in this book.

1.1.1 Index Notation and Transformation of Coordinates

On the course of the text, when dealing with the tensorial formulations, the index notation will be preferably used, and with the summation rule, for instance, X3 X3 yj ¼ aijxi ¼ aijxi, where i is a free index and j is a dummy in the sense i¼1 j¼1 that the sum is independent of the letter used, this expression takes the forms 8 8 9 2 38 9 <> y1 ¼ a11x1 þ a12x2 þ a13x3 <> y1 => a11 a12 a13 <> x1 => 6 7 y ¼ a x þ a x þ a x ) y ¼ 4 a a a 5 x :> 2 21 1 22 2 23 3 :> 2 ;> 21 22 23 :> 2 ;> y3 ¼ a31x1 þ a32x2 þ a33x3 y3 a31 a32 a33 x3

The transformation of the coordinates from a point in the coordinate system Xi to i the coordinate system X given by xi ¼ aijxj þ ai0 where the terms aij, ai0 are constants is called afﬁne transformation (linear). In this kind of transformation, the points of the space E3 are transformed into points, the straight lines in straight

© Springer International Publishing Switzerland 2016 1 E. de Souza Sa´nchez Filho, Tensor Calculus for Engineers and Physicists, DOI 10.1007/978-3-319-31520-1_1 2 1 Review of Fundamental Topics About Tensors lines, and the planes in planes. When ai0 ¼ 0, this transformation is called linear and homogeneous. The term ai0 represents only a translation of the origin of the referential.

1.2 Space of N Dimensions

The generalization of the Euclidian space at three dimensions for a number N of dimensions is prompt, defining a space EN. This expansion of concepts requires establishing a group of N variables xi, i ¼ 1, 2, 3, ...N, relative to a point i i PxðÞ2EN, related to a coordinate system X , which are called coordinates of the point in this reference system. The set of points associated in a biunivocal way to i the coordinates of the reference system X defines the N-dimensional space EN. In an analogous way a subspace EM EN is defined, with M < N, in which the i group of points PxðÞ2EM is related biunivocally with the coordinates defined in the coordinate system Xi. To make a few specific studies easier, at times the space is divided into subspaces. The space EN is called affine space, and if it is linked to the notion of distance between two points, then it is a metric space.

1.3 Tensors

1.3.1 Vectors

The structure of a vectorial space is defined by two algebraic operations: (a) the sum of the vectors and (b) the multiplication of vector by scalar. The conception of vectors u, v, w as geometric varieties is extended to a broad range of functions, as long as the set of these functions forms a vectorial space (linear space) on a set of scalars (numbers). The functions f, g, h, ...with continuous derivatives that fulfill certain axioms are assumed as vectors, and all the formulations and concepts developed for the geometric vectors apply to these formulations. A vectorial space is defined by the following axioms: 1. u þ v ¼ v þ u or f þ g ¼ g þ f . 2. ðÞþu þ v w ¼ u þ ðÞv þ w or ðÞþf þ g h ¼ f þ ðÞg þ h . 3. The null vector is such that u þ 0 ¼ u or 0 þ f ¼ f . 4. To every vector u there is a corresponding unique vector Àu, such that ðÞþÀu u ¼ 0 or ðÞþÀf f ¼ 0. 5. 1 Á kk¼u u or 1 Á kk¼f kkf . 6. mnðÞ¼u mnðÞu or mnfðÞ¼mnðÞ f , where m, n are scalars. 7. ðÞm þ n u ¼ mu þ nu or ðÞm þ n f ¼ mf þ nf . 8. mðÞ¼u þ v mu þ mv or mfðÞ¼þ g mf þ mg. 1.3 Tensors 3

1.3.2 Kronecker Delta and Permutation Symbol

The Kronecker delta is deﬁned by ( 1, i ¼ j δ ¼ δij ¼ δ i ¼ ð1:3:1Þ j 0, i 6¼ j that is symmetrical, i.e., δij ¼ δji, 8i, j. The Kronecker delta is the identity tensor. This tensor is used as a linear operator in algebraic developments, such as

∂xi ∂xk ∂xj ∂xj δ ¼ δ ¼ T δikuk ¼ T uk ¼ T ∂xj ki ∂xj ∂xi ki ∂xk ij kj j

The permutation symbol is deﬁned by 8 <> 1 is an even permutation of the indexes e ¼ eijk ¼ À1 is an odd permutation of the indexes ð1:3:2Þ ijk :> 0 when there are repeated indexes and the generalized permutation symbol is given by 8 <> 1 is an even permutation of the indexes e ¼ ei1i2i3ÁÁÁin ¼ À1 is an odd permutation of the indexes ð1:3:3Þ i1i2i3ÁÁÁin :> 0 when there are repeated indexes

Figure 1.1 shows an illustration how to obtain the values of this symbol.

1.3.3 Dual (or Reciprocal) Basis

The vector u expresses itself in the Euclidean space E3 by means of the linear combination of three linearly independent unit vectors, which form the basis of this space. For the case of oblique coordinate systems, there are two kinds of basis

Fig. 1.1 Values of the 11 permutation symbol

+1 -1

2233 4 1 Review of Fundamental Topics About Tensors

Fig. 1.2 Reciprocal basis X k

k X k e

e i O i X

i e

Xi called reciprocal or dual basis. Let vector u expressed by means of their components relative to a coordinate system with orthonormal covariant basis ej:

u ¼ ujej and with ei Á ek ¼ δjk, the dot product takes the form u Á ek ¼ ujej Á ek ¼ ujδjk ¼ uk, which are the components’ covariant of the vector u. These components are the projections of this vector on the coordinate axes. In the case of oblique coordinate system, the basis ej, ek is called reciprocal basis, k δ k i which fulfills the condition ej Á e ¼ j . In Fig. 1.2 the axes OX and OXk are k perpendicular, as are also the axes OX and OXi. This definition shows that the dot product of two reciprocal basis fulfills

i o i 1 kkei e cosðÞ¼ 90 À α 1 > 0 ) e ¼ kkei sin α

i k and with kkei ¼ 1 results in e > 1, then ei and e have different scales. Let the representation of the vector u in a coordinate system with covariant basis ei, ej, ek, where the indexes of the vectors of the basis indicate a cyclic permutation of i, j, k; thus, u ¼ uiei. These vectors do not have to be coplanar ei Á ek Â ek 6¼ 0; thus, the volume of the parallelepiped is given by the mixed product ei Á ek Â ek ¼ V and j with the relation between the two reciprocal basis ei Á e ¼ δij follows 1.3 Tensors 5

1 e Â e ¼ e i ¼ j k ei V

Then vector u in terms of reciprocal basis is deﬁned by u ¼ ujej where uj is the components of this vector in the new basis (contravariant), having these new components expressed in terms of the original components. j j Consider the representation u in terms of the two basis u ¼ uiei ¼ u e and with the dot product of both sides of this expression by ej, and applying the deﬁnition of j reciprocal basis e Á ei ¼ 1 provides

j u ¼ uiej Á ei

In an analogous way

i k V ¼ e Á ej Â e where V is the volume of the parallelepiped deﬁned by the mixed product of the unit vectors of the reciprocal basis. The height of the parallelepiped deﬁned by the mixed product of the unit vectors of a base is collinear with one of the unit vectors of the reciprocal basis (Fig. 1.3). The volume of the parallelepiped is determined by means of the mixed product of three vectors and allows assessing the relations between the same by means of the reciprocal basis in the levorotatory and dextrorotatory coordinates systems. Consider the mixed product of the vectors of the basis of a levorotatory coordinate system

V ¼ ei Á ej Â ek ¼ ei Á ðÞe123 e2e3ei which will cancel itself only if i ¼ 1, whereby

2 2 2 V V ¼ e123ðÞe1 e2e3 ¼ ðÞe1 e2e3 ) ðÞe1 ¼ e2e3

Fig. 1.3 Parallelepiped deﬁned by means of the reciprocal basis e3

e 2 3 e h e 2 e1 e 1 6 1 Review of Fundamental Topics About Tensors and for the reciprocal basis ÀÁ 1 2 V e2e3 e ¼ 2 ¼ ðÞe1 V ÀÁÀÁ i j k i 2 3 i 1 2 2 3 V ¼ e Á e Â e ¼ e Á e123e e e ¼ e e e 1 e1 ¼ e1 1 e e 1 1 V ¼ e2e3 ¼ 2 3 e2e3 ¼ ) VV ¼ 1e1 ¼ e1 V V e1

For a dextrorotatory coordinate system

1 1 V ¼ e2 Á ej Â e3 ¼ e2 Á ðÞe123 eie3e2 e ¼ e1 which cancels itself for i ¼ 1, so

2 1 1 V ¼ e2 ÁÀðÞ¼eie3e2 ðÞe2 e1e3e ¼ e1

1 e1e3 1 1 2 ¼À e ¼ ðÞe2 V e1 and for the case of reciprocal basis ÀÁÀÁ 2 i 3 2 i 3 2 2 2 1 3 V ¼ e Á e Â e ¼ e e123 e e e ¼À e e e 1 e ¼À 2 e2

In an analogous way

VV ¼ 1

If e1, e3, e2 are the unit vectors of an orthogonal coordinate system, then the reciprocal basis e1, e2, e3 also deﬁnes this coordinate system.

1.3.3.1 Orthonormal Basis

If the basis is orthonormal i j ei Á ej Â ek ¼ V ¼ V ¼ 1 e ¼ ej Â ek ¼ ei u ¼ ui 1.3 Tensors 7

This shows that for the Cartesian vectors, it is indifferent, covariant, or contravariant, of which the basis is adopted. The vector components in terms of this basis are equal, and the orthonormal basis is deﬁned by their unit vectors

ui ei ¼ kkui

The linear transformations 8m, u, v2E3:(a)FmðÞ¼u mFðÞu ;(b)FðÞ¼u Á v FmðÞ Ã defined in the Euclidean space E3 are also defined in the vectorial space E3, for there Ã is an intrinsic correspondence between these two spaces. The rules of calculus in E3 are analogous to those of E3, so these parameters are isomorphous. The existence of this duality is extended to the case of a vectorial space of finite Ã Ã ℜ * dimension EN, having EN or EN Z, for this space is dual to the Euclidean space EN.

1.3.3.2 Transformation Law of Vectors

The transformation of the coordinates from one point in the coordinate system Xi to i ∂xi ∂xi the coordinate system X is given by x ¼ j x , where j ¼ cos α are the matrix i ∂x j ∂x ij rotation elements, and its terms are the director cosines of the angles between the coordinate axes. In this linear and homogeneous transformation, the points of the space E3 are transformed into points expressed in terms of the new coordinates. Thus, the unit i i ∂xj vectors of X and of X transform according to the law e ¼ i e , where the values of i ∂x j ∂xi i j ¼ cos x x are the components of the unit vectors e¯ in the coordinate system X . ∂x i j i ∂xj For the position vector, u provides u ¼ i u . In the case of the inverse transfor- i ∂x j i i ∂xi mation, i.e., of X to X , provides analogously ej ¼ ∂xj ei, following for the ∂xj i components ∂xi ¼ cos xjxi of the unit vectors ej in the coordinate system X . i The determinant of the rotation matrix ∂x assumes the value þ1 in the case of ∂xj the transformation taking place between coordinate systems of the same direction, ∂xi which is then called proper transformation (rotation). Otherwise j ¼À1, and the ∂x transformation is called improper transformation (reﬂection).

1.3.3.3 Covariant and Contravariant Vectors

The representation of the vectors in oblique coordinate systems highlights various characteristics which are more general than the Cartesian representation. In these systems the vectors are expressed by means of two kinds of components. Let the representation of vector u in the plane coordinate system of oblique axes OXiXj that 8 1 Review of Fundamental Topics About Tensors

X abj

j j X u e j X j

j u j e e j

e j

O O i e ui e i X i i X e i

i X u e i i

Fig. 1.4 Vector components: (a) contravariant, (b) covariant

form an angle α, with basis vectors ei, ej (Fig. 1.4). The contravariant components are obtained by means of straight lines parallel to the axes OXi and OXj and graphed, respectively, as ui, uj (indicated with upper indexes). The covariant componentsare obtained by means of projection on the axes OXi and OXj given, respectively, by ui, uj (indicated with lower indexes). The projection of vector u on an axis provides its component on this axis, and by i j means of the dot product of u ¼ u ei and e : ÀÁ j i j i j i i u Á e ¼ u ei Á e ¼ u ei Á e ¼ u δij ¼ u that is the contravariant component of vector and in the same way by the covariant component ÀÁ i i u Á ej ¼ uie Á ej ¼ ui e Á ej ¼ uiδij ¼ ui

Thus, the vector is deﬁned by its components

i i u ¼ u ei ¼ uie

These components are not, in general, equal, and in the case of α ¼ 90o (Cartesian coordinate systems), the equality of these components is veriﬁed. 1.3 Tensors 9

e j e pro j e j e j i

j ∂x j ∂x

e i e i O i O ∂x ∂xi

Fig. 1.5 Transformation of coordinates: (a) covariant, (b) contravariant

1.3.3.4 Transformation Law of Covariant Vectors

i j The transformation law of base ei of an axis OX for a new axis OX , with base e¯j (Fig. 1.5a), is given by ÀÁ α ej ¼ projei ej ei ¼ ðÞ1 cos ei ∂xi cos α ¼ ∂xj ∂xi e ¼ e j ∂xj i that is the transformation law of the covariant basis. For the vector u the transfor- i mation of its covariant components is given by u ¼ ∂x u , where the variables j ∂xj i relative to the original axis in relation to which the transformation performed are found in the numerator of the equation.

1.3.3.5 Transformation Law of Contravariant Vectors

j The projection of the vector ei on the axis OX (Fig. 1.5b) provides α ej ¼ projej kkei ei ¼ ðÞ1 cos ei ∂xj cos α ¼ ∂xi ∂xj e ¼ e j ∂xi i that is the transformation law of the contravariant basis. For the vector u follows the j ∂xj i transformation law of its contravariant components u ¼ ∂xi u , where the variables relative to the new axis, for which the transformation is carried out, are found in the numerator of the expression. 10 1 Review of Fundamental Topics About Tensors

1.3.4 Multilinear Forms

The tensors of the order p are multilinear forms, which are vectorial functions, and linear in each variable considered separately. The concept of tensor is conceived by means of the following approaches: (a) the tensor is a variety that obeys a transformation law when changing the coordinate system; (b) this variety is invariant for any coordinate system; and (c) there is an equivalence between these definitions (equivalence law). A tensor of the order pÀÁis defined by a multilinear function with p N N components in the space EN, where R 1 À 2 ¼ 0 represents its order, which is maintained invariant if a change of the coordinate system occurs, and on the rotation of the reference axes (linear and homogeneous transformation) its coordinates modify according to a certain law. i Consider the space EN and the coordinate system X , i ¼ 1, 2, 3, ...N, defined in this space, where there are N equations that relate the coordinates of the points in EN, given by continuously differentiable functions ÀÁ xi ¼ xi xj i, j ¼ 1, 2, 3, ...N ð1:3:4Þ

i that transform these functions to a new coordinate system X . These transformations i j of coordinates require only that N functions x (x ) be independent. The necessary i and sufﬁcient condition for this transformation to be possible is that J ¼ ∂x 6¼ 0. ∂xj j ∂x The inverse function has an inverse Jacobian J ¼ ∂xi and implies that JJ ¼ 1.

1.3.4.1 Transformation Law of the Second-Order Tensors

i i i Let the position vector ui(x ) expressed in the coordinate system X of base e and a i new coordinate system X , with same origin, in which the vector is expressed by i ∂xk u ðÞx . Consider the elements i of the rotation matrix that relates the coordinates of i ∂x these two systems, then follow by means of the transformation law of covariant vectors

∂xk u ¼ u i, k ¼ 1, 2, 3 ð1:3:5Þ i ∂xi k

∂x‘ v ¼ v‘ j, ‘ ¼ 1, 2, 3 ð1:3:6Þ j ∂xj

i i The vectors ui(x ) and viðÞx deﬁne the transformation of the second-order tensor in terms of its covariant components 1.3 Tensors 11

∂xk ∂x‘ ∂xk ∂x‘ T ¼ u v ¼ u v‘ ¼ T ‘ ð1:3:7Þ ij i j ∂xi ∂xj k ∂xi ∂xj k and for the contravariant components provides an analogous manner

i j i j ij ∂x ∂x ‘ ∂x ∂x ‘ T ¼ uivj ¼ ukv ¼ Tk ð1:3:8Þ ∂xk ∂x‘ ∂xk ∂x‘ In a same way, it follows for the transformation law in terms of the mixed components

∂ i ∂ ‘ ∂ i ∂ ‘ i i x x k x x k T ¼ u v ¼ u v‘ ¼ T‘ ð1:3:9Þ j j ∂xk ∂xj ∂xk ∂xj

1.3.4.2 Transformation Law of the Third-Order Tensors

The transformations of the vectors u, v, w in terms of their covariant components are given by

∂xi ∂xj ∂xk u‘ ¼ ui vm ¼ vj wn ¼ wk ∂x‘ ∂xm ∂xn following by substitution

∂xi ∂xj ∂xk T‘mn ¼ u‘ vm wn ¼ uivjwk ∂x‘ ∂xm ∂xn that leads to the following transformation law for the covariant components of the third-order tensors

∂xi ∂xj ∂xk T‘mn ¼ Tijk ∂x‘ ∂xm ∂xn and for the contravariant components

‘ m n ‘mn ∂x ∂x ∂x T ¼ T ∂xi ∂xj ∂xk ijk and in an analogous way, for the mixed components

∂ ‘ ∂ m ∂ n ∂ ‘ ∂ m ∂ n mn x x x jk n x x x k T‘ ¼ T T‘ ¼ T ∂xi ∂xj ∂xk i m ∂xi ∂xj ∂xk ij ‘ m n i m n m ∂x ∂x ∂x j ‘m ∂x ∂x ∂x ij T‘ ¼ T T ¼ T n ∂xi ∂xj ∂xk ik n ∂x‘ ∂xj ∂xk k 12 1 Review of Fundamental Topics About Tensors

1.3.4.3 Inverse Transformation

i Let the inverse transformation of the vectors u and v of the coordinate system X for the coordinate system Xi, given by the covariant components of the vectors

∂xk ∂x‘ u ¼ u v ¼ v‘ ð1:3:10Þ i ∂xi k j ∂xj

It follows that

∂xk ∂x‘ ∂xk ∂x‘ T ¼ u v ¼ u v‘ ¼ T ‘ ð1:3:11Þ ij i j ∂xi ∂xj k ∂xi ∂xj k

Expression (1.3.11) allows concluding that a second-order tensor can be interpreted as a transformation in the linear space E3, which associates the vector u to the vector v by means of the tensorial product and that this linear and homogeneous transformation has an inverse transformation. The inverse transformations are deﬁned for the contravariant and mixed components in an analogous way. Expressions (1.3.7) and (1.3.11) show that if the components of a second- order tensor are null in a coordinates system, they will cancel each other in any other coordinate system. For the deﬁnition of the transformation law of second- order tensor to be valid, it is necessary that the transitive property apply to the linear operators (Fig. 1.6).

1.3.4.4 Transitive Property

i Let a second-order tensor Tk‘ deﬁned in the coordinate system X , that is expressed i in the coordinate system X by means of the expression (1.3.7), and with the i transformation of X for Xei

Fig. 1.6 Transitive k ¶x ¶x i property of the second-order = X Tij Tk tensors ¶x i ¶x j ¶ i ¶ j i ~ x x X T = T pq ¶~x p ¶~x q ij

T k X i k ~ ¶x ¶x T = T pq ¶~x p ¶x j k 1.3 Tensors 13

∂xi ∂xj Te ¼ T ð1:3:12Þ pq ∂exp ∂exq ij e However, the tensorT pq can be expressed in terms of tensor Tk‘, thereby avoiding the intermediary transformation, so substituting expression (1.3.7) in expression (1.3.12), it follows that

k ‘ i j e ∂x ∂x ∂x ∂x T ¼ T ‘ ð1:3:13Þ pq ∂xi ∂xj ∂exp ∂exq k and simplifying

∂xk ∂xi ∂xk ∂x‘ ∂xj ∂x‘ ¼ ¼ ð1:3:14Þ ∂xi ∂exp ∂exp ∂xj ∂exq ∂xj

Then

k ‘ e ∂x ∂x T ¼ T ‘ ð1:3:15Þ pq ∂exp ∂xj k

Expression (1.3.15) is the transformation law of the second-order tensor of the coordinate system Xi for the coordinate system Xei, which proves that the transitive property applies to these tensors. This property is also valid when using the contravariant and mixed components. The tensors studied in this book belong to metric spaces. If a variety is a tensor with respect to the linear transformations, it will be a tensor with respect to all the orthogonal linear transformations, but the inverse usually does not occur. The tensors are produced in spaces more general than the vectorial space. Table 1.1 shows the covariant, contravariant, and mixed tensors and their transformation laws for the space EN.

1.3.4.5 Multiplication of a Tensor by a Scalar

It is the multiplication that provides a new tensor as a result, which components are the components of the original tensor multiplied by the scalar. Let the tensor Tijk

Table 1.1 Kinds of tensors Tensor Expression Transformation law i j k Covariant TijÁÁÁk ∂x ∂x ∂x T ¼ ÁÁÁ T rsÁÁÁt ∂xr ∂xs ∂xt ijÁÁÁk ijÁÁÁk r s t Contravariant T rsÁÁÁt ∂x ∂x ∂x T ¼ ÁÁÁ TijÁÁÁk ∂xi ∂xj ∂xp k‘ÁÁÁh i j m n f h Mixed T mnÁÁÁh ∂x ∂x ∂x ∂x ∂x ∂x ‘ ijÁÁÁf T ¼ ÁÁÁ Tk ÁÁÁh rsÁÁÁt ∂xr ∂xs ∂xk ∂x‘ ∂xt ∂xh ijÁÁÁf 14 1 Review of Fundamental Topics About Tensors

and the scalar m which product Pijk is given by Pijk ¼ mTijk. For demonstrating this expression represents a tensor, all that is needed is to apply the tensor transformation law to the same.

1.3.4.6 Addition and Subtraction of Tensors

The addition of tensors of the same order and the same type is given by

k k k Tij ¼ Aij þ Bij

The addition of the mixed tensors given by the previous expression provides as a result a mixed tensor of the third order, which is twice covariant and once contravariant. To demonstrate this expression represents a tensor, all that is needed is to apply the tensor transformation law to the same. The subtraction is deﬁned in the same way as the addition, however, admitting k k k that a tensor is multiplied by the scalar À1. As an exampleTij ¼ Aij þÀðÞ1 Bij; thus, this expression provides as a result a mixed tensor of the third order, which is twice covariant and once contravariant. To demonstrate that previous expression represents a tensor all that is needed is to carry out the analysis developed for the addition considering the negative sign.

1.3.4.7 Contraction of Tensors

The contraction of a tensor is carried out when two of its indexes are made equal, a covariant index and a contravariant index, and thus reducing the order of this tensor k‘ ‘ in two. For instance, the tensor Tij contracted in the indexes and j results as kj kj k Ti‘ ¼ Tij ¼ Ti .

1.3.4.8 Outer Product of Tensors

The outer product is the product of two tensors that provide as a new tensor, which k ... order is the sum of the order of these two tensors. Let, for example, the tensor Aij ...... ‘m with variance index number ( p, q) and the tensor B... rs with variance index number k...‘m k...... ‘m (u, v), which if multiplied provides a tensor Tij...rs ¼ Aij...B...rs with variance index number ðÞp þ u, q þ v . The order of the tensor is given by the sum of these two indexes. To demonstrate that the previous expression is a tensor, all that is needed is to apply the tensor transformation law to the same. 1.3 Tensors 15

1.3.4.9 Inner Product of Tensors

The inner product of two tensors is deﬁned as the tensor obtained after the contracting of the outer product of these tensors. Let, for example, tensors Aij and ‘ ‘ ‘ Bk which the outer product is Pijk ¼ AijBk that provides as a result a tensor of the fourth order, which contracted in the indexes ‘ and k provide the inner product ‘ ‘ Pij‘ ¼ AijB‘ ¼ Pij. This shows that the resulting tensor is of the second order. To demonstrate that this expression represents a tensor, all that is needed is to apply the tensor transformation law to the same.

1.3.4.10 Quotient Law

This law allows verifying if a group of Np functions of the coordinates of the referential system Xi has tensorial characteristics. Its application serves to test if a variety is a tensor. The systematic for applying this law is to make the dot product of the variety that is to be tested by a vector, for the outer product of two tensors generates a tensor, and then carry out the contraction of this product and afterward, by means of applying the tensor transformation law, verify if the variety fulfills this law. Let, for example, the contravariant tensor of the first order Tk and the variety A(i, j, k) composed of 27 functions defined in the space EN,forwhichitis desired to verify if it is tensor. The fundamental premise is that the vector Tk is independent of A(i, j, k). If the inner product AiðÞ, j, k Tk ¼ Bij originates a contravariant tensor of the second order, then A(i, j, k) has the characteristics of a tensor. Applying the transformation law of tensors to the tensor Bij

p q p q pq ∂x ∂x ∂x ∂x B ¼ Bij ¼ AiðÞ; j; k Tk ∂xi ∂xj ∂xi ∂xj and for the vector Tk, it follows that

k ∂x r Tk ¼ T ∂xr

By substitution

p q k pq ∂x ∂x ∂x r B ¼ AiðÞ, j, k T ∂xi ∂xj ∂xr and in a new coordinate system, the tensor Bij is given by

pq r B ¼ ApðÞ, q, r T 16 1 Review of Fundamental Topics About Tensors following by substitution p q k ∂x ∂x ∂x r ApðÞÀ, q, r AiðÞ, j, k T ¼ 0 ∂xi ∂xj ∂xr

r As T is an arbitrary vector the result is

∂xp ∂xq ∂xk ApðÞ¼; q; r AiðÞ; j; k ∂xi ∂xj ∂xr that represents the transformation law of third-order tensors. This shows that the variety A(i, j, k) has tensorial characteristics.

1.4 Homogeneous Spaces and Isotropic Spaces

The isotropic space has properties which do not depend on the orientation being considered, and the components of isotropic tensors do not change on an orthogonal linear transformation. The sum of isotropic tensors results in an isotropic tensor, and the product of isotropic tensors is also an isotropic tensor. There is no isotropic tensor of the ﬁrst order. The isotropic tensor of the fourth order is given by

Tijk‘ ¼ λδijδk‘ þ μδikδj‘ þ νδi‘δjk ð1:4:1Þ where λ, μ, ν are scalars. The Kronecker delta δij is the only isotropic tensor of the second order. The homogeneous space has properties which are independent of the position of the point. The homogeneous tensors have constant components when the coordinate system is changed. A homogeneous tensor of the fourth order is given by ÀÁ Tij‘k ¼ λδijδk‘ þ μδikδj‘ þ δi‘δjk ð1:4:2Þ where λ, μ are scalars.

1.5 Metric Tensor

The study of tensors carried out in afﬁne spaces applies to another type, called metric space, in which the length of the curves is determined by means of a variety that deﬁnes this space, in which the basic magnitudes are the length of a curve and the vector’s norm, just as the angle between vectors and the angle between two curves. The distinction between these two types of spaces is of fundamental importance in the study of tensors. 1.5 Metric Tensor 17

Fig. 1.7 Elementary arc X 3 of a curve ds Q x i + dxi

g 3 P x i O

g 2 2 X g 1

X 1

The metric space is determined by the definition of its fundamental tensor which is related with its intrinsic properties. The conception of this metric tensor, which gives an arithmetic form to the space, considers the invariance of distance between two points, the concept of distance being acquired from the space E3. The geometry grounded in the concept of metric tensor is called Riemann geometry. The angle between two curves is calculated by means of the dot product between vectors using the metric tensor, which awards a generalization to this tensor’s formulation. Let the arc element length of a curve ds defined in the Cartesian i i coordinate system X with unit vectors g1, g2, g3 by means of its coordinates x (Fig. 1.7), with two neighboring pointsPxðÞi , QxðÞi þ dxi , which define the position vectors r and r þ dr, respectively. The coordinates of increment of the position vector dr are given by Q À P ¼ dxi; thus, lim ðÞ¼Q À P ds, and the dot product of Q!P this vector by himself takes the form

ds2 ¼ dr Á dr ¼ dxidxi ð1:5:1Þ

Consider a transformation of the coordinates xi ¼ xiðÞxi for a new coordinate i system X

∂xi dxi ¼ dxk ð1:5:2Þ ∂xk becomes

i i ∂x ∂x ‘ ds2 ¼ dxkdx ð1:5:3Þ ∂xk ∂x‘

Putting

∂xi ∂xi g ‘ ¼ ð1:5:4Þ k ∂xk ∂x‘ 18 1 Review of Fundamental Topics About Tensors thus the metric takes the form

2 k ‘ : : ds ¼ gk‘dx dx ð1 5 5Þ

The symmetry of the variety given by expression (1.5.4) is obvious, because gk‘ ¼ g‘k, then 2 3 2 3 g1g1 g1g2 g1g3 g11 g12 g13 4 5 4 5 : : gij ¼ gji ¼ g2g1 g2g2 g2g3 ¼ g21 g22 g23 ð1 5 6Þ g3g1 g3g2 g3g3 g31 g32 g33

Thehi analysis of expression (1.5.4) shows that gk‘ relates with the Jacobian i ½¼J ∂x of a linear transformation by means of the following expression ∂xk ∂ i T ∂ i x x T ½¼g ‘ ¼ ½J ½J ð1:5:7Þ k ∂xk ∂x‘

i For the coordinate system X , the variety gij is deﬁned by his unit vectors gi, gj. i Consider a new coordinate system X , with respect to which these unit vectors are expressed by

∂xi ∂xj g ¼ g g‘ ¼ g ð1:5:8Þ k ∂xk i ∂x‘ j

Thus ∂xi ∂xj ∂xi ∂xj ÀÁ∂xi ∂xj g ‘ ¼ g g ¼ g g ¼ g k ∂xk i ∂x‘ j ∂xk ∂x‘ i j ∂xk ∂x‘ ij then gij is a symmetric tensor of the second order. The arc length is invariable when changing the coordinate system. The coeffi- i 2 i i i cients of gk‘(x ) are class C , and the N equations x ¼ x ðÞx must satisfy the 1 2 NNðÞþ 1 partial differential equations given by expression (1.5.4). However, if i 1 gk‘(x ) is specified arbitrarily, this system of 2 NNðÞþ 1 partial differential equations, in general, has no solution. The fundamental tensor gk‘ related to a coordinate i system X , in a region of the space EN, must fulfill the following conditions: i 2 (a) gk‘(x ) is a class C function, i.e., its second-order derivatives exist and are continuous. (b) Be symmetrical, i.e., gk‘ ¼ g‘k. (c) detgk‘ ¼ g 6¼ 0, i.e., gk‘ is not singular. 2 k ‘ (d) ds ¼ gk‘dx dx is an invariant after a change of coordinate system. 1.5 Metric Tensor 19

Expression (1.5.5) is put under parametric form with the coordinates xi ¼ xiðÞt and i ¼ 1, 2, 3 ...N, and the parameter a t b provides sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb ‘ dxk dx s ¼ g ‘ dt ð1:5:9Þ k dt dt a

dxk Admit a functional parameter hi ¼Æ1, so as to allow the conditions gk‘ dt dx‘ > dxk dx‘ < dt 0 and gk‘ dt dt 0 to be be used instead of the absolute value shown in expression (1.5.9), because the use of hi is more adequate to the algebraic manipulations; thus,

ðb rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxk dx‘ s ¼ h g ‘ dt ð1:5:10Þ i k dt dt a

Φ k ‘ The quadratic and homogeneous form ¼ gk‘dx dx is called metric or fundamental form of the space, being invariant after a change of coordinate system. In space E3 with Φ > 0, which provides g > 0, and when Φ ¼ 0, the initial and final points of the arc coincide. If Φ ¼ 0 and dxi are not all null, the displacement between the two points is null. The possibility of Φ being undefined is admitted, for 2 2 instance, in the case Φ ¼ ðÞdx1 À ðÞdx2 , for which dx1 ¼ dx2 results in Φ ¼ 0. This case is interpreted as having a null displacement of the point. If dxi 6¼ 0, i.e., the displacements are not null, hi is adopted so that hiΦ > 0. The spaces EN (hyperspaces) are analyzed in an analogous way to the analysis of the space E3 by means of defining a metric, formalizing the Riemann geometry. The geometries not grounded on the concept of metric are called non-Riemann geometries. To demonstrate that expression (1.5.10) is invariant through a change in its parametric representation, let a curve of class C2 represented by means of the coordinates xi ¼ xiðÞt and a t b. Consider a transformation for the new 0 coordinates xi ¼ xiðÞt and a t b, where t ¼ ftðÞ with f ðÞt > 0, and in the new limits a ¼ faðÞ, b ¼ fbðÞ. Applying the chain rule to the function t ¼ ftðÞ:

dt 0 dt ¼ f ðÞ)t dt ¼ ð1:5:11Þ dt f 0 ðÞt and with expression (1.5.11) in expression (1.5.10) 20 1 Review of Fundamental Topics About Tensors

ðbrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðbrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i j i j hi i j dx dx dx dx 0 2 dx dx 0 L ¼ h g dt ¼ h g f ðÞt dt ¼ h g f ðÞt dt i ij dt dt i ij dt dt i ij dt dt a a a ðbrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxi dxj ¼ h g dt ¼ L i ij dt dt a then the value of this expression does not vary with the change of the curve’s parameterization. The metric can be written in matrix form so as to make the usual calculations easier hi ds 2 dxk T dx‘ ¼ g ð1:5:12Þ dt dt ij dt

In the space E3, the metric is deﬁned by

dt 0 dt ¼ f ðÞ)t dt ¼ ð1:5:13Þ dt f 0 ðÞt

2 1 1 1 2 1 3 ds ¼ g11dx dx þ g12dx dx þ g13dx dx 2 1 2 2 2 3 : : þ g21dx dx þ g22dx dx þ g23dx dx ð1 5 14Þ 3 1 3 2 3 3 þ g31dx dx þ g32dx dx þ g33dx dx or ÀÁ ÀÁ 2 i 2 k 2 i k : : ds ¼ gii dx þ gkk dx þ 2gikdx dx ð1 5 15Þ

For the particular case in which the coordinate systems are orthogonal (Fig. 1.8), i the segments on the coordinate axes X are deﬁned by the unit vectors gi of these axes

i : : dsðÞi ¼ gidx ð1 5 16Þ which provide the metric ÀÁÀÁÀÁ 2 1 2 2 2 3 2 ds ¼ h1g1dx þ h2g2dx þ h3g3dx ÀÁÀÁÀÁ 1 2 2 2 3 2 ¼ h1dx þ h2dx þ h3dx ð1:5:17Þ then the metric tensor is deﬁned by the elements of the diagonal of the matrix 1.5 Metric Tensor 21

Fig. 1.8 Orthogonal g coordinate systems 3

3 X 2 X

O g2

1 X

2 3 2 h1 00 4 2 5 : : gij ¼ 0 h2 0 ð1 5 18Þ 2 00h3 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi where h1 ¼ g11, h2 ¼ g22, h3 ¼ g33, and detgij ¼ g ¼ g11g22g33.

i j i j Exercise 1.1 Let gijx x ¼ 0, 8x , x show that gk‘ þ g‘k ¼ 0. Putting

Φ i j ¼ gijx x ¼ 0 and differentiating with respect to xk

∂Φ ∂xi ∂xj ¼ g xj þ g xi ¼ g δ ixj þ g δ jxi ¼ g xj þ g xi ¼ 0 ∂xk ij ∂xk ij ∂xk ij k ij k kj ik

Differentiating with respect to x‘

∂2Φ ∂ j ∂ i x x j i ¼ g þ g ¼ g δ þ g δ ¼ g ‘ þ g‘ ¼ 0 Q:E:D: ∂xk∂x‘ kj ∂x‘ ik ∂x‘ kj ‘ ik ‘ k k

Exercise 1.2 Calculate the length of the curve of class C2 given by the parametric equations x1 ¼ 3 À t, x2 ¼ 6t þ 3, and x3 ¼ ‘nt, in the space deﬁned by the metric tensor 22 1 Review of Fundamental Topics About Tensors 2 3 12 4 0 4 5 gij ¼ 41 1 2 01ðÞx1

The metric of the space in matrix form stays hi ds 2 dxk T dx‘ ¼ g dt dt ij dt and with the derivatives

dx1 dx2 dx3 1 ¼À1 ¼ 6 ¼ dt dt dt t it follows 2 38 9 > > 2 12 4 0 < À1 = 2 ds 1 4 5 ðÞt þ 3 ¼ À1; 6; 41 1 6 ¼ dt 2 :> 1 ;> t2 t 01ðÞ 3À t t

Making h1 ¼ 1 in expression

ðb rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe dxk dx‘ t þ 3 s ¼ h g ‘ dt ) L ¼ dt ¼ e þ 2 i k dt dt t a 1

1.5.1 Conjugated Tensor

Let the increment of the position vector expressed by means of their covariant i j components dr ¼ rig in any coordinate system, where g is the basis vector of this referential and with the dot product ÀÁ i j dr Á dr ¼ dxidxj g g ð1:5:19Þ and ÀÁ gigj ¼ gij ¼ gji ð1:5:20Þ whose symmetry comes from the commutative property of the dot product. 1.5 Metric Tensor 23

This variety with properties analogous to the properties of the metric tensor is represented by nine components of a symmetrical matrix 3 Â 3, which form a second-order contravariant tensor. It is called conjugated metric tensor; thus, 2 3 g11 g12 g13 gij ¼ 4 g21 g22 g23 5 ð1:5:21Þ g31 g32 g33

The deﬁnition of the conjugated of the metric tensor is given by

gij ¼ gigj ð1:5:22Þ and with the relations between the reciprocal basis

g Â g‘ g Â g gi ¼ k gj ¼ m n ð1:5:23Þ V V results for the conjugated metric tensor

ij 1 g ¼ ðÞg Â g‘ ðÞg Â g ð1:5:24Þ V2 k m n but with the fundamental formula of the vectorial algebra

: : ðÞÁgk Â g‘ ðÞ¼gm Â gn ½ÁðÞÂgk Â g‘ gm gn ð1 5 25Þ and developing the double-cross product in brackets

: : ½ÁðÞÂgk Â g‘ gm gn ¼ ðÞgk Á gm g‘ À ðÞg‘ Á gm gk ð1 5 26Þ

ij 1 g ¼ ½ððÞg Á g ðÞÀg‘ Á g ðÞg‘ Á g ðÞg Á g 1:5:27Þ V2 k m n m k n

The term in brackets in expression (1.5.27) is the development of the determinant

g Á g g Á g g g Gij ¼ k m k n ¼ km kn ð1:5:28Þ g‘ Á gm g‘ Á gn g‘m g‘n

Then

G gij ¼ ij ð1:5:29Þ V2 24 1 Review of Fundamental Topics About Tensors

Summarizing these analyses by means of the transcription of the following expressions

ij ij Gij Gij ij G G : : gij ¼ 2 ¼ g ¼ 2 ¼ ð1 5 30Þ V g V g

Thus qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ij : : V ¼Æ g ¼Æ detg V ¼Æ g ¼Æ detgij ð1 5 31Þ

The sign ðÞþ in expressions (1.5.31) corresponds to a levorotatory coordinates, and the sign ðÞÀ corresponds to a dextrorotatory coordinates. Knowing that VV ¼ 1, it follows that gg ¼ 1. n n ∂g ... Exercise 1.3 Let detgijðÞ¼x gxðÞ. Calculate the derivative ∂xn, n ¼ 1, 2, . The matrix linked to the determinant g is a function of the variables xn: ÀÁ i gij ¼ gij x and this determinant being a function of the matrix elements

g ¼ ggij by the chain rule

∂g ∂g ∂g ¼ ij ∂ n ∂ ∂ n x gij x

As det g is expressed by its cofactors

g ¼ g1kGk1 ¼ g11G11 þ g12G21 þ g13G31 þÁÁÁ and the terms Gk1 do not contain the terms g1k,so

∂g ∂g ∂g ∂ ¼ G11 ∂ ¼ G21 ∂ ¼ G31 ÁÁÁ g11 g12 g13

Generalizing provides

∂g ∂ ¼ Gji gij 1.5 Metric Tensor 25

By substitution ∂g ∂g ¼ G ij ∂xn ji ∂xn

1 2 1 2 x x ðÞx Exercise 1.4 Calculate the derivative of detg ¼ 2 with respect to the ðÞx1 2x1 1 variable x . From Exercise 1.3 ∂g ∂g ¼ G ij ∂xi ji ∂xi

This expression is the sum of n determinants. Each of these determinants differs from the determinant g only in the lines and columns which are being differentiated, so

∂ ∂ g g ∂ g11 g12 11 12 x2 2x1 1 2 1 2 g x x ðÞx ¼ ∂x1 ∂x1 þ ∂g ∂g ¼ 2 þ ∂x1 21 22 ðÞx1 2x1 2x1 2 g21 g22 ∂x1 ∂x1

k Exercise 1.5 Let g ¼ detgij the determinant of the metric tensor gij and x an ∂ðÞ‘ng ∂ðÞ‘ng arbitrary variable. Calculate (a) ∂ and (b) ∂ k . gij x (a) From Exercise 1.3

∂g ∂ ¼ Gji gij

but as gij ¼ gji it follows that

∂g ∂ ¼ Gij gji

Expression (1.5.30) provides Gij gij ¼ g G g ¼ ij ) G ¼ gg ij g ij ij

By substitution

∂ g ij ∂ ¼ ggij ¼ gg gji 26 1 Review of Fundamental Topics About Tensors whereby

∂ ‘ ∂ ∂ ‘ ðÞng 1 g ðÞng ij ∂ ¼ ∂ ) ∂ ¼ g gij g gij gij

(b) By the chain rule

∂ðÞ‘ng ∂ðÞ‘ng ∂g ¼ ij ∂ k ∂ ∂ k x gij x

and substituting the result obtained in the previous item in this expression

∂ðÞ‘ng ∂g ¼ gij ij ∂xk ∂xk

Exercise 1.6 Calculate the metric tensor, its conjugated tensor, and the metric for the Cartesian coordinate system. Let the Cartesian coordinates (x1, x2, x3), and by the deﬁnition of the distance between two points ÀÁ ÀÁ ÀÁ 2 2 2 ds2 ¼ dx1 þ dx2 þ dx3 which is the square of the metric, thus

2 i j ds ¼ δijdx dx

By the deﬁnition of the metric tensor and the conjugated metric tensor, then 2 3 100 δ 4 5 gij ¼ ij ¼ 010 001 2 3 100 1 gij ¼ ¼ 4 0105 g ij 001

Exercise 1.7 Calculate the metric tensor, its conjugated tensor, and the metric for the cylindrical coordinate system given by r x1, θ x2; and z x3 where À1 r 1,0 θ 2π, and À1 z 1, which relations with the Cartesian coordinates are x1 ¼ x1 cos x2,x2 ¼ x1 sin x2, and x3 x3. 1.5 Metric Tensor 27

With the deﬁnition of metric tensor

∂xk ∂xk ∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g ¼ ¼ þ þ ij ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj

– i ¼ j ¼ 1

1 1 2 2 3 3 ÀÁÀÁ ∂x ∂x ∂x ∂x ∂x ∂x 2 2 g ¼ þ þ ¼ cos x2 þ sin x2 þ 0 ¼ 1 11 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1

– i ¼ j ¼ 2

1 1 2 2 3 3 ÀÁÀÁÀÁ ∂x ∂x ∂x ∂x ∂x ∂x 2 2 2 g ¼ þ þ ¼Àx1 sin x2 þ x1 cos x2 þ 0 ¼ x1 22 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2

– i ¼ j ¼ 3

∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g ¼ þ þ ¼ 0 þ 0 þ 1 ¼ 1 33 ∂x3 ∂x3 ∂x3 ∂x3 ∂x3 ∂x3

– i ¼ 1, j ¼ 2

∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g ¼ þ þ 12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ÀÁÀÁ 2 2 ¼ cos x2 Àx1 sin x2 þ sin x2 x1 cos x2 þ 0 ¼ 0

For the other terms g21 ¼ g13 ¼ g31 ¼ g23 ¼ g32 ¼ 0, then ÀÁ ÀÁÀÁ ÀÁ 2 1 1 2 2 2 2 1 2 1 2 2 2 3 2 ds ¼ g11dx dx þ g22dx dx þ g11dx dx ¼ dx þ x dx þ dx ¼ ðÞdr 2 þ ðÞrdθ 2 þ ðÞdz 2

The metric tensor and its conjugated tensor are given, respectively, by 2 3 2 3 100 100 6 7 1 6 1 7 4 2 5 ij 4 5 gij ¼ 0 r 0 g ¼ ¼ 0 2 0 gij r 001 001 and with the base vectors

∂xj g ¼ e i ∂xi j 28 1 Review of Fundamental Topics About Tensors 8 > ∂xj > i ¼ 1 ) g ¼ e <> 1 ∂x1 j ∂x1 ∂x2 ∂x3 > j ¼ 1, 2, 3 ) g ¼ e þ e e > 1 ∂ 1 1 ∂ 1 2 ∂ 1 3 :> x x x g ¼ cos x2e þ sin x2e 8 1 1 2 > ∂xj > i ¼ 2 ) g ¼ e <> 2 ∂x2 j ∂x1 ∂x2 ∂x3 > j ¼ 1, 2, 3 ) g ¼ e þ e e > 2 ∂ 2 1 ∂ 2 2 ∂ 2 3 :> x x x g ¼Àx1 sin x2e þ x1 cos x2e 8 2 1 2 > ∂xj > i ¼ 3 ) g ¼ e <> 3 ∂x3 j ∂x1 ∂x2 ∂x3 > j ¼ 1, 2, 3 ) g ¼ e þ e e > 3 ∂ 3 1 ∂ 3 2 ∂ 3 3 :> x x x g3 ¼ 0 þ 0 þ 1 Á e3 ¼ e3

By means of the dot products

δ δ gi Á gj ¼ ij ei Á ej ¼ ij it follows for the components of the metric tensor ÀÁÀÁ 2 2 2 2 g11 ¼ g1 Á g1 ¼ cos x e1 þ sin x e2 Á cos x e1 þ sin x e2 ¼ 1 ÀÁÀÁÀÁ 1 2 1 2 1 2 1 2 2 2 g22 ¼ g2 Á g2 ¼Àx sin x e1 þ x cos x e2 ÁÀx sin x e1 þ x cos x e2 ¼ x g33 ¼ g3 Á g3 ¼ ðÞÁe3 ðÞ¼e3 1

The other components of this tensor are null. Exercise 1.8 Calculate the metric tensor, its conjugated tensor, and the metric for the spherical coordinate system r x1, φ x2, θ x3, À1 r 1, and 0 φ π, where 0 θ 2π, which relations with the Cartesian coordinates are x1 ¼ x1 sin x2 cos x3,x2 ¼ x1 sin x2 sin x3,and x3 x1 cos x2. With the deﬁnition of metric tensor

∂xk ∂xk ∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g ¼ ) g ¼ þ þ ij ∂xi ∂xj ij ∂xi ∂xj ∂xi ∂xj ∂xi ∂xj – i ¼ j ¼ 1 1.5 Metric Tensor 29

∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g ¼ þ þ 11 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ÀÁÀÁÀÁ 2 2 2 ¼ sin x2 cos x3 þ sin x2 sin x3 þ cos x2 ¼ 1

– i ¼ j ¼ 2

∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g22 ¼ þ þ ÀÁ∂x2 ∂x2 ∂x2 ∂x2 ÀÁ∂x2 ∂x2 ÀÁÀÁ 2 2 2 2 ¼ x1 cos x2 cos x3 þ x1 cos x2 sin x3 þÀx1 sin x2 ¼ x1

– i ¼ j ¼ 3

∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g33 ¼ þ þ ÀÁ∂x3 ∂x3 ∂x3 ∂x3 ÀÁ∂x3 ∂x3 ÀÁ 2 2 2 ¼Àx1 sin x2 sin x3 þ x1 sin x2 cos x3 þ 0 ¼ x1 sin x2

For the other terms g12 ¼ g21 ¼ g13 ¼ g31 ¼ g23 ¼ g32 ¼ 0, then ÀÁ ÀÁÀÁ ÀÁÀÁ 2 1 1 2 2 3 3 1 2 3 2 2 2 1 2 2 3 2 ds ¼ g11dx dx þ g22dx dx þ g33dx dx ¼ dx þ x dx þ x sin x dx ¼ ðÞdr 2 þ ðÞrdφ 2 þ ðÞr sin θ dθ 2

The metric tensor and its conjugated tensor are given, respectively, by 2 3 2 3 10 0 10 0 6 7 6 1 7 6 7 ij 1 6 0 0 7 g ¼ 4 0 r2 0 5 g ¼ ¼ 6 r2 7 ij g 4 5 2 2 ij 1 00r sin φ 00 r2 sin 2φ

Exercise 1.9 Calculate the metric tensor, its conjugated tensor, and the metric for the cylindrical elliptical coordinate system ξ x1, η x2, and z x3, where ξ 0, 0 η 2π, À1 z 1, which relations with the Cartesian coordinates are x1 ¼ coshx2 cos x2, x2 ¼ sinhx2 sin x2,x3 x3. 3 : 1 : With x ¼ const , the elliptical cylinder is x0 ¼ const : 1 2 2 2 ÀÁÀÁ x x 2 2 2 2 1 þ 1 ¼ cos x þ sin x ¼ 1 chx0 shx0 dx1 ¼ sinhx1 cos x2dx1 dx2 ¼ coshx1 sin x2dx1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ ds ¼ ðÞdx1 2 þ ðÞdx2 2 ¼ cosh2x1 À cos 2x2 dx1 ÀÁ 2 1 2 2 g11 ¼ cosh x À cos x 30 1 Review of Fundamental Topics About Tensors

1 : 2 : With x ¼ const the hyperbolic cylinder is x0 ¼ const : 1 2 2 2 ÀÁÀÁ x x 1 2 1 2 2 À 2 ¼ coshx À sinhx ¼ 1 cos x0 sin x0

dx1 ¼Àcoshx1 sin x2dx2 dx2 ¼ sinhx1 cos x2dx2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ ds ¼ ðÞdx1 2 þ ðÞdx2 2 ¼ cosh2x1 À cos 2x2 dx2 ÀÁ 2 1 2 2 g22 ¼ cosh x À cos x

3 3 2 3 For x x provides dx ¼ dx , whereby g33 ¼ 1, following ÀÁÀÁ ÀÁÀÁ ÀÁ 2 2 2 2 2 ds2 ¼ cosh2x1 À cos 2x2 dx1 þ cosh2x1 À cos 2x2 dx2 þ dx3

The metric tensor and its conjugated tensor are given, respectively, by 2 3 cosh2x1 À cos 2x2 00 6 7 2 1 2 2 gij ¼ 4 0 cosh x À cos x 0 5 001 2 3 1 6 007 6 cosh2x1 À cos 2x2 7 gij ¼ 6 1 7 4 0 0 5 cosh2x1 À cos 2x2 001

1.5.2 Dot Product in Metric Spaces

Let the vectors u and v contained in the metric space EN deﬁned by the fundamental i j tensor gk‘. The dot product u Á v with u ¼ u ei and v ¼ v ej depends only on the vectors and is independent of the coordinate system in relation to which the same is speciﬁed. It is observed that only when the coordinates of the vectors are covariant and contravariant, this product is like to the dot product in Cartesian coordinates. The dot product is invariant in view of the transformation of coordinates

i j i j i: j ij i : j u Á v ¼ u ei Á v ej ¼ giju v ¼ uie vje ¼ g uivj ¼ u ei vje : : j i i j ð1 5 32Þ ¼ gi u vj ¼ u vj ¼ uiv 1.5 Metric Tensor 31

1.5.2.1 Vector Norm

The generalization of the dot product of vectors for a metric space EN allows obtaining the norm of a vector. Let vector v with norm (modulus) pffiffiffiffiffiffiffiffi pffiffiffiffiffi kkv ¼ v Á v ¼ v2 that is equal to the distance between the extreme points, thus, with the expression of the metrics

2 k ‘ v ¼ higk‘v v results for the norm of the vector in terms of its contravariant components qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k kkv ¼ higkkv v

In an analogous way for the covariant components vk pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kk kkv ¼ hig vkvk and for Cartesian coordinates pffiffiffiffiffiffiffiffi kkv ¼ vkvk

If v is a unit vector, the expressions provide

k k kk k k higkkv v ¼ 1 hig vkvk ¼ 1 v v ¼ 1 The properties of the vectors norm are: (a) kkv 0, which is a trivial property, for the norm will only cancel itself if v is null. (b) kk¼mv kkm kkv , where m is a scalar. (c) kku þ v kku þ kkv . (d) kku Á v kku Á kkv , Cauchy–Schwarz inequality. For the case of non-null vectors, the equality of the relation (d) exists only if u ¼ mv, where m is a scalar.

Exercise 1.10 Calculate the modulus of vector u(1; 1; 0; 2) in space E4, deﬁned by the metric tensor 2 3 À10 0 0 6 0 À10 07 g ¼ 6 7 ij 4 00À105 000c2 32 1 Review of Fundamental Topics About Tensors

2 i j For the line element ds ¼ giju u , and developing this expression

2 i j 1 1 2 2 3 3 4 4 ds ¼ giju u ¼ g11u u þ g22u u þ g33u u þ g44u u ¼À1 Â 1 Â 1 À 1 Â 1 Â 1 þ 0 þ c2 Â 2 Â 2 ¼À2 þ 4c2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ 22ðÞc2 À 1

1.5.2.2 Lowering of a Tensor’s Indexes

By means of analysis referent to the transformation of the covariant components of the vector in their contravariant components, and vice versa, it is veriﬁed that inner product of a tensor by the metric tensor allows raising or lowering the indexes of this tensor. For multiplying the contravariant tensor of the ﬁrst order, i.e., the contravariant i i i ‘ vector T by the tensor gk‘, results in Tk‘ ¼ gk‘T , and for the contraction i ¼ , then i i Tki ¼ gkiT ¼ Tk that is a covariant vector. The index of the original vector was lowered and its order reduced in two units.

1.5.2.3 Raising of a Tensor’s Indexes

ik Let the covariant vector Tk, which multiplied by g , provides as a result the tensor g ik Tk, and changing the covariant coordinates of the vector by its contravariant coordinates ÀÁ ik ik ‘ δ i ‘ i g Tk ¼ g gk‘T ¼ ‘T ¼ T that is a contravariant vector. The index of the original vector was raised. Then a covariant vector is obtained by means of the inner product of a contravariant vector, i this indexes transformation process as being reciprocal. The vectors T and Ti are called associated vectors, and it refers to the contravariant and covariant components of the vector. For the case of second-order tensors, an analysis is carried out that is analogous to the one developed for the vectors. Let the covariant tensor of the second-order Tk‘ ij ik j‘ and its associated tensor T ¼ g g Tk‘. It is veriﬁed in the general case that these tensors are not conjugated tensors, for example, when Tij ¼ mgij, where m is a scalar, the tensor Tk‘ will be a multiple of gk‘, ik j‘ ik j‘ ikδ ‘ ‘k k‘ Tk‘ ¼ g g Tij ¼ g g mgij ¼ mg i ¼ mg ¼ mg 1.5 Metric Tensor 33

The raising and lowering operations of the indexes of tensors are carried out adopting, ﬁrstly, a point for indicating where the position to be left empty in the j index that will be raised or lowered. For example, for the tensor Ti, the empty j rs position is indicated by means of the notation T i, and in an equal manner Ap rs exists for the tensor Ap . i i Let the inner product of the tensor T jk by the metric tensor g‘i, g‘iTjk ¼ T‘jk in kj k which the upper index was lowered, and gijT ¼ Ti that had an index lowered, or j‘ further, gijgk‘T ¼ Tik, which two upper indexes were lowered. ij For raising the indexes, in an analogous way to the raising of an index g Tjk i kj k ij k‘ ik ¼ Tk or g Tij ¼ Ti , thus g g Tj‘ ¼ T . In the case in which the index is lowered and then raised, the original tensor is ij i kj i ij obtained gkjT ¼ Tk and next g Tk ¼ T .

1.5.2.4 Tensorial Equation

If a term of a tensorial equation contains a dummy index, it can be raised or lowered, i.e., change the position without changing the value of the equation. The following example illustrates this assertion ÀÁÀÁ i ki ‘ ki ‘ δ k ‘ ‘ i AjBi ¼ g Akj gi‘B ¼ g gi‘AkjB ¼ ‘ AkjB ¼ A‘jB ¼ AijB where the index i was lowered in one tensor and raised in the tensor. If a free index is a part of the tensorial expression, a new tensorial expression equivalent to this one can be obtained, lowering or raising this index in the members of the original expression. To illustrate this assertion, the following tensorial equation is admitted

Tijk ¼ AijBk which is equivalent to

i‘ i‘ g T‘jk ¼ g A‘jBk so it results in

i i Tjk ¼ AjBk where the index i was raised. 34 1 Review of Fundamental Topics About Tensors

Exercise 1.11 Raise and lower the indexes of vector u, for the metric tensor and its conjugated tensor: 8 9 2 3 < 3 = 10 0 2 j 4 0 ðÞx1 0 5 (a) u ¼ : 4 ; gij ¼ 1 2 2 8 5 9 200ðÞx sin x 3 < 5 = 10 0 4 1 À2 5 (b) uj ¼ 4 gij ¼ 0 ðÞx 0 : ; 2 3 00ðÞx1 sin x2 À (a) Carrying out the following matrix multiplication provides the covariant components of the vector 2 38 9 8 9 10 0 < 3 = < 3 = j 4 1 2 5 1 2 ui ¼ giju ¼ 0 ðÞx 0 4 ¼ 4ðÞx 2 : ; : 2 ; 00ðÞx1 sin x2 5 5ðÞx1 sin x2

(b) Carrying out the following matrix multiplication provides the contravariant components of the vector 2 38 9 8 9 10 0 < 5 = < 5 = i 4 1 À2 5 1 À2 u ¼ gijuj ¼ 0 ðÞx 0 4 ¼ 4ðÞx 2 : ; : 2 ; 00ðÞx1 sin x2 À 3 3ðÞx1 sin x2 À

; ; Exercise 1.12 Given the2 covariant basis3 g1 ¼ e1 g2 ¼ e1 þ e2 g3 ¼ e3 and the 111 4 5 tensor of the space gij ¼ 122, calculate the vectors of the contravariant 123 basis and the conjugated metric tensor. The determinant of the metric tensor is given by

111

g ¼ detgi ¼ 122 ¼ 1 123 which indicates that the system is dextrorotary. For the vectors of the contravariant basis, it follows that

e1 e2 e3 1 g2 Â g3 g ¼ ¼ 110 ¼ e1 À e2 g 111

e1 e2 e3 2 g3 Â g1 g ¼ ¼ 111 ¼ e2 À e3 g 100 1.5 Metric Tensor 35

e1 e2 e3 3 g1 Â g2 g ¼ ¼ 100 ¼ e3 g 110 then the conjugated metric tensor is given by 2 3 2 À10 gij ¼ gi Á gj ¼ 4 À12À1 5 0 À11

The veriﬁcation of the operation is carried out by means of the expression ij δ i gijg ¼ j , thus 2 32 3 2 3 111 2 À10 100 4 12254 À12À1 5 ¼ 4 0105 123 0 À11 001

1.5.2.5 Associated Tensors

ij The metric tensor gij and its conjugated tensor g relate intrinsically to each other, which allows using them for analyzing the relations between the covariant and contravariant components of the vector u

j i ij : : ui ¼ giju u ¼ g uj ð1 5 33Þ

These expressions generate two linear equation systems, which unknown quan- 1 2 3 tities are u1, u2, u3, and u , u , u . The solution of the system given by means of Cramer’s rule and with the determinant of the metric tensor

g g g 11 12 13 g ¼ detg ¼ g g g ij 21 22 23 g31 g32 g33 and its cofactor

g g Gij ¼ km kn g‘m g‘n thus

G uj G u ¼ g uj ¼ ij ) g ¼ ij i ij g ij g 36 1 Review of Fundamental Topics About Tensors

In an analogous way for the system given by expression 1.5.33

Giju Giju ui ¼ giju ¼ j ) gij ¼ j j g g

ij The linear operators gij, g allow relating the covariant and contravariant components of vector u. Deﬁning this vector by means of its covariant components and performing the dot product of this vector by the basis unit vectors gi: ÀÁ ÀÁ i j i j i ij i u Á g ¼ ujg Á g ¼ uj g Á g ¼ g uj ¼ u that are the contravariant components of vector u. In an analogous way for the transformation of the contravariant components in the covariant components ÀÁ ÀÁ i i j u Á gi ¼ u gj Á gi ¼ u gj Á gi ¼ giju ¼ ui

These expressions relate to each other in a kind of coordinate as a function of the ij other, where the tensors gij, g are the operators responsible for these transformations. The covariant components and the contravariant components of the vector u are given, respectively, by 8 8 8 < 1 < 1 11 < 11 u1 ¼ g11u u ¼ g u1 g11g ¼ 1 2 i 2 22 22 ui ¼ : u2 ¼ g22u u ¼ : u ¼ g u2 ) : g22g ¼ 1 3 3 33 33 u3 ¼ g33u u ¼ g u3 g33g ¼ 1

ij j The linear operators gij, g , gi are useful in the explanation of the more general properties of tensors.

Exercise 1.13 For the vector v ¼ 4g1 þ 3g2 referenced to a coordinate system, calculate their contravariant and covariant components in the referential system that have the basis vectors g1 ¼ 3g1, g2 ¼ 6g1 þ 8g2, and g3 ¼ g3. The metric tensor of the space is given by 2 3 2 3 g g g g g g 9180 4 1 1 1 2 1 3 5 4 5 gij ¼ gi Á gj ¼ g2g1 g2g2 g2g3 ¼ 18 100 0 ) detgij ¼ 576 g3g1 g3g2 g3g3 001

For the conjugated metric tensor 2 3 hi 100 À18 0 À1 1 g gij ¼ δ ) gij ¼ g ) gij ¼ 4 À18 9 0 5 ij ij ij 576 0 0 576 1.5 Metric Tensor 37

The vectors of the contravariant basis are given by

i ij g ¼ g gj 8 9 8 9 2 38 9 > 1 1 > < 1 = < = <> g1 À g2 => g 100 À18 0 3g1 3 4 i 2 1 4 5 g ¼ : g ; ¼ À18 9 0 : 6g1 þ 8g2 ; ¼ > 1 > 3 576 > g2 > g 0 0 576 g3 : 8 ; g3

With the contravariant components of v

vi ¼ v Á gi 7 3 v1 ¼ ðÞÁ4g þ 3g g1 ¼ v2 ¼ ðÞÁ4g þ 3g g2 ¼ v3 ¼ ðÞÁ4g þ 3g g3 ¼ 0 1 2 12 1 2 8 1 2 so

7 3 v ¼ g1 þ g2 12 8

With the covariant components of v

vi ¼ v Á gi ¼ ðÞÁ4g1 þ 3g2 gi : : v1 ¼ ðÞ4g1 þ 3g2 g1 ¼ 12 v2 ¼ ðÞ4g1 þ 3g2 g2 ¼ 48 v3 ¼ ðÞÁ4g1 þ 3g2 g3 ¼ 0 so

v ¼ 12g1 þ 48g2 ‘j Exercise 1.14 Show that in the space EN exists g‘jgik À g‘igjk g ¼ ðÞN À 1 gik, where gij is the metric tensor. Developing the given expression ‘j ‘j ‘j ‘j ‘j g‘jgik À g‘igjk g ¼ g‘jgikg À g‘igjkg ¼ g‘jg gik À g‘ig gjk ‘j δ ‘ ‘j δ j ¼ g‘jg gik À g‘i k ¼ g‘jg gik À gki ¼ j gik À gki

δ j δ1 δ2 δ n as j ¼ 1 þ 2 þÁÁÁþ n ¼ N for the space EN, the result is ‘j g‘jgik À g‘igjk g ¼ Ngik À gki 38 1 Review of Fundamental Topics About Tensors

but gik ¼ gki; thus, ‘j : : : g‘jgik À g‘igjk g ¼ ðÞN À 1 gik Q E D

∂ ij ij gij ∂g Exercise 1.15 Show that in space EN exists g ∂xk þ gij ∂xk ¼ 0, where gij is the metric tensor. The relation between the metric tensor and its conjugated tensor is given by

ij δ j δ1 δ2 δ n gijg ¼ j ¼ 1 þ 2 þÁÁÁþ n ¼ N

Differentiating this expression with respect to xk ∂ ij gijg ∂g ∂ðÞgij ∂N ¼ ij gij þ g ¼ ¼ 0 ∂xk ∂xk ij ∂xk ∂xk so

∂g ∂gij gij ij þ g ¼ 0 Q:E:D: ∂xk ij ∂xk

Exercise 1.16 For the symmetric tensorTij, that fulﬁlls the condition gijT‘kÀ gi‘Tjk þ gjkT‘i À gk‘Tij ¼ 0, show that Tij ¼ mgij, where m 6¼ 0 is a scalar. Multiplying the expression given by gij follows

ij ij ij ij δ i δ ‘ δ i ij g gijT‘k À g gi‘Tjk þ g gjkT‘i À g gk‘Tij ¼ i T‘k À j Tjk þ kT‘i À g gk‘Tij ¼ 0

δ j δ1 δ2 δ n ‘ As j ¼ 1 þ 2 þÁÁÁþ n ¼ N, and for j ¼ and i ¼ k

ij NTji ¼ g gijTij

2 ij As ds ¼ g Tij ¼ m1, where m1 is a scalar, and with Tij ¼ Tji follows m NT ¼ m g ) T ¼ 1 g ji 1 ij ji N ij

m1 Putting m ¼ N : : : Tij ¼ mgij Q E D 1.6 Angle Between Curves 39

1.6 Angle Between Curves

The angle between two curves is deﬁned by the angle formed by their tangent unit vectors g1, g2 (Fig. 1.9), by means of the dot product

α g1 Á g2 cos ¼ ¼ g1 Á g2 kkg1 kkg2

In differential terms this angle is calculated supposing that in the space E3 two curves intersect in a point R, and admitting a third curve that intersects the other two at points A1 and A2, which distances from the point R are, ds(1) and ds(2) (Fig. 1.9). i i i i i The points M, A1, A2 have coordinates x , x þ dxðÞ1 and x þ dxðÞ2 , respectively. With the cosine law

ðÞRA 2 þ ðÞRA 2 À ðÞA A 2 cos α ¼ lim 1 2 1 2 2ðÞRA1 ðÞRA2 which in differential terms stays ÀÁÀÁÀÁ 2 2 2 ds 1 þ ds 2 À ds 3 cos α ¼ ðÞ ðÞ ðÞ 2dsðÞ1 dsðÞ2 and using the basic expressions for the length of the arcs of the curves ÀÁ ÀÁ 2 i j 2 i j dsðÞ1 ¼ gijdxðÞ1 dxðÞ1 dsðÞ2 ¼ gijdxðÞ2 dxðÞ2 hi ÀÁ 2 2 2 i i i i i i dsðÞ3 ¼ gij x þ dxðÞ1 À x þ dxðÞ2 ¼ gij dxðÞ1 À dxðÞ2 i i j j ¼ gij dxðÞ2 À dxðÞ1 dxðÞ2 À dxðÞ1

Fig. 1.9 Angle between R two curves T2 T1

ds()2

ds()1

A2 C 3 A1 C2

C1 ds()3 40 1 Review of Fundamental Topics About Tensors then g dxi dxj þ dxj dxi i j ij ðÞ1 ðÞ2 ðÞ2 ðÞ1 gijdxðÞ1 dx 2 cos α ¼ ¼ ðÞ 2dsðÞ1 dsðÞ2 dsðÞ1 dsðÞ2

dx i dx j Considering ui ¼ ðÞ1 and vj ¼ ðÞ2 , which are, respectively, the contravariant dsðÞ1 dsðÞ2 unit vectors of the tangents T1 and T2 to the curves C1 and C2, respectively, provides ! ! dxi dxj α ðÞ1 ðÞ2 cos ¼ gij dsðÞ1 dsðÞ2

α π If two vectors are orthogonal, then ¼ 2, so the condition of orthogonality for i j two directions is giju v ¼ 0. The necessary and sufficient condition so that a coordinate system is orthonormal is that gij ¼ 0 8i 6¼ j at the points of this space. The null vector has the peculiar characteristic of being normal to itself. Figure 1.10 illustrates the components of the differential element of arc ds with respect to the coordinate system Xi with origin at point P. The lengths of the arc elements measuredffiffiffiffiffiffi with respectffiffiffiffiffiffi to the coordinate axesffiffiffiffiffiffi of the referential system are p 1 p 2 p 3 dsðÞ1 ¼ g11dx , dsðÞ2 ¼ g22dx , and dsðÞ3 ¼ g33dx . To prove that α is real and that cos α 1, consider the expression

α i j ij i j cos ¼ giju v ¼ g uivj ¼ u vj ¼ uiv where u, v are unit vectors. Admit that these vectors are multiplied by two non-null real numbers ‘, m, originating ðÞ‘ui þ mvi , as the metric of the space is positive deﬁnite, then for all the values of this pair of numbers ÀÁÀÁ ‘ i i ‘ j j gij u þ mv u þ mv 0

3 Fig. 1.10 Components of X the differential arc element with respect to the Xi = 3 coordinate system ds()3 g33 d x ds

a 2 23 ds = g d x 2 2 22 X a 13 P a 12 = 1 ds()1 g11 d x 1 X 1.6 Angle Between Curves 41

Developing this inequality

‘2 þ 2‘m cos α þ m2 0

i j for u vj ¼ uiv ¼ cos α, which can be written under the form ÀÁ ðÞ‘ þ m cos α 2 þ m2 1 À cos 2α 0 that will be positive definite if cos 2α 1 orkk cos α 1, so α is real. Let the modulus of a vector in terms of their contravariant components qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε k ‘ : : v ¼ gk‘v v ð1 6 1Þ and in terms of their covariant components pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kk v ¼ εg vkvk ð1:6:2Þ thus the angle between two curves is determined when calculating the angle between their tangent unit vector ui, vj, then

i j giju v cos α ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ ð1:6:3Þ i j ðÞu ui v vj

Exercise 1.17 Let the orthogonal unit vectors ui and vj, calculate the norm of vector wi ¼ ui þ vi. The condition of orthogonality between two vectors is given by

i j giju v ¼ 0 and as ui and vj are unit vectors

i j i j giju u ¼ 1 gijv v ¼ 1

For the vector wi

2 i j i j j i i j i i j j j i kkw ¼ gijw w ¼ gijðÞu þ v ðÞ¼u þ v giju u þ giju v þ gijv u þ gijv v ¼ 1 þ 0 þ 0 þ 1 ¼ 2 then pffiffiffi kkw ¼ 2 42 1 Review of Fundamental Topics About Tensors

i j Exercise 1.18 The vectorsu and v are orthogonal, and each one of them has ‘ p j k i ‘4 modulus , show that gpjgki À gpkgji u v u v ¼ . The square of the modulus of the vectors is given by

i j ‘2 i j ‘2 giju u ¼ gijv v ¼ and the condition of orthogonality between these vectors is given by

i j giju v ¼ 0

Developing the given expression g g À g g upvjukvi ¼ g g upvjukvi À g g upvjukvi pj ki pk ji pj ki pk ji p j k i p k j i ¼ gpju v gkiu v À gpku u gjiv v ¼ 0 À ‘2 Â ‘2 so p j k i ‘4 : : : gpjgki À gpkgji u v u v ¼ Q E D

i j Exercise 1.19 Given the symmetric tensorTij and the unit vectors u and v k i i i i orthogonal to the vector w , show that Tiju À m1giju þ n1gijw ¼ 0 and Tijv À i i m2gijv þ n2gijw ¼ 0, where m1 6¼ m2 and n1 6¼ n2 are scalars, then these unit vectors are orthogonal. As ui and vj are unit vectors, then

i j i j giju u ¼ 1 gijv v ¼ 1 and the conditions of orthogonality of these unit vectors with respect to the vector wi are

i j i j giju w ¼ 0 gijv w ¼ 0

Multiplying by vj both the members of the ﬁrst expression

i j i j i j i j i j Tiju v À m1giju v þ n1gijw v ¼ 0 ) Tiju v ¼ m1giju v and multiplying by uj both the members of the second expression

i j i j i j i j i j Tijv u À m2gijv u þ n2gijw u ¼ 0 ) Tijv u ¼ m2gijv u 1.6 Angle Between Curves 43

The indexes i and j are dummies, so their position can be changed

i j i j Tiju v ¼ m2giju v thus

i j i j i j m1giju v ¼ m2giju v ) ðÞm1 À m2 giju v ¼ 0

i j i As by hypothesis m1 6¼ m2, then giju v ¼ 0; this shows that the unit vectors u and vj are orthogonal.

1.6.1 Symmetrical and Antisymmetrical Tensors

If the change of position of two indexes, covariant or contravariant, does not modify the tensor’s components, then this is a symmetrical tensor

pqrs pqrs pqrs pqrs pqrs pqrs Tijk ¼ Tikj ¼ Tjik ¼ Tjki ¼ Tkij ¼ Tkji . The symmetry, a priori, does not ensure that the new variety is a tensor. Admit pqrs qprs ’ that Tijk‘ ¼ Tijk‘ , whereby by the hypothesis of this tensor s symmetry, it follows pqrs qprs pqrs that Tijk‘ À Tijk‘ ¼ 0. As Tijk‘ is a tensor, the result of the difference between the two varieties being null, and as the referential system is arbitrary, it is concluded that this result will always be null for any coordinate system, i.e., it always has the pqrs qprs tensor null. WritingTijk‘ þ 0 ¼ Tijk‘ , and as the summation of tensors is a tensor, it pqrs is concluded that Tijk‘ is a tensor. A tensor is called antisymmetrical with respect to two of its indexes, if it changes signs on the change of position between these two indexes: Tijk‘ ¼ÀT‘jki. The number of independent components of an antisymmetric tensor of order p in the space EN is given by

N! n ¼ ð1:7:1Þ p!ðÞN À p !

ijk Let the space EN in which the antisymmetric pseudotensor of the third order ε is defined (a general definition of pseudotensors will be presented in item 1.8), and by the definition of antisymmetry, it provides six components of εijk which are numerically equal:

εijk ¼ εjki ¼ εkij ¼Àεikj ¼Àεjik ¼Àεkji

This variety has 27 components, having 21 null, for it is veriﬁed that only the six components ε123 ¼ ε231 ¼ ε312 ¼Àε132 ¼Àε213 ¼Àε321 are non-null. Let, for example, a linear and homogeneous transformation be applied to the component ε123: 44 1 Review of Fundamental Topics About Tensors

∂x1 ∂x2 ∂x3 ε123 ¼ ε123 ð1:7:2Þ ∂xi ∂xj ∂xk

Developing expression (1.7.2) ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 ε123 ¼ þ þ ∂x1 ∂x2 ∂x3 ∂x2 ∂x3 ∂x1 ∂x3 ∂x1 ∂x2 ð1:7:3Þ ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 À À À ε123 ∂x1 ∂x3 ∂x2 ∂x3 ∂x2 ∂x1 ∂x2 ∂x1 ∂x3

i In compact form for the component ε123 in the coordinate system X

k ∂x ε123 ¼ ε123 ð1:7:4Þ ∂x‘ and with

∂xi ∂xj g ‘ ¼ g k ∂xk ∂x‘ ij

It follows by means of product of determinants

i j i 2 i 2 ∂x ∂x ∂x ∂x jjg ‘ ¼ Á Á g ) jjg ‘ ¼ g ) g ¼ g k ∂xk ∂x‘ ij k ∂xk ij ∂xk

i 1 ∂x 1 pffiffiffi ¼ pffiffiffi ð1:7:5Þ g ∂xk g

Comparing expression (1.7.5) with the expression (1.7.4)

1 ε123 ¼ pffiffiffi ð1:7:6Þ g

So as to generalize expression (1.7.6), this analysis is made for the other contravariant components of the pseudotensor εijk. As this variety assumes the values 0, Æ 1 as a function of the position of their indexes, it is linked to the permutation symbol eijk by means of the following relations: 8 > þ1 ijk > pffiffiffi e is an even permutation of the indexes < g εijk 1 : : ¼ > Àffiffiffi ijk ð1 7 7Þ > p e is an odd permutation of the indexes :> g 0 when there are repeated indexes 1.6 Angle Between Curves 45

Expression (1.7.7) represents the components of the Ricci pseudotensor, also called Levi-Civita pseudotensor. The covariant components of this pseudotensor are obtained by means of the metric tensor, whereby using the approaches presented in item 1.5, it is provided for the lowering of the indexes of the pseudotensor εpqr:

ε εpqr : : ijk ¼ gipgjqgkr ð1 7 8Þ and with the deﬁnition of the determinant of the metric tensor, and with the deﬁnition of εijk given by the relations (1.7.7), it follows that

1 pffiffiffi εijk ¼ g pffiffiffi ¼ g ð1:7:9Þ ij g

In terms of the permutation symbol eijk, it is provided as the covariant coordinates of the Ricci pseudotensor 8 ffiffiffi < p geffiffiffiijk is an even permutation of the indexes ε p : : ijk ¼ : À geijk is an odd permutation of the indexes ð1 7 10Þ 0 when there are repeated indexes

The definition of the Ricci pseudotensor presented for the space E3 is generalized for the space EN, in which the contravariant components and covariant of this variety are given, respectively, in terms of the permutation symbol by 8 > 1 > þ i1i2i3ÁÁÁin > pffiffiffi e is an even permutation of the indexes < g εi1i2i3ÁÁÁin ¼ À1 ð1:7:11Þ > ffiffiffi i1i2i3ÁÁÁin > p e is an odd permutation of the indexes :> g 0 when there are repeated indexes 8 pffiffiffi < ge is an even permutation of the indexes pffiffiffii1i2i3ÁÁÁin ε ¼ À ge is an odd permutation of the indexes ð1:7:12Þ i1i2i3ÁÁÁin : i1i2i3ÁÁÁin 0 when there are repeated indexes

The conception of permutation symbol is associated to the value of a determinant, with no link to the space EN, whereby it refers only to a symbol that seeks to simplify the calculations. With the definition of the Ricci pseudotensor in terms of this symbol, it is verified that in the relation between these two varieties exists the pffiffiffi term g linked to the metric of the space. This shows the fundamental difference between the same, for the change of sign of the Ricci pseudotensor as a function of the permutations of their indexes (sign defined by the permutation symbol) indicates the orientation of the space. With relation (1.7.10) it follows that

εijkεjki ¼ 3! ¼ 6 ð1:7:13Þ 46 1 Review of Fundamental Topics About Tensors

The deﬁnitions and deductions presented next seek to complement the relations between the generalized Kronecker delta and the Ricci pseudotensor in the space EN. These expressions are called δ À ε relations.

1.6.1.1 Generalization of the Kronecker Delta

∂xi ∂xj ∂xk The Ricci pseudotensor represents the mixed product of three vectors ∂x‘ , ∂x‘ , ∂x‘, where ‘ ¼ 1, 2, 3 indicates the components of these vectors, which comprise the lines and columns of the determinant that expresses this product, called Gram determinant, that in terms of their covariant components stays

i j k ∂x ∂x ∂x

∂x1 ∂x1 ∂x1 δ δ δ ∂ i ∂ j ∂ k ∂ i ∂ j ∂ k 1i 1j 1k ε x x x x x x δ δ δ ijk ¼ ∂ ‘ Â ∂ ‘ Á ∂ ‘ ¼ ¼ 2i 2j 2k x x x ∂x2 ∂x2 ∂x2 δ δ δ ∂ i ∂ j ∂ k 3i 3j 3k x x x ∂x3 ∂x3 ∂x3 and in terms of their contravariant components

p q r ∂x ∂x ∂x

∂x1 ∂x1 ∂x1 δ1p δ1q δ1r ∂ p ∂ q ∂ r εpqr x x x δ2p δ2q δ2r ¼ ¼ ∂x2 ∂x2 ∂x2 δ3p δ3q δ3r ∂ p ∂ q ∂ r x x x ∂x3 ∂x3 ∂x3

The product of these two determinants being given by

1p 1q 1r δ1i δ1j δ1k δ δ δ pqr 2p 2q 2r εijkε ¼ δ2i δ2j δ2k Á δ δ δ 3p 3q 3r δ3i δ3j δ3k δ δ δ ÀÁÀÁÀÁ δ δ1p δ δ2p δ δ3p δ δ1q δ δ2q δ δ3q δ δ1r δ δ2r δ δ3r ÀÁ1i þ 2i þ 3i ÀÁ1i þ 2i þ 3i ÀÁ1i þ 2i þ 3i pqr 1p 2p 3p 1q 2q 3q 1r 2r 3r εijkε ¼ δ1jδ þ δ2jδ þ δ3jδ δ1jδ þ δ2jδ þ δ3jδ δ1jδ þ δ2jδ þ δ3jδ ÀÁÀÁÀÁ δ δ1p þ δ δ2p þ δ δ3p δ δ1q þ δ δ2q þ δ δ3q δ δ1r þ δ δ2r þ δ δ3r 1k 2k 3k 1k 2k 3k 1k 2k 3k mp mq mr δmiδ δmiδ δmiδ pqr mp mq mr εijkε ¼ δmjδ δmjδ δmjδ mp mq mr δmkδ δmkδ δmkδ

δ p δ q δ r i i i ε εpqr δ p δ q δ r : : ijk ¼ j j j ð1 7 14Þ δ p δ q δ r k k k 1.6 Angle Between Curves 47

With the expressions (1.7.10) and (1.7.7), it follows that 8 < 1 ε εrst δrst δst : : r‘m ¼ r‘m ¼ ‘m ¼ : 0 ð1 7 15Þ À1

The contraction of the indexes k and r of the product of two pseudotensors, given by expression (1.7.15), provides

δ p δ q δ k i i i ε εpqr δpqk δ p δ q δ k ijk ¼ ijk ¼ j j j δ p δ q δ k k k k

δ k and as k ¼ 3 it follows that

δ p δ q δ k i i i δ p δ q ε εpqr δpqk δ p δ q δ k i i δ pδ q δ qδ p : : ijk ¼ ijk ¼ j j j ¼ δ p δ q ¼ i j À i j ð1 7 16Þ δ p δ q j j k k 3

Analogously, and with the contraction of the indexes j and p:

δ j δ q δ k i i i δ q δ r pqr jqr δ q δ k i i r q q r εijkε ¼ δ ¼ 3 ¼À q ¼ δ δ À δ δ ijk j j δ δ r i k i k δ j δ q δ k k k k k k

ε εpqr δpq The product ijr ¼ ij leads to the generalization of the Kronecker delta that has its value defined as a function of the number of permutations of their indexes. For the covariant components of this operator, δijpq ¼ δpijq is provided, where the number of permutations of the indexes is even, so it is verified that this operator is symmetrical, and δijpq ¼Àδjipq ¼Àδijqp is antisymmetric for an odd number of permutations of the indexes. The deltas with repeated indexes are null, for example, δ11pq ¼ δ22pq ¼ δij33 ¼ 0. This analysis allows defining the generalized Kronecker delta in space EN: 8 < þ1 is an even permutation of i1i2i3ÁÁÁ, j j j ÁÁÁ j j j ÁÁÁj 1 2 3 1 2 3 n δ ¼ À1 is an odd permutation of i1i2i3ÁÁÁ, j j j ÁÁÁ ð1:7:17Þ i1i2i3ÁÁÁin : 1 2 3 0 when there are repeated indexes

1.6.1.2 Fundamental Expressions with the Generalized Kronecker Delta

The generalized Kronecker delta in terms of the Ricci pseudotensor is given by 48 1 Review of Fundamental Topics About Tensors

δ δ δ j i1 j i2 ÁÁÁ j in 1 1 1 j ÁÁÁj j ÁÁÁj δ δ δ j j2j3ÁÁÁjm 1 m mþ1 n j2i1 j2i2 ÁÁÁ j2in εi i i ÁÁÁi ε 1 ¼ δ ¼ ð1:7:18Þ 1 2 3 m i1ÁÁÁimimþ1ÁÁÁin ÁÁÁ ÁÁÁ ÁÁÁ ÁÁÁ δ δ δ δ jni1 jni2 j1i1 jnin

δpq Various fundamental expressions are obtained with the Kronecker delta ij that are useful in Tensor Calculus. Let, for example, the contraction of the indexes j and q of this tensor

δpq δpj δ pδ j δ qδ j δ pδ j δ jδ p δ p δ jδ p δ p ij ¼ ij ¼ i j À i j ¼ i j À i j ¼ 3 i À i j ¼ 2 i whereby 1 1 δ p ¼ δpj ¼ δp1 þ δp2 þ δp3 i 2 ij 2 i1 i2 i3

It is also veriﬁed for the contractions j ¼ q and k ¼ r 1 1 δpqr ¼ δpjk ¼ δ p ¼ δp12 þ δp23 þ δp31 ijk 2 ijk i 2 i12 i23 i31

The generalization of these expressions that involve Kronecker deltas for the space EN is given by the following expression:

p p p ÁÁÁp ðÞN À n ! p ÁÁÁp p ÁÁÁp δ 1 2 3 m ¼ δ 1 m mþ1 n ð1:7:19Þ i1i2i3ÁÁÁim ðÞN À m ! i1ÁÁÁimimþ1ÁÁÁin

p p p ÁÁÁp The Kronecker delta tensor δ 1 2 3 m provided by expression (1.7.19) is of order i1i2i3ÁÁÁim p ÁÁÁp p ÁÁÁp 2ðÞn À m inferior to the order Kronecker delta tensor δ 1 m mþ1 n , from which it i1ÁÁÁimimþ1ÁÁÁin was obtained by means of contractions of the indexes. In this expression for m ¼ 1, n ¼ 3

1 δ p ¼ δpjk i ðÞN À 2 ðÞN À 1 ijk

Putting m ¼ 1, n ¼ 2 in expression (1.7.19) results in

1 δ p ¼ δpj i ðÞN À 1 ij

δp These two examples show that i can be obtained by two contractions of the δpqr indexes of the sixth-order tensor ijk or by means of only a contraction of δpq the indexes of the fourth-order tensor ij . 1.6 Angle Between Curves 49

In expression (1.7.19) for m ¼ 1, i ¼ p

ðÞN À n ! δi1 ¼ δi1i2ÁÁÁin i1 ðÞN À 1 ! i1i2ÁÁÁin and as δi1 n it results in i1 ¼

ðÞN À n ! nNðÞÀ 1 ! n ¼ δi1i2ÁÁÁin ) δi1i2ÁÁÁin ¼ ðÞN À 1 ! i1i2ÁÁÁin i1i2ÁÁÁin ðÞN À n ! n! δi1i2ÁÁÁin ¼ ð1:7:20Þ i1i2ÁÁÁin ðÞN À n !

ε εi1i2ÁÁÁin For the inner product of the Ricci pseudotensors i1i2ÁÁÁin and with N ¼ n expression (1.7.20) provides

ε εi1i2ÁÁÁin δi1i2ÁÁÁin n! 1:7:21 i1i2ÁÁÁin ¼ i1i2ÁÁÁin ¼ ð Þ

Expression (1.7.19) with N ¼ n provides

p p ÁÁÁp ε εp1p2ÁÁÁpmpmþ1ÁÁÁpn N m !δ 1 2 n 1:7:22 i1i2ÁÁÁimimþ1ÁÁÁin ¼ ðÞÀ i1i2ÁÁÁin ð Þ

Expression (1.7.22) relates in space EN the inner product of two Ricci pseudotensors with the generalized Kronecker delta tensor.

1.6.1.3 Product of the Ricci Pseudotensor by the Generalized Kronecker Delta

δ pijk The deﬁnition of the generalized Kronecker delta shows that q123 ¼ 0, for a dummy index will always occur when these vary. With expression (1.7.18)

δpq δp1 δp2 δp3 δ δ δ δ ε εpijk δ pijk iq i1 i2 i3 q123 ¼ q123 ¼ ¼ 0 δjq δj1 δj2 δj3

δkq δk1 δk2 δk3

Developing this determinant in terms of the ﬁrst column

δpqεijkε123 À δiqεpjkε123 þ δjqεpikε123 À δkqεpijε123 ¼ 0 and with εpik ¼Àεipk

εijkδpq ¼ εpjkδiq þ εipkδjq þ εijpδkq ð1:7:23Þ 50 1 Review of Fundamental Topics About Tensors

The symmetry of δpq allows changing the position of these indexes

εijkδqp ¼ εqjkδip þ εiqkδjp þ εijqδkp ð1:7:24Þ

1.6.1.4 Norm of the Antisymmetric Pseudotensor of the Second Order

A vector is represented by an oriented segment of a straight line, and its norm is given by the length of this segment. For an antisymmetric pseudotensor of the second order A associated to an axial vector u provides that its norm is linked to the area of the parallelogram which sides are the vectors that define the vectorial product u ¼ v Â w. Let α the angle between the vectors v and w, the square of the modulus of the cross product of these vectors is given by ÀÁ kku 2 ¼ kkv Â w 2 ¼ kkv 2 kkw 2 sin 2α ¼ kkv 2 kkw 2 1 À cos 2α ¼ kkv 2 kkw 2 À ðÞv Á w 2 thus qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kku ¼ kkv 2 kkw 2 À ðÞv Á w 2 ð1:7:25Þ

This norm can be expressed in terms of the components of the pseudotensor A. Let the components of the vectors v and w in the coordinate system Xk, so with the expression (1.7.25) ÀÁ 2 i ‘ j m i m j ‘ i ‘ j m kku ¼ gi‘v v gjmw w À ðÞgimv w gj‘v w ¼ gi‘gjm À gimgj‘ v v w w

g ‘ g i im i ‘ j m ¼ v v w w gj‘ gjm

This determinant allows writing

ÀÁ gi‘ gim ‘ 1 gi‘ gim ‘ viv wjwm ¼ viwj À vjwi v wm gj‘ gjm 2 gj‘ gjm and as ÀÁ ‘ 1 ‘ ‘ A m ¼ v wm À vmw 2 it follows that 1.6 Angle Between Curves 51

ÀÁ gi‘ gim ‘ 1 gi‘ gim ‘ ‘ Aijv wm ¼ Aij v wm À vmw gj‘ gjm 2 gj‘ gjm then

gi‘ gim ‘ ‘ 1 gi‘ gim ‘ viw v wm ¼ AijA m ð1:7:26Þ gj‘ gjm 2 gj‘ gjm

1.6.1.5 Generation of Tensors from the Ricci Pseudotensor

The Ricci pseudotensor generates an antisymmetric tensor from a pseudotensor (axial vector), and this pseudotensor generates an antisymmetric tensor from a pseudotensor (axial vector). This characteristic of the Ricci pseudotensor in space

E3 is generalized for the space EN, where the known antisymmetric tensor A½i1i2ÁÁÁin provides

1 Tj1j2ÁÁÁjnÀm ¼ εj1j2ÁÁÁjnÀmi1i2ÁÁÁim A ð1:7:27Þ m! ½i1i2ÁÁÁim

Tensor Tj1j2ÁÁÁjnÀm is generated by the Ricci pseudotensor, which works as an operator applied to the antisymmetric tensor to produce this associated tensor. ε Multiplying both the members of expression (1.7.27)by j1j2ÁÁÁjnÀmi1i2ÁÁÁim results in 1 ε Tj1j2ÁÁÁjnÀm ¼ ε εj1j2ÁÁÁjnÀmi1i2ÁÁÁim A j1j2ÁÁÁjnÀmi1i2ÁÁÁim m! j1j2ÁÁÁjnÀmi1i2ÁÁÁim ½i1i2ÁÁÁim

With expressions (1.7.19), (1.7.21), and (1.7.22), it follows that the expression for the antisymmetric tensor A½i1i2ÁÁÁim in terms of the Ricci pseudotensor is given by

1 A ¼ ε Tj1j2ÁÁÁjnÀm ð1:7:28Þ ½i1i2ÁÁÁim ðÞn À m ! j1j2ÁÁÁjnÀmi1i2ÁÁÁim

To illustrate the application of expression (1.7.28), let the antisymmetric tensor of the fourth order A[ijk‘] with i, j, k, ‘ ¼ 1, 2, 3, ÁÁÁn, to which the following ﬁve varieties are associated

1 ijk‘ i 1 ijk‘p ij 1 ijk‘pq T ¼ ε A ‘ T ¼ ε A ‘ T ¼ ε A ‘ 4! ½ijk 4! ½jk p 4! ½k pq

ijk 1 ijk‘pqr ijk‘ 1 ijk‘pqrs T ¼ ε A ‘ T ¼ ε A 4! ½pqr 4! ½pqrs 52 1 Review of Fundamental Topics About Tensors

1.7 Relative Tensors

The tensors defined in the previous items are called absolute tensors. However, other varieties with properties that are analogous to those of these tensors can be defined. The relative tensors are more general varieties, the absolute tensors being a particular case of the same. In solving various problems that involve integration processes, the need of generalizing the concept of tensor is verified. This generalization leads to the concept of relative tensor. To exemplify the concept of the relative tensor, let a covariant tensor of the second order be defined in the space E3, which transforms by means of the expression

∂xi ∂xj Tk‘ ¼ Tij ð1:8:1Þ ∂xk ∂x‘ ∂xi The determinants of the terms of this function are given by detT ‘, det , k ∂xk ∂xj det ‘ , and det T . Applying the determinant product rule to the determinant ∂x ij terms of this expression ∂xi ∂xj detTk‘ ¼ det det detTij ð1:8:2Þ ∂xk ∂x‘ the Jacobian of the inverse transformation of tensor Tk‘ is given by ∂xm J ¼ det > 0 ð1:8:3Þ ∂xn so

2 detTk‘ ¼ J detTij ð1:8:4Þ

Expression (1.8.4) shows that detTk‘ of a second-order tensor is not a scalar and also is not a second-order tensor of the type Tpq. This expression is the new transformation. Assuming that detTij > 0 and detTk‘ > 0 provides

ÀÁ1 ÀÁ1 2 2 detTk‘ ¼ J detTij ð1:8:5Þ

This shows that the deﬁnition of tensors can be expanded introducing the concept of relative tensor. Consider the mixed tensor

∂ i ∂ j ∂ p ∂ r ∂ s ∂ v ij...p W x x x x x x ab...d T ... ¼ ðÞJ ÁÁÁ ÁÁÁ T ... ð1:8:6Þ rs v ∂xa ∂xb ∂xd ∂xe ∂xf ∂xh ef h 1.7 Relative Tensors 53 that is called relative tensor of weight W or with weighing factor W. This weight is an integer number, and J is the Jacobian of the transformation. For the particular case in which W ¼ 0, an absolute tensor exists. The concept of relative tensor allows distinguishing a relative invariant of a scalar, which is an absolute invariant. To differentiate these concepts, let the relative invariant A of weight W, which transforms according to the expression

A ¼ JWA ð1:8:7Þ

For the particular case in which W ¼ 0, an absolute tensor exists A ¼ A that is a scalar. For W ¼ 1 provides the scalar density A ¼ JA. The deﬁnition of scalar density will be presented in detail in later paragraphs. To illustrate the concept of relative tensor, let the metric tensor gij with detgij ¼ g. Applying a linear and homogeneous transformation to this tensor

∂xi ∂xj ge‘ ¼ g ð1:8:8Þ m ∂ex‘ ∂exm ij e e with detgij ¼ g, and by means of the property of the product of determinants, provides the relative scalar of weight W ¼ 2 pffiffiffi pffiffiffi ge ¼ J2g ) eg ¼ J g ð1:8:9Þ pffiffiffi For J ¼ 1 provides g that is a relative tensor of unit weight, being, therefore, an invariant. With expression (1.8.9) and the condition gg ¼ 1, having detgij ¼ g,itis pffiffiffi verified that g is a relative tensor of weight À1. Let the Jacobian J of weight W ¼ 1, which is an invariant and when changing to a new coordinate system provides for this determinant J ¼ αJ being α a scalar (invariant). Raising both members of this expression to the power W

W J ¼ αWJW ð1:8:10Þ where JW is an invariant of weight W, thus

W αW ¼ J JÀW ð1:8:11Þ

i Consider the relative tensor Tjk of weight W that transforms by means of the expression

m n i i ∂x ∂x ∂x ‘ T ¼ αW T ð1:8:12Þ jk ∂xj ∂xk ∂x‘ mn and substituting in expression (1.8.12), the value of αW given by expression (1.8.11) provides 54 1 Review of Fundamental Topics About Tensors

m n i i W ∂x ∂x ∂x ‘ T ¼ J JÀW T jk ∂xj ∂xk ∂x‘ mn

It follows that

m n i ÀÁ ÀW i ∂x ∂x ∂x ‘ J T ¼ JÀWT ð1:8:13Þ jk ∂xj ∂xk ∂x‘ mn ÀÁ ÀW ‘ As J Tmn is an absolute tensor, and by means of the transformation law of ÀW i tensors, it is concluded that J Tjk is also an absolute tensor. This shows that the transformation of a relative tensor of weight W in an absolute tensor is carried out multiplying it by the invariant of unit weight raised to the power ÀW. The invariant pffiffiffi g of unit weight is used to carry out this kind of transformation. This systematic k allows, for instance, transforming the relative tensor Tij of weight W into the k absolute tensor Aij by means of ÀÁffiffiffi k p ÀW k : : Tij g ¼ Aij ð1 8 14Þ

The operations multiplying by a scalar, addition, subtraction, contraction, outer product, and inner product are applicable to the relative tensors. These operations provide new relative tensors; as a result, the proof is analogous to the demonstrations performed for the absolute tensors.

Exercise 1.20 Show that δij is an absolute tensor. It is admitted firstly that δij is a relative tensor of unit weight, being detδij ¼ 1, δ∗ which transforms into the absolute tensor ij by means of the expression ÀÁffiffiffi δ∗ p À1δ ij ¼ g ij ÀÁffiffiffi δ∗ δ p À1 As ij is an isotropic tensor so ij is also isotropic, then g ¼ 1, which shows that δij is an absolute tensor.

1.7.1 Multiplication by a Scalar

This operation provides as a result a relative tensor of weight W, which components are the components of the original relative tensor multiplied by the scalar. W Let, for example, the relative tensor (J) Tij and the scalar m which product Pij is W W given by ðÞJ Pij ¼ mJðÞ Tij. To demonstrate that this expression represents a tensor, it is enough to apply the transformation law of tensors to this expression. 1.7 Relative Tensors 55

1.7.1.1 Addition and Subtraction

This operation is deﬁned for relative tensors of the same order and of the same kind, such as in the case of the following mixed tensors

W k W k W k ðÞJ Tij ¼ ðÞJ Aij þ ðÞJ Bij

Subtraction is deﬁned in the same way as addition, however, admitting that one W k W k W k of the tensors be multiplied by the scalar À1: ðÞJ Tij ¼ ðÞJ Aij þÀðÞ1 ðÞJ Bij.To demonstrate that these expressions represent relative tensors, the transformation law of tensors is applied to this expression.

1.7.1.2 Outer Product

This operation is deﬁned in the same way as the outer product of absolute tensors. W1 k... Let, for example, the relative tensor ðÞJ Aij... of variance ( p, q) and weight W1, and W2 ...‘m the relative tensor ðÞJ B...rs of variance (u, v) and weight W2, which multiplied provide hihi W k...‘m W1 k... W2 ...‘m ðÞJ Tij...rs ¼ ðÞJ Aij... ðÞJ B...rs that is a relative tensor of variance ðÞp þ u, q þ v and weight W ¼ W1 þ W2.To demonstrate that this product is a relative tensor, the transformation law of tensors is applied to this expression.

1.7.1.3 Contraction

This operation is deﬁned in the same way as the contraction of the absolute tensors. W ij Let, for example, the relative tensor (J) Tk‘m, in which contracting the upper index j provides

W ij W i ðÞJ Tj‘m ¼ ðÞJ T‘m that shows that the resulting relative tensor has its order reduced in two, but maintains its weight W. To demonstrate that this contraction is a relative tensor, the transformation law of tensors is applied to this expression. 56 1 Review of Fundamental Topics About Tensors

1.7.1.4 Inner Product

This operation is deﬁned in the same manner as the inner product of the absolute tensors. W1 W2 ‘ Let, for example, two relative tensors ðÞJ Aij and ðÞJ Bk , whereby it follows for the outer product of these tensors hihi W1þW2 ‘ W1 W2 ‘ ðÞJ Pijk ¼ ðÞJ Aij ðÞJ Bk that represents a relative tensor of the fourth order and weight W ¼ W1 þ W2, and with the contraction of the index ‘ the inner product is given by hihi W1þW2 ‘ W1 W2 ‘ W1þW2 ðÞJ Pij‘ ¼ ðÞJ Aij ðÞJ B‘ ¼ ðÞJ Pij

This shows that the resulting relative tensor is of the second order and weight W ¼ W1 þ W2. To demonstrate that this product is a relative tensor, the transformation law of tensors is applied to this expression.

1.7.1.5 Pseudotensor

The varieties that present a few tensorial characteristics, for example, when changing the coordinate system they follow a transformation law that differs from the transformation law of tensors by the presence of the Jacobian, are called pseudotensor (relative tensors). However, these varieties are not maintained invariant when the coordinate system is transformed. ijk The deﬁnitions of the antisymmetric pseudotensors εijk and ε are associated, respectively, to the permutation symbols in the covariant form eijk or in the contravariant form eijk, to which correspond the values þ1orÀ1 relative to the even or odd number of permutations, respectively. The Ricci pseudotensors εijk and εijk are associated to the concept of space orientation. These varieties, when changing the coordinate system, transform in the same way as the tensors, but are not invariant after these transformations. This shows that a few characteristics are similar to the tensors but vary with the change of referential, for they assume the values Æ1, so they are not tensors in the sensu stricto of the term. In expression (1.7.27) it is veriﬁed that εj1j2ÁÁÁjnÀmi1i2ÁÁÁim has weight þ1, and the tensor Tj1j2ÁÁÁjnÀm has weight superior to the weight of the antisymmetric tensor

A½i1i2ÁÁÁim . This expression illustrates the applying of the pseudotensors. Exercise 1.21 Show that (a) εijk is a covariant pseudotensor of the third order and weight À1. (b) εijk is a contravariant pseudotensor of the third order and weight þ1. (c) The absolute pseudotensors can be obtained from these pseudotensors. 1.7 Relative Tensors 57

(a) The deﬁnition of determinant allows writing Jεpqr, and as the pseudotensor εijk assume the values 0,Æ1, on being applied to this variety, it provides a linear and homogeneous transformation

∂xi ∂xj ∂xk ∂xi ∂xj ∂xk Jε ¼ ε ) ε ¼ JÀ1ε ¼ ε pqr ∂xp ∂xq ∂xr ijk pqr ∂xp ∂xq ∂xr ijk pqr

then εijk is a covariant pseudotensor of the third order and weight À1. (b) In a way that is analogous to the previous case, for deﬁning the determinant Jε‘mn, and for the transformation law of tensors

p q r p q r ∂x ∂x ∂x ∂x ∂x ∂x À1 Jεpqr ¼ εijk ) εpqr ¼ J εijk ∂xi ∂xj ∂xk ∂xi ∂xj ∂xk

As JJ ¼ 1 it results in

∂xp ∂xq ∂xr εpqr ¼ Jεijk ∂xi ∂xj ∂xk

then εijk is a contravariant pseudotensor of the third order and weight +1. (c) As the pseudotensor εijk has weight À1, it follows by the transformation law of relative tensors into absolute tensors, where the upper asterisk indicates the absolute tensor hi ÀÁffiffiffi À1 ffiffiffi ε* p À1 ε p ε ijk ¼ g ijk ¼ g ijk

For the relative pseudotensor εijk the absolute pseudotensor indicated by the lower asterisk exists

1 εijk ¼ pffiffiffi εijk * g

Exercise 1.22 Show that gij is an absolute tensor. Rewriting expression (1.8.9) pffiffiffi pffiffiffi g ¼ J g and with the cofactor of the matrix of tensor gij given by

1 ‘ Gij ¼ eik ejpqg g 2 kp ‘q and in terms of Ricci’s pseudotensor 58 1 Review of Fundamental Topics About Tensors

1 ‘ Gij ¼ εik εjpqg g 2 kp ‘q it follows that

ij ij G 1 ik‘ jpq g ¼ ¼ ε ε g g‘ g 2 kp q

The term to the right of this expression. is the product of two pseudotensors and the tensors, being gij the inner product of these two varieties. This expression has weight W ¼ 0, then gij is an absolute tensor.

1.7.1.6 Scalar Capacity

Let an antisymmetric pseudotensor Cijk in an afﬁne space, for which according to expression (1.7.1) for N ¼ 3 and p ¼ 3, there is only one independent component. Writing Cijk as a function εijk follows

Cijk ¼ εijkc where c is a component of the variety and with the change of the coordinate system

i j k ∂x ∂x ∂x pqr Cijk ¼ C ∂xp ∂xq ∂xr and as the antisymmetry is maintained when the reference system is changed

pqr C ¼ εpqrc ð1:8:15Þ

Considering the component C123:

1 2 3 ∂x ∂x ∂x pqr C123 ¼ C ð1:8:16Þ ∂xp ∂xq ∂xr and substituting expression (1.8.15) in expression (1.8.16)

∂x1 ∂x2 ∂x3 c ¼ εpqr c ð1:8:17Þ ∂xp ∂xq ∂xr

Let

∂x1 ∂x2 ∂x3 J ¼ εpqr ð1:8:18Þ ∂xp ∂xq ∂xr 1.7 Relative Tensors 59 results in the following expressions

1 c ¼ Jc ) c ¼ c ¼ Jc ð1:8:19Þ J

Function c is the only independent component of the antisymmetric pseudotensor Cijk, which is called scalar capacity. Then a scalar capacity is a pseudotensor of weight À1. To illustrate the concept of scalar capacity, let, for example, the antisymmetric variety dVijk that defines an elementary volume in space E3. This analysis follows the same routine presented when defining the scalar capacity. The elementary volume is obtained by means of the mixed product of three vectors that define the three reference axes in this space

dx1 00 1 2 3 dVijk ¼ 0 dx2 0 ¼ dx dx dx ¼ dV ) dVijk ¼ dxidxjdxk ð1:8:20Þ

00dx3 and with the transformation law of tensors

∂xi ∂xj ∂xk ∂x1 ∂x2 ∂x3 dVijk ¼ dxpdxqdxr dV ¼ dxpdxqdxr ð1:8:21Þ ∂xp ∂xq ∂xr ∂xp ∂xq ∂xr

The antisymmetry of the pseudotensor is maintained when changing the coordinate system

dxpdxqdxr ¼ εijkdV ð1:8:22Þ and substituting expression (1.8.21) in expression (1.8.22)

∂x1 ∂x2 ∂x3 dV ¼ εijk dV ∂xp ∂xq ∂xr results in the following expressions

1 1 dV ¼ JdV ) dV ¼ dV ∴dx1dx2dx3 ¼ dx1dx2dx3 ð1:8:23Þ J J

This shows that the elementary volume in an afﬁne space is a pseudoscalar of weight À1. In a more restricted manner, it says that the volume is a scalar capacity. The term capacity comes from the association of the volume (capacity, content) to the variety being analyzed. It is concluded that the integration of expression (1.8.23), which represents a scalar ﬁeld, is a pseudoscalar. 60 1 Review of Fundamental Topics About Tensors

1.7.1.7 Scalar Density

Let the antisymmetric pseudotensor Dijk, for which an analysis analogous to the one developed when deﬁning the scalar capacity is carried out

∂xi ∂xj ∂xk D ¼D¼ε D 123 ijk ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 ∂xi ∂xj ∂xk J ¼ ε ¼ ε pqr ∂xp ∂xq ∂xr ijk ∂x1 ∂x2 ∂x3 then

1 D¼JD)D¼ D)D¼JD J

Function D is the unique component of the antisymmetric pseudotensor Dijk, which is called scalar density. Then a scalar density is a pseudotensor of weight þ1. To illustrate the concept of scalar density, let, for example, a body of elementary mass dm in the afﬁne space E3. This mass is determined by means of density (speciﬁc mass) ρ(x1, x2, x3) and the elementary volume dV, thus dm ¼ ρðÞx1; x2; x3 dV. Considering that the mass is invariable (is a scalar) ÀÁÀÁ dm ¼ ρ x1; x2; x3 dV ¼ ρ x1; x2; x3 dV ð1:8:24Þ and as dV is a scalar capacity of weight À1, substituting expression (1.8.23)in expression (1.8.24) provides

ÀÁÀÁ1 ÀÁÀÁ ρ x1; x2; x3 dV ¼ ρ x1; x2; x3 dV ) ρ x1; x2; x3 ¼ J ρ x1; x2; x3 ð1:8:25Þ J

This shows that the density in an affine space is a pseudoscalar of weight þ1, this variety being called scalar density. The term density is not physically correct, for in truth ρ(x1, x2, x3) such as it is presented defines the body’s specific mass. The concepts shown for the elementary volume and for density in the affine space E3 can be generalized for the space EN. The varieties that transform by means of the expressions with structure analogous to the structures of expressions (1.8.23) and (1.8.25) are called, respectively, scalar capacity and scalar density in space EN.

1.7.1.8 Tensorial Capacity

k Let the space E3 where product of a scalar capacity c by the tensor Tij exists, which k deﬁnes a tensorial density Cij given by 1.7 Relative Tensors 61

k k : : Cij ¼ cTij ð1 8 26Þ and for a new coordinate system

p q r r ∂x ∂x ∂x C ¼ C k ð1:8:27Þ pq ∂xi ∂xj ∂xk ij it follows that

p q r p q r r ∂x ∂x ∂x ∂x ∂x ∂x r r C ¼ cT k ¼ JcT k ) C ¼ JcT ð1:8:28Þ pq ∂xi ∂xj ∂xk ij ∂xi ∂xj ∂xk ij pq pq

k Expression (1.8.28) shows that Cij transforms in accordance with a law that is similar to the transformation law of scalar capacity; however, it does not represent a relative scalar but a relative tensor of weight þ1. The generalization of the concepts of tensorial capacity for the space EN is immediate.

1.7.1.9 Tensorial Density

Let, for example, the space E3 where the product of a scalar density D by the tensor k k Tij exists, which deﬁnes a tensorial density Dij given by

k k : : Dij ¼DTij ð1 8 29Þ and for a new coordinate system

p q r p q r p q r r ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x 1 D ¼ D k ¼ DT k ¼ DT k pq ∂xi ∂xj ∂xk ij ∂xi ∂xj ∂xk ij ∂xi ∂xj ∂xk J ij r À1 r : : Dpq ¼ J DTpq ð1 8 30Þ

k Expression (1.8.30) shows that Dij transforms in accordance with a law that is to the scalar density transformation law. However, it does not represent a relative scalar but a relative tensor of weight À1. The generalization of the concepts of tensorial density for the space EN is immediate. The outer products between these varieties (pseudotensors and tensors) result in

Scalar capacity Â scalar density ¼ scalar Scalar capacity Â tensor ¼ tensorial capacity Scalar density Â tensor ¼ tensorial density Pseudotensor Â pseudotensor ¼ tensor Tensor Â pseudotensor ¼ pseudotensor 62 1 Review of Fundamental Topics About Tensors

1.8 Physical Components of a Tensor

In mathematics the approach to the problems, in general, is carried out by means of nondimensional parameters. In physics and engineering the parameters have magnitude and dimensions, for example, N/mm2, m/s, etc. The analysis of a physical problem by tensorial means requires that the parameters being studied be invariant when changing the coordinate system. It happens that the axes of the coordinate systems generally do not have the same dimensions. A Cartesian coordinate system has axes that deﬁne lengths, but, for example, a spherical coordinate system has two axes that express nondimensional coordinates, the same occurring in the cylindrical coordinate system with one of their axes. Therefore, the components of a tensor have dimensions, and when the coordinate system is changed, these components vary in magnitude and dimension. To express the transformation of tensors in a consistent way (in magnitude and dimension), and that these varieties can be added after a change of the coordinate systems, the same must be expressed in terms of their physical components.

1.8.1 Physical Components of a Vector

The concept of geometric vector is associated to the idea of displacement, its k ∂xk i ∂xk transformation law being dx ¼ i dx where the coefﬁcients i are constants. ∂x ∂x With respect to a Cartesian coordinates, the term dxjgj represents a displacement in terms of the unit vectors of the coordinate axes

dxjgj ¼ dx1i þ dx2j þ dx3k ð1:9:1Þ

However, this term in a curvilinear coordinates does not represent a displacement, so gk will not be a unit vector in this coordinate system. This shows that the vector must be written in terms of components that express a displacement, called the physical components of the vector. Ã Consider the vector u with physical components uj , which can be written in terms of these components and of their contravariant components

* k : : u ¼ ukek ¼ u gk ð1 9 2Þ

Comparing expressions (1.9.1) and (1.9.2)

j j *j dx g ¼ dx ej ð1:9:3Þ where dx* j are the physical components which by analogy correspond to the k k Ã displacement dx , and with the unit vectors of base gk, ek, the components u , uk 1.8 Physical Components of a Tensor 63

pffiffiffiffiffiffiffiffi are obtained in terms of the unit vector gk. Let gi Á gj ¼ gij then gj ¼ gðÞjj , where the indexes shown in parenthesis do not indicate a summation in j. As the unit vector ej is collinear with gi, thus pffiffiffiffiffiffiffiffi : : gj ¼ gðÞjj ej ð1 9 4Þ and with expression (1.9.4) in expression (1.9.3) pffiffiffiffiffiffiffiffi *j j : : dx ¼ gðÞjj dx ð1 9 5Þ

In an analogous way, by means of expression (1.9.2) pffiffiffiffiffiffiffiffiffi * k : : uk ¼ gðÞkk u ð1 9 6Þ

Ã The physical components uk have the characteristics of displacement, so they can be added vectorially (parallelogram rule), denoting the contravariant physical components of the vector. These components are not unique. Let another variety of components u˜k that represents the projection of vector u on the direction of the unit vector ek. Consider e˜k the reciprocal unit vector of ek, whereby, for this reciprocal basis,

uek ¼ u Á ek ð1:9:7Þ k u ¼ ukg ¼ uekek ð1:9:8Þ

k but e˜k is collinear with g thus 1 ek ¼ ek ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pg pg ðÞkk e ðÞkk gk ¼ gðÞkk ek ¼ e ) ek ¼ ek gk pffiffiffiffiffiffiffiffiffi e k : : ek ¼ gðÞkk g ð1 9 9Þ where the indexes shown in parenthesis do not indicate summation in k, and with expression (1.9.9) in expression (1.9.7) pffiffiffiffiffiffiffiffiffi e : : uk ¼ gðÞkk uk ð1 9 10Þ

The physical components u˜k are the covariant components of vector u. Putting k uk ¼ gðÞkk u and with the expressions (1.9.6) and (1.9.10) then in an orthogonal * e coordinate system uk ¼ uk. This shows that the distinction between the covariant and contravariant basis disappears when the coordinate system is orthogonal. 64 1 Review of Fundamental Topics About Tensors

3 Fig. 1.11 Physical X components of the vector u in the curvilinear coordinate system Xi u

u 3 u2

g33 g22 2 X

g2 g3

u1 g1 g 1 P 11 X

Figure 1.11 shows the physical components of vector u in the curvilinear coordinate system Xi. The components puffiffiffiffiffiffiffik (expression (1.9.10)) represent the gðÞkk lengths of the projections which are orthogonal to the coordinate axes of the pffiffiffiffiffiffiffiffiffi k referential system. The components gðÞkk u (expression (1.9.6)) represent the lengths the of the sides of the parallelepiped, which diagonal is the vector u. Exercise 1.23 Calculate the contravariant, covariant, and physical components of i dxi the velocity vector of a point v ¼ dt , in terms of the cylindrical coordinates of the point xi(r, θ, z).

Cartesian to cylindrical Cylindrical to Cartesian qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ ÀÁ x1 ¼ x1 cos x2 ¼ r cos θ 2 2 x1 ¼ x1 þ x2 ¼ r

2 x2 ¼ x2 sin x2 ¼ r sin θ x2 ¼ arctg x ¼ θ x1 x3 ¼ x3 ¼ zx3 ¼ x3 ¼ z

i dxi The Cartesian coordinates of the vector are v ¼ dt , and for the cylindrical coordinates

∂xi ∂xi dxi dxi vi ¼ vj ) vi ¼ ¼ ∂xj ∂xj dt dt

This shows that the contravariant components of vector v are derivatives with respect to the time of the position vector deﬁned by the coordinates xi, then ÈÉ dx1 dx2 dx3 dr dθ dz vi ¼ , , ¼ , , dt dt dt dt dt dt

For the covariant components in terms of the cylindrical coordinates 1.8 Physical Components of a Tensor 65

∂xi v ¼ vj ¼ g vj i ∂ i ij 2x 3 100 4 2 5 gij ¼ 0 r 0 001

Developing the expression of vi

dx1 dr v ¼ g v1 þ g v2 þ g v3 ¼ v1 ¼ ¼ 1 11 12 13 dt dt ÀÁ 2 2 dx dθ v ¼ g v1 þ g v2 þ g v3 ¼ x1 ¼ r2 2 12 22 23 dt dt dx3 dz v ¼ g v1 þ g v2 þ g v3 ¼ ¼ 3 31 23 33 dt dt whereby ÈÉ 1 ÀÁ 2 3 dx 2 dx dx dr dθ dz vi ¼ , x1 , ¼ , r2 , dt dt dt dt dt dt

The vector norm is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx1 2 dx2 2 dx3 2 kkv ¼ þ ðÞx1 2 þ dt dt dt whereby for its physical components dx1 dx2 dx3 dr dθ dz fgv ¼ , x1 , ¼ , r , dt dt dt dt dt dt

Exercise 1.24 Let the vector u ¼ 3g1 þ g2 þ 2g3, having g1 ¼ 2e1, g2 ¼ 2e1 þ e2, and g3 ¼ 2e1 þ e2 þ 3e3, where e1, e2, e3 are orthonormal vectors, calculate their contravariant physical components. From the covariant basis ffiffiffiffiffiffi : p g1 Á g1 ¼ 2e1 2e1 ¼ 4 ) g11 ¼ 2 ffiffiffiffiffiffi pffiffiffi : p g2 Á g2 ¼ 2e1 2e1 þ e2 Á e2 ¼ 4 þ 1 ¼ 5 ) g22 ¼ 5 ffiffiffiffiffiffi pffiffiffiffiffi : p g3 Á g3 ¼ 2e1 2e1 þ e2 Á e2 þ 3e3 Á 3e3 ¼ 4 þ 1 þ 9 ¼ 14 ) g33 ¼ 14 follows 66 1 Review of Fundamental Topics About Tensors

ffiffiffiffiffiffi *1 1p u ¼ u g11 ¼ 3 Â 2 ¼ 6 ffiffiffiffiffiffi pffiffiffi pffiffiffi *2 2p u ¼ u g22 ¼ 2 Â 5 ¼ 2 5 ffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi *3 3p u ¼ u g33 ¼ 1 Â 14 ¼ 14

1.8.1.1 Physical Components of the Second-Order Tensor

The contravariant physical components of the vectors u and v are given by expression (1.9.10) ffiffiffiffiffiffiffiffi p e e ui ui ¼ gðÞii ui ) ui ¼ pffiffiffiffiffiffiffiffi gðÞii ffiffiffiffiffiffiffiffi p e e vj vj ¼ gðÞjj vj ) vj ¼ pffiffiffiffiffiffiffiffi gðÞjj

For the second-order tensor e T ij ¼ euievj ÀÁ e ui vj 1 Tij T ij ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffi uivj ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðÞii gðÞjj gðÞii gðÞjj gðÞii gðÞjj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e : : Tij ¼ gðÞii gðÞjj T ij ð1 9 11Þ

In a related manner, for the contravariant physical components pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * ij eij : : T ¼ gðÞii gðÞjj T ð1 9 12Þ

The obtaining of the physical components of tensors of a higher order follows the analogous way to that of the second-order tensors.

1.9 Tests of the Tensorial Characteristics of a Variety

The transformation law of the tensors and the quotient law allow establishing a group of functions Np of the coordinates of the referential system Xi which are the components of a tensor. The tensorial nature of the functions that fulﬁll these requirements is highlighted by the invariance that this variety has when there is a change of the coordinate system. However, the evaluation if a variety has tensorial characteristics by means of the quotient law is not wholly complete, as it will be 2 shown next applying to the group of N components of a variety Tpq, for which it is desired to search if it has the characteristics of a tensor. Multiplying Tpq by an 1.9 Tests of the Tensorial Characteristics of a Variety 67

p p q arbitrary vector v and admitting by hypothesis that Tpqv v ¼ m, where m is a i j scalar, it provides for a new coordinate system Tijv v ¼ m, and as m is an invariant, then m ¼ m, by means of the transformation law of vectors

∂xi ∂xj T vpvq ¼ T vpvq pq ∂xp ∂xq ij

Then ∂xi ∂xj T À T vpvq ¼ 0 ð1:10:1Þ pq ∂xp ∂xq ij

The summation rule is applied varying the indexes p and q, so the product vpvq is not, in general, null. Consider the vectors vi with unit componentsðÞ 1, 0, 0ÁÁÁ0 , ðÞ0, 1, 0ÁÁÁ0 , andðÞ 0, 0, 0ÁÁÁ1 , the term in parenthesis of expression (1.10.1) stays ∂xi ∂xj T À T v1v1 ¼ 0 11 ∂x1 ∂x1 ij and as v1v1 6¼ 0

∂xi ∂xj T À T ¼ 0 ð1:10:2Þ 11 ∂x1 ∂x1 ij

In an analogous way it results in

∂xi ∂xj T À T ¼ 0 ð1:10:3Þ 22 ∂x2 ∂x2 ij and so successively for the other values assumed for the indexes. This shows that for p ¼ q the terms in parenthesis from expression (1.10.1) cancel each other. However, for p 6¼ q the complementary analysis of this expression behavior becomes necessary. Let vector vi with components ðÞv1, v2,0,ÁÁÁ0 , whereby from expression (1.10.1) for p, q ¼ 1, 2, it follows that ∂xi ∂xj ∂xi ∂xj T À T v1v1 þ T À T v1v2 11 ∂x1 ∂x1 ij 12 ∂x1 ∂x2 ij ð1:10:4Þ ∂xi ∂xj ∂xi ∂xj þ T À T v2v1 þ T À T v2v2 ¼ 0 21 ∂x2 ∂x1 ij 22 ∂x2 ∂x2 ij

Expressions (1.10.2) and (1.10.3) simplify expression (1.10.4), for the coefﬁ- p q cients of the terms v v are null for p ¼ q. For p 6¼ q with Tij ¼ Tji 68 1 Review of Fundamental Topics About Tensors

∂xi ∂xj ∂xi ∂xj T ¼ T ∂x1 ∂x2 ij ∂x2 ∂x1 ij and with the hypothesis of symmetry results in

∂xi ∂xj ∂xi ∂xj T ¼ T ∂x1 ∂x2 ij ∂x2 ∂x1 ji

Expression (1.10.4) is rewritten as ÀÁ∂xi ∂xj ðÞÀT þ T T þ T v1v2 ¼ 0 12 21 ij ji ∂x1 ∂x2 and as the components v1 and v2 are arbitrary, for v1 ¼ v2 ¼ 1

ÀÁ∂xi ∂xj T þ T ¼ T þ T ð1:10:5Þ 12 21 ij ji ∂x1 ∂x2

Generalizing expression (1.10.5) for the variation of the indexes p, q ¼ 1, 2, 3, ..., it results in

ÀÁ∂xi ∂xj T þ T ¼ T þ T ð1:10:6Þ pq qp ij ji ∂xp ∂xq

ExpressionÀÁ (1.10.6) is the transformation law of second-order tensors, for the term Tpq þ Tqp represents the symmetric part of tensor 2Tpq. However, the antisymmetric part of this tensor is not contained in this analysis, whereby it cannot be concluded that this portion has tensorial characteristics. It is concluded that only 2 the symmetric part of the N components of variety Tpq is a tensor, for when applying the quotient law to this portion it transforms according to the transformation law of second-order tensors. This is the reason why the quotient law must be applied with caution, so as to avoid evaluation errors when checking the tensorial characteristics of a variety. The transformation law of tensors and the consideration of invariance of the variety when having a linear transformation form the criterion that is most appropriate to evaluate if the Np components of this variety have tensorial characteristics.

Problems

1.1 Use8 the index9 notation to write: 1 <> dx => dt a11 a12 x1 Φ 2 2 (a) 2 ¼ ; (b) ¼ x1 þ x2 þ 2x1x2 :> dx ;> a21 a22 x2 dt Answer: (a) x, t ¼ aijxj; (b) Φ ¼ xixj. 1.9 Tests of the Tensorial Characteristics of a Variety 69

∂ ðÞaijxixj 1.2 Let aij constant 8i, j, calculate ) where aij ¼ aji. ∂xk ∂ Answer: ðÞ2aikxi 2a ∂x‘ ¼ ik i j k 1 2 n 1.3 If aijkx x x ¼ 0 8x , x , ÁÁÁ, x and aijk are constant values, show that akji þ ajki þ aikj þ aijk þ akij þ ajik ¼ 0. 1.4 Calculate for i, j ¼ 1, 2, 3: (a) δijAi,(b)δijAij,(c)δii,(d)δijδji,(e)δijδjkδk‘, (f) C ¼ aijkaijk δ δ δ i δ δijij jiji Answer:(a) ijAi ¼ Aj,(b) ijAi ¼ Aii ¼ Ajj,(c) i ¼ 3, (d) ij ji ¼ 3, δ δijij δjkjk k‘k‘δ (e) ij jk k‘¼ i‘, and (f) 64. 1.5 Calculate the Jacobian of the linear transformations between the coordinate systems (a) x1 ¼ x1; x2 ¼ x1x2; x3 ¼ x1x2x3 ; (b) x1 ¼ x1 cos x2 sin x3; x2 ¼ 1 2 3; 3 1 3 x sin x sin x xÀÁ¼ x cos x . ÀÁ 1 2 2 1 2 3 Answer: (a) J ¼ x x2; (b) J ¼À3 x sin x . 100 4 5 i 1.6 Given the tensor Tk‘ ¼ 021in the coordinate system X , calculate the 013 i components of this tensor in the coordinate system X , with the relations between the coordinates of these systems given by 1 1 3 2 1 2 3 x ¼ x þ x , x2 ¼ x þ x3, x ¼ x . 322 4 5 Answer: Tij ¼ 221 2142 3 11 5 1.7 Given the tensor Tij ¼ 4 12À1 5 in the coordinate system Xi, calculate 5 À13 i the components of this tensor in the coordinate system X , with the relations between the coordinates of these systems given by x1 ¼ x1 þ 2x2, x2 ¼ 3x3, x ¼ x3. 2 3 25 8 2 ij Answer: T ¼ 4 84105 2103 1.8 Show that (a) trðÞ¼T trðÞS , where T and S are, respectively, a symmetric and an antisymmetric tensor, both of the second order; (b) Tijk‘ ¼ 0, being Tijk‘ one symmetric tensor in the indexes i, j and antisymmetric in the indexes j, ‘. 1.9 Decompose the second-order tensor in two tensors, one symmetric and another antisymmetric 70 1 Review of Fundamental Topics About Tensors 2 3 À12 0 4 5 Tij ¼ 30À2 101 2 3 2 3 À12:50:5 0 À0:5 À0:5 Answer: 4 2:50À1 5 4 0:50 À1 5. 0:5 À11 0:51 0 1.10 Consider the tensor Tij that satisfies the tensorial equation mTij þ nTji ¼ 0, where m > 0 and n > 0 are scalars. Prove that if Tij is a symmetric tensor, then m ¼Àn, and m ¼ n if this is an antisymmetric tensor. 1.11 Let the Cartesian coordinate system with basis vectors e1, e2, e3. Calculate the metric tensor of the space with basis vectors g1 ¼ e1, g2 ¼ e1 þ e2, and g3 ¼ e1 þ e2 þ2e3. 3 111 4 5 Answer: gij ¼ 122 123 i 1.12 Let the basis vectors e1, e2 of the coordinate system X with metric tensor gij and the basis vectors e1 ¼ 3e1 þ e2 and e2 ¼Àe1 þ 2e2 of the coordinate ei e system X . Calculate the covariant components of the metric tensor gij in terms of the components of gij. e e e Answer: g11 ¼ 9g11 þ 6g12 þ g22 ; g12 ¼ g21 ¼À3g11 þ 5g12 þ 2g22 ; e g22 ¼ g11 À 4g12 þ 4g22. 1.13 Calculate the contravariant components of the vector u ¼ g1 þ 2g2 þ g3, where the covariant base vectors are g1 ¼ e1, g2 ¼ e1 þ e2, g3 ¼ e3, being e1, e2, e3 base vectors. Answer: u ¼ 4g1 þ 7g2 þ 8g3. ; ; 1.14 Let the contravariant base vectors g1 ¼ e1 g2 ¼ e1 þ e2 and g3 ¼ e1 þ e2 þ e3, where e1, e2, e3 are the vectors of the one orthonormal base. Calculate the: (a) Vectors g1, g2, g3 of the contravariant base (b) Metric tensor8 and the conjugated tensor.2 3 2 3 1 < g ¼ e1 À e2 111 2 À10 2 4 5 ij 4 5 Answer: (a) : g ¼ e2 À e3 ; (b) gij ¼ 122 g ¼ À122 3 g ¼ e3 123 0 À11 1.15 Consider the coordinate system x1 ¼ x1 cos x2, x2 ¼ x1 sin x2, x3 ¼ x3. Calcu- late the arc length along the parametric curve x1 ¼ a cos t, x2 ¼ a sin t, x3 ¼ b t in the intervalp 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t c, being a, b, c positive constants. Answer: L ¼ c a2 þ b2 1.16 Calculate the angle between the vectors (a) uðÞ2; À3; 1 , vðÞ3; À1; À2 ; (b) uðÞ2; 1; À5 , vðÞ5; 0; 2 . Answer: (a) 60o; (b) 90o 1.17 With i, j, k ¼ 1, 2, 3 calculate the following expressions: (a) uivjδji À vkuiδki; (b) δijδji; (c) eijkuiujuk 1.9 Tests of the Tensorial Characteristics of a Variety 71

Answer: (a) zero, (b) 3, (c) zero 1.18 Show that the followings expressions are invariants (a) Tijuivj; (b) Tii; (c) det Tij. ij ij ij ji ij ij 1.19 Let the vector (a) ui show that if A uiuj ¼ÀÁB uiuj, then A þ A ¼ B þ B ; ij ij ji (b) ui and if A uiuj is invariant, show that A þ A is a tensor. 1.20 Let Tpqrs an absolute tensor, show that if Tijk‘ þ Tij‘k ¼ 0 in the coordinate i i system X , then Tijk‘ þ Tij‘k ¼ 0 in another coordinate system X . 1.21 Let the vector u ¼ g1 þ 2g2 þ g3, having g1 ¼ e1, g2 ¼ e1 þ e2, and g3 ¼ e1 þ e2 þ e3, where e1, e2, e3 are orthonormal vectors, calculate their contravariant physical components. pffiffiffi pffiffiffi Answer: u*1 ¼ 1, u*2 ¼ 2 2, u*3 ¼ 3 1.22 Calculate the value of the permutation symbol e321546. Answer: e321546 ¼ 1. 1.23 Show that 8 < 1 rst δrst ‘ (a) eijkejki ¼ 6; (b) eijkujuk ¼ 0; (c) er‘me ¼ r‘m ¼ : 0 r, , m, s, t ¼ 1, À1 2, 3; rst δ sδ t δ tδ s ij ! ijk‘ ! ijk‘ÁÁÁ ! (d) er‘me ¼ ‘ m À ‘ m; (e) e eij ¼ 2 ; (f) e eijk‘ ¼ 4 ; (g) e eijk‘ÁÁÁ ¼ N where N is the index number. ‘ 1.24 For i, j, k, ¼ 1, 2, 3 show that δ i δ i δ i i i r s t ij δ δ‘ j j ijk ij ijk (a) δ ¼ k ¼ δ iδ À δ iδ , (b) δ ¼ δ j δ j δ j ¼ δ , (c) δ ¼ 6. k‘ δ j δ j k ‘ ‘ k rst r s t k‘ ijk k ‘ δ k δ k δ k r s t 1.25 Calculate the determinant by means of the expansion of the permutation symbol

1101

À10 11 ¼ eijk‘a1ia2ja3ka4‘ 01À11

1 À110

Answer: À3. ÀÁ 1.26 Show that (a) εijk ¼ ej Â ek Á ei where ei, ej, ek are unit vectors of one i i ijk coordinate system; (b) if ui and u are associate tensors and u ¼ ε ujvk, j k ijk i j k then ui ¼ εijku v ; (c) ε uivjwk ¼ εijku v w . 1.27 Simplify the expression F ¼ εijkεpqrAipAjqAkr. 1.28 Verify if the expression εmnpεmij þ εmnjεmpi ¼ εmniεmpj is correct or false. Justify the answer. ij 1 εijk ‘ 1.29 Write the tensor components T ¼ 2 Ak‘ with i, j, k, ¼ 1, 2, 3, 4, and ij show that if ijk‘ is an even permutation for the pair of 1234, then T ¼ Ak‘. Chapter 2 Covariant, Absolute, and Contravariant Derivatives

2.1 Initial Notes

The curve represented by a function ϕ(xi) in a closed interval is continuous if this function is continuous in this interval. If the curve is parameterized, i.e., ϕ[xi(t)] being t2½a; b , then it will be continuous if xi(t) are continuous functions in this interval, and it will be smooth if it has continuous and non-null derivatives for a value of t2½a; b . The smooth curves do not intersect, i.e., the conditions xiðÞ¼a xiðÞb will only be satisfied if a ¼ b. This condition defines a curve that can be divided into differential elements, forming curve arcs. For the case in which the initial and final points coincide, expressed by condition a ¼ b, the curve is closed. The various differential elements obtained on the curve allow calculating its line integral. The curves can be smooth by part, i.e., they are composed of a finite number of smooth parts (arc elements), connected in their initial and final point. This kind of curve can intersect in one or more points, and if their extreme points coincide, it is called a closed curve. The differentiation condition of a function is associated to the concept of neighborhood and limit. The neighborhood of a point P(xi) is defined admitting that the very small radius ε, with which a sphere is traced, is centered on it. The interior of this sphere is this point’s neighborhood of radius ε. This definition is valid in the plane, changing the sphere for a circle, and is complemented admitting a set of points, which is called an open set. The points interior to the cube shown in Fig. 2.1 form an open set, for in each point P(xi) a sphere of radius ε can be drawn in its interior, which will be contained in the cube’s interior. If the cube’s edges are included, the result is a closed set.

© Springer International Publishing Switzerland 2016 73 E. de Souza Sa´nchez Filho, Tensor Calculus for Engineers and Physicists , DOI 10.1007/978-3-319-31520-1_2 74 2 Covariant, Absolute, and Contravariant Derivatives

Fig. 2.1 Neighborhood of a point

2.2 Cartesian Tensor Derivative

The tensors derivative study begins with the research of what happens when a scalar function is differentiated. The behavior of this kind of function leads to the study of more general cases, such as those of the vectorial and tensorial functions. Consider a scalar function ϕ(xi) deﬁned in a coordinate system Xi, which derivative with respect to the variable xi is given by

∂ϕðÞxi ¼ ϕ ¼ G ∂xi , i i

i and in another coordinate system X has as derivative

∂ϕðÞxi ∂ϕðÞxi ∂ϕðÞxi ∂xi ∂xi Gi ¼ i ¼ i ¼ i ) Gi ¼ i Gi ∂x ∂x ∂xi ∂x ∂x whereby

∂ϕðÞxi ϕ ¼ ϕ ,i ∂xi , i

∂ϕðÞxi that is the transformation law of vectors, so ∂xi is a vector and deﬁnes the gradient of the scalar function. In vectorial notation, it is graphed as u ¼ gradϕðÞxi . 2.2 Cartesian Tensor Derivative 75

Differentiating the scalar function again with respect to the variable xj results in ∂xk ∂xm ∂2ϕ ϕ ¼ ,ij ∂xj ∂xi ∂xm∂xk and for i ¼ j ∂x ∂x ∂2ϕ ϕ ¼ k m ,ii ∂xi ∂xi ∂xm∂xk

∂xk ∂xm δ ¼ km ∂xi ∂xi

∂2ϕ ∂2ϕ ϕ ¼ δ ¼ ,ii km ∂xm∂xk ∂xk∂xk

i then ϕ(x ),ii is a scalar.

2.2.1 Vectors

In the study of vectors, there are two distinct manners to carry out a derivative: the derivative of a vector and the derivative of a point. In this text, a few conceptual considerations are made before calculating the derivative of a vector. Consider the vectorial space EN in which the scalar variable t2ðÞa; b is deﬁned, where for each value of this variable limited in the open interval there is a vector u (t) embedded in this space which has a metric tensor. Let the vector u(t) deﬁned by a continuous function of the variable t, and that admits continuous derivatives, then by means of an elementary increment Δt this vector will have an elementary increment

ΔuðÞ¼t uðÞÀt þ Δt uðÞt and when Δt ! 0 a vector v will exist such that ΔuðÞt lim À v ! 0 Δt then the vector v is derived from the vector u(t) by means of the variation of the parameter t, whereby

du v ¼ dt 76 2 Covariant, Absolute, and Contravariant Derivatives

The rules of Differential Calculus are applicable to this kind of differentiation. The other method of calculating a vector by means of a derivative is associated to the concept of punctual space EN, with respect to which a variable t2ðÞa; b is associated in a univocal way to an arbitrary point P(t). Taking a ﬁxed point O contained in EN as reference origin, the vector rðÞ¼t OPðÞt is determined. Let the vector r(t) deﬁned by a continuous and derivable function, then the result drðÞt is the vector dt that does not depend on the origin O, but only on the point P(t). This statement can be demonstrated admitting a new arbitrary point O* for which the result is the vector

OP ¼ OO* þ O*P

Fixing the vector OO* and calculating the derivative of these vectors with respect to the variable t, the result is ÀÁÀÁ ÀÁ dðÞOP d OO* d O*P dðÞOP d O*P ¼ þ ) ¼ dt dt dt dt dt

drðÞt This equality proves the independence of the vector dt with respect to the arbitrated origin. This vector is derived from the point P(t), and in the case of the points deﬁning a smooth curve C contained in the space EN, continuous and differentiable, and will be tangent to the curve in each point for which this derivative was calculated. In the general case of the vector being a function of various scalar variables rðÞxi , i ¼ 1, 2, ..., N, the result is by means of the differentiation rules of Differ- ential Calculus

∂2rðÞt ∂2rðÞt ¼ ∂xi∂xk ∂xk∂xi For a vectorial function of scalar variables, the vector derivative is given by

∂rðÞt drðÞ¼t dxi ∂xi If the curve C is a function of the variables xi, and these depend on the variable t2ðÞa; b , i.e., xi ¼ xiðÞt , the curve is parameterized, then the hypothesis of Differ- ential Calculus is applicable, and the total differential of the vector r[xi(t)] is

drðÞt ∂rðÞt dxi ¼ dt ∂xi dt The rules applicable to the differentiation of vectors, and to the vectors obtained by means of differentiation of a point are the same. The concept of vector calculated by differentiation of one point can be extended to the study of tensors, where the points to be analyzed are contained in the tensorial space EN. 2.2 Cartesian Tensor Derivative 77

Exercise 2.1 Show that derivative of the vector r with constant direction maintains its direction invariable. Let

rðÞ¼t ψðÞt u where ψ(t) is a parameterized scalar function and u is a constant vector, thus

drðÞt 0 du 0 ¼ ψ ðÞt u þ ψðÞt ¼ ψ ðÞt u dt dt

drðÞt then r and dt have the same direction.

2.2.2 Cartesian Tensor of the Second Order

The tensorial functions deﬁned by Cartesian tensors are often found in applications of physics and the areas of engineering. Let, for instance, the derivative of this kind of tensorial function that is a particular case of the derivative of tensors expressed in curvilinear coordinate systems. For the analysis of this derivative a Cartesian tensor of the second order Tij is admitted, which components are functions of the coordinates xi. The tensor with this characteristic is a function of the point considered in the space E3. The transformation law of the tensors of the second order is given by

∂xp ∂xq T ¼ T ij ∂xi ∂xj pq

For the Cartesian coordinate systems the coefﬁcients ∂xp and ∂xq are constants, for ∂xi ∂xj they represent the variation rates for the linear transformations, so they do not depend on the point’s coordinates. The derivative of this expression is given by ∂T ∂xp ∂xq ∂T ∂xm ∂T ∂xp ∂xq ∂xm ∂T ij ¼ pq ) ij ¼ pq ∂x‘ ∂xi ∂xj ∂xm ∂x‘ ∂x‘ ∂xi ∂xj ∂x‘ ∂xm that is the transformation law of tensors of the third order, concluding that the derivative of tensor Tij increased the order of this tensor in one unit. This conclusion is general and applicable to any Cartesian tensor. This kind of derivative is not valid for the more general tensors. The concept of tensors derivative must, therefore, be generalized for tensors which components are given in curvilinear coordinate systems.

2 ∂ Tij Exercise 2.2 Show that if Tij is a Cartesian tensor of the second order, then ∂xk∂xm will be a tensor of the fourth order. 78 2 Covariant, Absolute, and Contravariant Derivatives

The tensor is a function of the coordinates Tij(x1,x2,x3), whereby the result is

2 ∂xi ∂ xi ¼ δij ) ¼ 0 ∂xj ∂xj∂xk and by transformation law

∂xi ∂xj Tpq ¼ Tij ∂xp ∂xq then 2 2 ∂ Tpq ∂ ∂xi ∂xj ∂xk ∂ ∂xm ∂ ∂xi ∂xj ¼ Tij ¼ Tij ∂xr∂xs ∂xr∂xs ∂xp ∂xq ∂xr ∂xk ∂xs ∂xm ∂xp ∂xq ∂x ∂x ∂x ∂x ∂ ∂T ¼ k m i j ij ∂xr ∂xs ∂xp ∂xq ∂xk ∂xm

In a mnemonic manner

∂xk ∂xm ∂xi ∂xj Tpq,rs ¼ Tij, mk ∂xr ∂xs ∂xp ∂xq that is the transformation law of tensors of the fourth order as Tij,mk ¼ Tij, km the 2 ∂ Tij result is that Tij,km ¼ is a tensor of the fourth order. ∂xk∂xm

2.3 Derivatives of the Basis Vectors

Consider the contravariant vector uk ¼ ukðÞxi deﬁned in terms of the parametric curve xi ¼ xiðÞt , expressed with respect to the Cartesian coordinate system Xi.By means of the transformation law of vectors, the result for the curvilinear coordinate i system X is

i ‘ ∂x u ¼ uk ∂xk and with the techniques of differentiation with respect to the parameter t results in

du‘ ∂xi duk ∂2xi dx‘ ¼ þ uk ð2:3:1Þ dt ∂xk dt ∂xk∂x‘ dt 2.3 Derivatives of the Basis Vectors 79

i Fig. 2.2 Cartesian X and X 3 i curvilinear X coordinates with basis vectors e and g , i i g respectively 3

g 1 P X 1 g X 3 2

2 e X 3

e 2 2 X e 1 X 1 that only represents a contravariant vector if, and only if, xi is a linear function of xk. The ﬁrst term on the right of this expression represents an ordinary differentiation of a vectorial function expressed in Cartesian coordinates, but the second term contains the derivatives curvilinear coordinates xi, relative to a coordinate system that varies as a function of the points of the space. The study of this term is carried out considering the Cartesian coordinate system i i X and the curvilinear coordinate system X , with unit vectors ei and gi, respectively (Fig. 2.2). Whereby deﬁning the position vector r of point P with respect to the coordinate system Xi by means of their contravariant components

i r ¼ x ei the differential total of this vector is given by

∂r dr ¼ dxi ð2:3:2Þ ∂xi

As the basis vectors ei do not depend on point P:

i dr ¼ dx ei so

∂r e ¼ ð2:3:3Þ i ∂xi 80 2 Covariant, Absolute, and Contravariant Derivatives

i With respect to the local coordinate system X the result is the differential total of the position vector r:

i dr ¼ r, i dx ð2:3:4Þ Whereby the base vectors of the curvilinear coordinate system results : : gi ¼ r, i ð2 3 5Þ that shows that the unit vectors gi are tangent to the curves that deﬁne the curvilinear coordinate system Xi, that varies for each point of the vectorial space E3, and as the unit vectors ei do not vary

∂r ∂xi ¼ ei ∂xk ∂xk

Comparing with expression (2.3.5)

∂xi g ¼ ei ð2:3:6Þ k ∂xk then

∂xj e ¼ g ð2:3:7Þ i ∂xi j

The covariant derivative of the base vector deﬁned by the expression (2.3.6)is given by

∂2xi g ‘ ¼ ei ð2:3:8Þ k, ∂xk∂x‘ and substituting expression (2.3.7) in this expression

∂xj ∂2xi g ‘ ¼ g k, ∂xi ∂xk∂x‘ j Deﬁning the variety

j 2 i j ∂x ∂ x Γ ‘ ¼ ð2:3:9Þ k ∂xi ∂xk∂x‘ with which the covariant derivatives of the basis vectors of the curvilinear coordinate system can be written as linear combination of the base vector gj:

Γ j : : gk, ‘ ¼ k‘gj ð2 3 10Þ 2.3 Derivatives of the Basis Vectors 81

2.3.1 Christoffel Symbols

The coefﬁcients determined by expression (2.3.9) can be expressed in terms of the derivatives of the metric tensor and its conjugated tensor. For the derivatives of Γ i the contravariant basis vectors considering another variety jm:

i Γ i m : : g,j ¼ jmg ð2 4 1Þ

Writing ÀÁ i δ i i i : : g Á gj ¼ j ¼ g, k Á gj þ gj , k Á g ¼ 0 ð2 4 2Þ , k , k and substituting the expressions (2.3.10) and (2.4.1) in expression (2.4.2)

Γ i k Γ i i : : jkg Á gj þ jkgj Á g ¼ 0 ð2 4 3Þ then

Γ i Γ i jk ¼À jk and with the expressions (2.3.9), (2.3.10), (2.4.1) and with the prior expression it follows that

∂ m ∂2 k Γ m x x ij ¼ ∂xk ∂2xi∂xj Γ m gi, j ¼ ij gm i Γ i m : : g, j ¼À jmg ð2 4 4Þ

The relation between the covariant and contravariant unit vectors is deﬁned by

j gi ¼ gijg and the derivative of this expression with respect to the coordinate xk is given by

j j gi, k ¼ gij,kg þ gijg, k

Replacing expressions (2.3.10) and (2.4.4) in this last expression

Γ m j Γ j m ik gm ¼ gij,kg À gij kmg and with the multiplying by gn 82 2 Covariant, Absolute, and Contravariant Derivatives

Γ m n j n Γ j m n ik gm Á g ¼ gij,kg Á g À gij kmg Á g Γ mδ n jn Γ j mn ik m ¼ gij, kg À gij kmg Γ n jn mnΓ j ik ¼ gij, kg À gijg km

The multiplying of this last expression by gnp provides

Γ n mn Γ j jn Γ n δ mΓ j δ j gnp ik þ gijg gnp km ¼ gij,kg gnp ) gnp ik þ gij p km ¼ gij,k p whereby

Γ n Γ j : : gnp ik þ gij kp ¼ gip, k ð2 4 5Þ and with the cyclic permutation of the free indexes i, p, k of expression (2.4.5)

Γ n Γ j : : gnk pi þ gpj ik ¼ gpk, i ð2 4 6Þ Γ n Γ j : : gni kp þ gkj pi ¼ gki, p ð2 4 7Þ

Multiplying expression (2.4.5)byÀ1/2 and expressions (2.4.6) and (2.4.7)by 1/2 and adding 1 1 1 À g Γ n þ g Γ j þ g Γ n þ g Γ j þ g Γ n þ g Γ j 2 np ik ij kp 2 nk pi pj ik 2 ni kp kj pi 1 ¼ g þ g À g 2 pk, i ki, p ip, k and with the change of the index n for the index j, and considering the symmetry of the metric tensor 1 g Γ j ¼ g þ g À g kj ip 2 pk,i ki,p ip,k

The term to the right of the expression shows the existence of coefficients that are functions only of the partial derivatives of the metric tensor that define the Christoffel symbol of first kind 1 1 ∂g ∂g ∂g ½¼p; k Γ ¼ g þ g À g ¼ pk þ ki À ip ð2:4:8Þ ip,k 2 pk, i ki, p ip, k 2 ∂xi ∂xp ∂xk

Multiplying expression (2.4.8)bygkm: 1 1 gkmg Γ j ¼ gkm g þ g À g ) δ mΓ j ¼ gkm g þ g À g kj ip 2 pk,i ki,p ip,k j ip 2 pk, i ki, p ip,k 2.3 Derivatives of the Basis Vectors 83 whereby m 1 ¼ Γ m ¼ gkm g þ g À g ip ip 2 pk,i ki,p ip,k 1 ∂g ∂g ∂g ¼ gkm pk þ ki À ip ð2:4:9Þ 2 ∂xi ∂xp ∂xk

The term to the right of this expression shows the existence of coefficients that depend on the partial derivatives of the metric tensor and the conjugate metric Γi tensor. The coefficients represented by k‘ given by expressions (2.3.9) and (2.4.9) define Christoffel symbol of second kind. Expression (2.4.9) is more convenient for calculating these coefficients than expression (2.3.9). Multiplying the terms of expression (2.3.10)bygi:

i Γ j i δ iΓ j g Á gk, ‘ ¼ k‘g Á gj ¼ j k‘ Γ i i : : k‘ ¼ g Á gk, ‘ ð2 4 10Þ

2.3.2 Relation Between the Christoffel Symbols

Expressions (2.4.8) and (2.4.9) relate the two Christoffel symbols, i.e.:

Γ m kmΓ : : ij ¼ g ij, k ð2 4 11Þ then the Christoffel symbol of second kind is the raising of the third index of the Christoffel symbol of ﬁrst kind. Expression (2.4.10) written in terms of the Christoffel symbol of ﬁrst kind is given by

Γ k kpΓ k ij ¼ g ij, p ¼ g Á gi, j and multiplying the members by gkp

kpΓ k Γ k Γ k gkpg ij, p ¼ gkpg Á gi, j ) ij, p ¼ gkpg Á gi, j ) ij, p ¼ gk Á gp Á g Á gi, j then

Γ : : ij, p ¼ gp Á gi, j ð2 4 12Þ 84 2 Covariant, Absolute, and Contravariant Derivatives

2.3.3 Symmetry

For the Christoffel symbol of ﬁrst kind 1 ∂g ∂g ∂g 1 ∂g ∂g ∂g Γ ¼ jk þ ik À ij Γ ¼ ik þ jk À ji ij,k 2 ∂xi ∂xj ∂xk ji,k 2 ∂xj ∂xi ∂xk and considering the symmetry of the metric tensor gik ¼ gki, gjk ¼ gkj, gij ¼ gji it results in

Γij,k ¼ Γji, k then the Christoffel symbol of ﬁrst kind is symmetrical with respect to the ﬁrst two indexes, and with considering this symmetry results to the Christoffel symbol of second kind

Γ m kmΓ kmΓ Γ m ij ¼ g ij, k ¼ g ji,k ¼ ji that is symmetrical in regard to a permutation of the lower indexes.

2.3.4 Cartesian Coordinate System

δ For the Cartesian coordinate systems the elements of the metric tensor are gij ¼ ij, whereby for p ¼ 1, 2, ..., N it results in

∂g ip ¼ 0 ∂xp

By means of the deﬁnition of the Christoffel symbol of ﬁrst kind 1 ∂g ∂g ∂g Γ ¼ jk þ ik À ij ¼ 0 ij,k 2 ∂xi ∂xj ∂xk

It is veriﬁed that the Christoffel symbol of second kind is cancelled, for

Γ p pkΓ ij ¼ g ij,k ¼ 0 then for the Cartesian, orthogonal, or oblique coordinate systems, all the terms of Γ Γp ij,k and ij are null. 2.3 Derivatives of the Basis Vectors 85

2.3.5 Notation ij The oldest notations for the Christoffel symbols are for the symbol of first k ij kind, and for the symbol of second kind. Improving this notations Levi-Civita k adopted the spelling [ij, k] and {ij, k}, which second symbol was later improved by k various authors to , where the indexes were placed in more adequate and ij logical positions. This notation is well adopted, using the representation [ij, k] for the Christoffel symbol of first kind. Hermann Weyl used the Greek letter Γ to denote these symbols, which positions Γ Γk of the indexes indicates the kind it represents: ij,k and ij. This symbology is known as the notation of the Princeton School. A few authors invert the position of Γ the indexes and write k,ij. k The argument for adopting the notations [ij, k] and is that the Princeton ij School notation leads to confusing these coefficients with a tensor. However, this argument does not make its adoption valid, for the use of brackets or keys could also lead to confusion with a matrix or column matrix. That is not the case. The use of Γ Γk the notations ij,k and ij, even not being universally accepted, has in its favor the economy of characters in a text with many expressions containing these symbols. Several authors do not use the comma for indicating the differentiation with respect to one of the indexes in the Christoffel symbol of first kind, and write Γijk.

2.3.6 Number of Different Terms

For the tensorial space EN where i, j ¼ 1, 2, ..., N, it is veriﬁed that the metric 2 tensor gij has N terms 2 3 g g ÁÁÁ g 6 11 12 1N 7 6 g21 g22 ÁÁÁ g2N 7 gij ¼ 4 ⋮⋮⋮⋮5

gN1 gN2 ÁÁÁ gNN NÂN ÀÁ 2 This matrix has N diagonal termsÀÁgii,so N À N terms remain in their sides. As g is symmetrical, the result is 1 N2 À N terms for i 6¼ j. The total of different ij 2 ÀÁ 2 1 2 NNðÞþ1 terms in the metric tensor is N þ 2 N À N ¼ 2 . For each kind of Christoffel symbol N derivatives of gij are calculated, so the NNðÞ2þ1 number of different terms for these coefﬁcients is given by 2 . 86 2 Covariant, Absolute, and Contravariant Derivatives

2.3.7 Transformation of the Christoffel Symbol of First Kind

i Let Γpq,r defined in the coordinate system X and Γij,k expressed in the coordinate i system X , then using the expression that defines the Christoffel symbol of first kind, ∂ gij and adopting the notation gij, k ¼ ∂xk it follows that ∂ ∂xp ∂xq g ¼ g ij,k ∂xk pq ∂xi ∂xj ð2:4:13Þ ∂g ∂xp ∂xq ∂2xp ∂xq ∂xp ∂2xq ¼ pq þ g þ g ∂xk ∂xi ∂xj pq ∂xk∂xi ∂xj pq ∂xi ∂xk∂xj

By the chain rule

∂g ∂g ∂xr ∂xr pq ¼ pq ¼ g ð2:4:14Þ ∂xk ∂xr ∂xk pq, r ∂xk ! ! ∂xp ∂xq ∂xr ∂2xr ∂xs ∂2xs ∂xr g ¼ g þ g þ g ð2:4:15Þ ij,k pq, r ∂xi ∂xj ∂xk pq ∂xk∂xi ∂xj pq ∂xk∂xj ∂xi and with cyclic permutation of the indexes in each term of the previous expression and with gqp ¼ gpq ! ! ∂xq ∂xr ∂xp ∂2xq ∂xp ∂2xp ∂xq g ¼ g þ g þ g ð2:4:16Þ jk, i qr, p ∂xj ∂xk ∂xi pq ∂xi∂xj ∂xk pq ∂xi∂xk ∂xj ! ! ∂xr ∂xp ∂xq ∂2xp ∂xq ∂2xp ∂xq g ¼ g þ g þ g ð2:4:17Þ ki, j rp, q ∂xk ∂xi ∂xj pq ∂xj∂xk ∂xi pq ∂xj∂xi ∂xk ∂ ∂ ∂ 1 gjk gki gij Γij,k ¼ þ À ð2:4:18Þ 2 ∂xi ∂xj ∂xk

By substitution

∂xp ∂xq ∂xr ∂2xp ∂xq Γij,k ¼ Γpq,r þ g ð2:4:19Þ ∂xi ∂xj ∂xk pq ∂xi∂xj ∂xk that is the transformation law of the Christoffel symbol of ﬁrst kind. The second term to the right of this expression shows that these coefﬁcients are not the components of a tensor. 2.3 Derivatives of the Basis Vectors 87

2.3.8 Transformation of the Christoffel Symbol of Second Kind

Writing the Christoffel symbol of second kind in terms of the components of a new i coordinate system X : j p i ∂x ∂x Γ ¼ gipΓ ¼ gqr Γ ð2:4:20Þ jk jk,p ∂xq ∂xr jk,p

As the Christoffel symbol of ﬁrst kind transforms by mean of expression (2.4.19)

∂x‘ ∂xm ∂xn ∂2x‘ ∂xm Γjk,p ¼ Γ‘m, n þ g‘ ð2:4:21Þ ∂xj ∂xk ∂xp m ∂xj∂xk ∂xp and substituting expression (2.4.21) in expression (2.4.20) it follows that ! ∂ ‘ ∂ m ∂ n ∂ i ∂ p ∂2 ‘ ∂ m ∂ i ∂ p i qr x x x x x qr x x x x Γ ¼ g Γ þ g g‘ jk rn ∂xj ∂xk ∂xp ∂xq ∂xr m ∂xj∂xk ∂xp ∂xq ∂xr ! ∂ ‘ ∂ m ∂ i ∂2 ‘ ∂ i i qr n x x x qr m x x Γ ¼ g Γ‘m,nδ þ g g‘ δ jk r ∂xj ∂xk ∂xq m r ∂xj∂xk ∂xq

∂ ‘ ∂ m ∂ i ∂2 ‘ ∂ i i qr n x x x qr m x x Γ ¼ g Γ‘m,nδ þ g g‘ δ jk r ∂xj ∂xk ∂xq m r ∂xj∂xk ∂xq ∂ ‘ ∂ m ∂ i ∂2 ‘ ∂ i i qr n x x x qr x x Γ ¼ g Γ‘m,nδ þ g g‘ jk r ∂xj ∂xk ∂xq r ∂xj∂xk ∂xq qrΓ Γ q g ‘m,r ¼ ‘m qr δ q g g‘r ¼ ‘ ‘ m i 2 ‘ i i ∂x ∂x ∂x ∂ x ∂x Γ ¼ Γ q þ jk ‘m ∂xj ∂xk ∂xq ∂xj∂xk ∂xq

Replacing the dummy indexes ‘ ! q, q ! p, m ! r results in

q r i 2 q i i ∂x ∂x ∂x ∂ x ∂x Γ ¼ Γ p þ ð2:4:22Þ jk qr ∂xj ∂xk ∂xp ∂xj∂xk ∂xp

Expression (2.4.22) is the transformation law of the Christoffel symbol of second kind. The second term to the right of this expression shows that these coefﬁcients are not the components of tensor. The Christoffel symbols do not depend only on the coordinate system, but depend also on the rate with which this coordinate system varies in each point of the space. This variation rate is not present in the transformation law of tensors. 88 2 Covariant, Absolute, and Contravariant Derivatives

2.3.9 Linear Transformations

Consider the transformation of coordinates between two coordinate systems given by linear relation

j j i j x ¼ ai x þ b

j j where ai and b are constants, and with the techniques of successive differentiation

∂xj ∂2xj ¼ a j ) ¼ 0 ∂xi i ∂xi∂xk then for this kind of transformation of coordinates the Christoffel symbols transform as tensors.

2.3.10 Orthogonal Coordinate Systems

In the orthogonal coordinate systems, the tensorial space EN is deﬁned by the metric tensor gij 6¼ 0 for i ¼ j and gij ¼ 0 for i 6¼ j. 2 Putting hi ¼ gii, where gii does not indicate the summation of the terms, with the Christoffel symbol of ﬁrst kind 1 ∂g ∂g ∂g Γ ¼ jk þ ik þ ij ij,k 2 ∂xi ∂xj ∂xk and with the components of the tensor gij given by ( ‘ 1 ! i ¼ j g k ¼ 0 ! i 6¼ j 1 g ¼ ij gij it results for the relation between the Christoffel symbols ( gkkΓ ! ‘ ¼ k Γ ‘ ‘kΓ ij, k ij ¼ g ij,k ¼ 0 !! ‘ 6¼ k

Varying the indexes: – i ¼ j ¼ k 2.3 Derivatives of the Basis Vectors 89 1 ∂g ∂g ∂g 1 ∂g ∂h Γ ¼ ii þ ii À ii ¼ ii ¼ h i ii, i 2 ∂xj ∂xj ∂xj 2 ∂xj i ∂xj ÀÁpffiffiffiffiffi 1 ∂g 1 ∂ðÞ‘ng ∂ ‘n g 1 ∂h Γ k ¼ Γ i ¼ giiΓ ¼ ii ¼ ii ¼ ii ¼ i ij ii ii, i ∂ j ∂ j ∂ j ∂ j 2gii x 2 x x hi x

– i ¼ j 6¼ k 1 ∂g ∂g ∂g Γ ¼ ik þ ik À ii ii,k 2 ∂xi ∂xi ∂xk

as i 6¼ k it implies by deﬁnition of the metric tensor that gik ¼ 0, so

1 ∂g ∂h Γ ¼À ii ¼Àh i ii,k 2 ∂xk i ∂xk 1 ∂g h ∂h Γ k ¼ Γ k ¼ gkkΓ ¼À ii ¼À i i ij ii ii, k ∂ k 2 ∂ k 2gkk x ðÞhk x

– i ¼ k 6¼ j 1 ∂g ∂g ∂g Γ ¼ ji þ ii þ ij ij, i 2 ∂xi ∂xj ∂xi

and in an analogous way to the previous case where gji ¼ gij ¼ 0, so

1 ∂g ∂h Γ ¼ ii ¼ h i ij,i 2 ∂xj i ∂xj ÀÁpffiffiffiffiffi 1 ∂g ∂ ‘n g 1 ∂h Γ k ¼ Γ i ¼ giiΓ ¼ ii ¼ ii ¼ i ij ij ij, i ∂ j ∂ j ∂ j 2gii x x hi x

– for i 6¼ j, j 6¼ k, i 6¼ k it results in Γij,k ¼ 0, for by the deﬁnition of the metric ij Γ k tensor it implies gij ¼ g ¼ 0ifi 6¼ j, whereby ij ¼ 0.

2.3.11 Contraction

The tensorial expressions at times contain Christoffel symbols. However, the calculation of their components can be avoided, for an expression can be obtained that relates the derivative of the natural logarithm of the metric tensor with these symbols. 90 2 Covariant, Absolute, and Contravariant Derivatives

Let the Christoffel symbol of second kind 1 ∂g ∂g ∂g Γ j ¼ gmj km þ mi À ik ik 2 ∂xi ∂xk ∂xm and with contraction of the indexes j and k 1 ∂g ∂g ∂g Γ j ¼ gmj jm þ mi À ij ij 2 ∂xi ∂xj ∂xm

The symmetry of the metric tensor provides

∂g ∂g ∂g gmj mi ¼ gjm ji ¼ gmj ij ∂xj ∂xm ∂xm where the second equality was obtained by means of indexes interchanging the m $ j. ∂g Substituting gmj mi in the expression of the contracted Christoffel symbol ∂xj

1 ∂g Γ j ¼ gjm jm ij 2 ∂xi The conjugate metric tensor can be written as

Gjm gjm ¼ ) g ¼ Gjmg g jm jm being G the cofactor of this matrix and g ¼ detgjm, it follows that ÀÁpffiffiffi 1 ∂g 1 ∂g 1 ∂ðÞ‘ng ∂ ‘n g Γ j ¼ Gjm jm ¼ ¼ ¼ ij 2g ∂xi 2g ∂xi 2 ∂xi ∂xi and with the contracted form of the Christoffel symbol of second kind ÀÁpffiffiffi 1 ∂ g Γ j ¼ pffiffiffi ð2:4:23Þ ij g ∂xi that is of great use in manipulations of tensorial expressions, for it reduces the algebrism in calculating the Christoffel symbol. < For g ¼jgijj 0 the analysis is analogous, having only to change the sign of the determinant in the expression shown in the previous demonstration ÀÁpffiffiffiffiffiffiffi ÀÁpffiffiffiffiffiffiffi ∂ ‘n Àg 1 ∂ Àg Γ i ¼ ¼ pffiffiffiffiffiffiffi ð2:4:24Þ ip ∂xp Àg ∂xp 2.3 Derivatives of the Basis Vectors 91

2.3.12 Christoffel Relations

Consider the transformation of the Christoffel symbol of second kind from one j coordinate system Xi to another coordinate system X , whereby rewriting expression (2.4.22)

r i j r 2 j r ∂x ∂x ∂x ∂x ∂ x Γ ¼ Γ m þ pq ij ∂xm ∂xp ∂xq ∂xj ∂xq∂xp

∂xs and multiplying by ∂xr it follows that

s s r i j s r 2 j ∂x r ∂x ∂x ∂x ∂x ∂x ∂x ∂ x Γ ¼ Γ m þ ∂xr pq ij ∂xr ∂xm ∂xp ∂xq ∂xr ∂xj ∂xq∂xp

∂xs ∂xr ∂xs ∂xr ¼ δ s ¼ δ s ∂xr ∂xm m ∂xr ∂xj j

s i j 2 j s i j 2 s ∂x r ∂x ∂x ∂ x ∂x r ∂x ∂x ∂ x Γ ¼ Γ mδ s þ δ s ) Γ ¼ Γ s þ ∂xr pq ij m ∂xp ∂xq j ∂xq∂xp ∂xr pq ij ∂xp ∂xq ∂xq∂xp

2 s s i j ∂ x ∂x r ∂x ∂x ¼ Γ À Γ s ð2:4:25Þ ∂xq∂xp ∂xr pq ij ∂xp ∂xq

Expression (2.4.25) shows that the second derivative of the coordinate xs can be decomposed into terms with the ﬁrst derivatives of this coordinate and the coordinates xi, xj, and with the Christoffel symbols of second kind. This important expression was deducted in 1869 by Elwin Bruno Christoffel. Let an inverse transformation for the Christoffel symbol of second kind of the j coordinate system X to another referential system Xi given by

p q j j 2 r r ∂x ∂x ∂x ∂x ∂ x Γ j ¼ Γ þ ‘k pq ∂x‘ ∂xk ∂xr ∂xr ∂x‘∂xk

∂xm and multiplying both members by ∂xj and proceeding in a manner that is analogous to the previous one

2 m m p q ∂ x ∂x m ∂x ∂x ¼ Γ j À Γ ð2:4:26Þ ∂x‘∂xk ∂xj ‘k pq ∂x‘ ∂xk

The transformation of the Christoffel symbols from one coordinate system Xi to j another coordinate system X , and from this one to a third coordinates system Xek is identical to the transformation from Xi directly to Xek, so the transitive property is valid for the transformations of the Christoffel symbols. This shows that these symbols form a group. 92 2 Covariant, Absolute, and Contravariant Derivatives

The Christoffel relation given by expression (2.4.25) can be written as

2 k s s k r ∂ x ∂x s ∂x ∂x ∂x ¼ Γ À Γ p ∂xj∂xm ∂xk jm ∂xp ∂xj ∂xm kr and contracting the terms in the indexes s and m

2 k m m k r ∂ x ∂x m ∂x ∂x ∂x ¼ Γ À Γ p ∂xj∂xm ∂xk jm ∂xp ∂xj ∂xm kr 2 k m k ∂ x ∂x m ∂x ¼ Γ À δ r Γ p ∂xj∂xm ∂xk jm p ∂xj kr k 2 k m m ∂x ∂ x ∂x Γ ¼ Γ r þ jm ∂xj kr ∂xj∂xm ∂xk that is the transformation law of the contracted Christoffel symbol of second kind.

2.3.13 Ricci Identity

Another usual expression in Tensor Calculus is obtained by means of deﬁning the Christoffel symbol of ﬁrst kind 1 ∂g ∂g ∂g 1 ∂g ∂g ∂g Γ ¼ ik þ jk À ij Γ ¼ ij þ kj À ik ji,k 2 ∂xj ∂xi ∂xk ki,j 2 ∂xk ∂xi ∂xj

The sum of these two expressions provides the Ricci identity

∂g jk ¼ Γ þ Γ ð2:4:27Þ ∂xi ji, k ki,j

In an analogous way, subtracting the second expression from the ﬁrst expression of the Christoffel symbol of ﬁrst kind provides

∂g ∂g ij À jk ¼ Γ À Γ ð2:4:28Þ ∂xk ∂xi kj, i ij,k

Expressions (2.4.27) and (2.4.28) are very useful in manipulations of tensorial equations. ij Exercise 2.3 If T and gik are the components of a symmetric tensor and the metric ∂ jkΓ 1 jk gjk tensor, respectively, show that T ij,k ¼ 2 T ∂xi . 2.3 Derivatives of the Basis Vectors 93

With the Ricci identity

∂g ik ¼ Γ þ Γ ∂xi ji, k ki,j 1 ∂g 1 ÀÁ1 ÀÁ Tjk jk ¼ Tjk Γ þ Γ ¼ TjkΓ þ TjkΓ 2 ∂xi 2 ji,k ki,j 2 ji, k ki,j

Interchanging the indexes j $ k in the last term to the right of the expression and considering the tensor’s symmetry then

1 ∂g 1 ÀÁ1 ÀÁ1 Tjk jk ¼ Tjk Γ þ Γ ¼ TjkΓ þ TkjΓ ¼ Â 2TjkΓ 2 ∂xi 2 ji,k ki,j 2 ji, k ji,k 2 ji,k

1 ∂g Tjk jk ¼ TjkΓ Q:E:D: 2 ∂xi ji,k

2.3.14 Fundamental Relations

The derivative of the metric tensor with respect to an arbitrary variable can be placed in terms of Christoffel symbols of second kind and the metric tensor, thus from the deﬁnition of this symbol

Γ p pjΓ Γ p piΓ ik ¼ g ik, j jk ¼ g jk,i

Multiplying these two expressions by gpj and gpi, respectively:

Γ p pjΓ Γ p δ Γ gpj ik ¼ gpjg ik,j ) gpk ik ¼ pj ik,j Γ p piΓ Γ p δ Γ gpi jk ¼ gpig jk,i ) gpi jk ¼ pi jk,i whereby

Γ Γ p Γ Γ p ik, j ¼ gpj ik jk,i ¼ gip jk Adding these two expressions and considering the Ricci identity

∂g ik ¼ Γ þ Γ ð2:4:29Þ ∂xi ji, k ki,j and as gip ¼ gpi it results in

∂g ij ¼ g Γ p þ g Γ p ð2:4:30Þ ∂xk pj ik ip jk 94 2 Covariant, Absolute, and Contravariant Derivatives

With analogous analysis this derivative can be placed in terms of Christoffel symbols of the second kind and the conjugate metric tensor, and with

ij δ i g gkj ¼ k the derivative is

∂gij ∂g ∂gij ∂g g þ gij kj ¼ 0 ) g ¼Àgij kj ∂xp kj ∂xp ∂xp kj ∂xp

Multiplying both members of this last expression by gkq:

∂gij ∂g ∂gij ∂g ∂giq ∂g gkqg ¼Àgkqgij kj ) δ q ¼Àgkqgij kj ) ¼Àgijgkq kj kj ∂xp ∂xp j ∂xp ∂xp ∂xp ∂xp and with the Ricci identity

∂g kj ¼ Γ þ Γ ∂xp kp, j jp,k that substituted in the previous expression provides

∂giq ÀÁ ¼Àgijgkq Γ þ Γ ¼ÀgijgkqΓ À gijgkqΓ ∂xp kp,j jp, k kp, j jp,k

For the Christoffel symbol of second kind the result is the following relations

kq ijΓ kqΓ i ij kqΓ ijΓ q g g kp,j ¼ g kp g g jp, k ¼ g jp whereby

∂giq ¼ÀgkqΓ i À gijΓ q ∂xp kp jp Γ q Γ q As jp ¼ pj, the result is

∂giq ¼ÀgijΓ q À gkqΓ i ð2:4:31Þ ∂xp pj kp

Expressions (2.4.30) and (2.4.31) are well used in the development of tensorial expressions. Γ Γk Exercise 2.4 Calculate the Christoffel symbols ijk and ij for the polar coordinates systems, which metric tensor is given by 2.3 Derivatives of the Basis Vectors 95 10 g ¼ 2 ij 0 ðÞx1

The Christoffel symbol of ﬁrst kind is given by 1 ∂g ∂g ∂g Γ ¼ jk þ ik À ij ij,k 2 ∂xi ∂xj ∂xk so

g11 ¼ 1 ) g11,1 ¼ 0, g11, 2 ¼ 0

1 2 1 g22 ¼ ðÞx ) g22,1 ¼ 2x , g22,2 ¼ 0

g12,1 ¼ g12, 2 ¼ g21,1 ¼ g21, 2 ¼ g22,2 ¼ 0

It follows that

Γ11, 1 ¼ Γ11,2 ¼ Γ12, 1 ¼ Γ21, 1 ¼ Γ22,2 ¼ 0 1 Γ12, 2 ¼ Γ21,2 ¼ x 1 Γ22, 1 ¼Àx

In matrix form the result is 00 0 x1 Γ ¼ Γ ¼ ij,1 0 Àx1 ij,2 x1 0

For the Christoffel symbol of second kind it follows that 2 3 hi À1 10 ij 4 1 5 gij ¼ g ¼ 0 ðÞx1 2 Γ k k2Γ 12 ¼ g 12,2 Γ k ¼ gk1Γ ( 22 22,1 Γ1 12Γ 12 ¼ g 12, 2 ¼ 0 k ¼ 1 ) Γ1 11 1 1 22 ¼ g ðÞ¼ÀÀx x

In matrix form the result is 2 3 1 0 00 6 1 7 Γ1 ¼ Γ2 ¼ 4 x 5 ij 0 Àx1 ij 1 0 x1 96 2 Covariant, Absolute, and Contravariant Derivatives

Exercise 2.5 Calculate the Christoffel symbols for the cylindrical coordinates system, deﬁned by r x1 , θ x2 ,z x3, where À1 r 1,0 θ 2π , À1 z 1, which metric tensor and its conjugated tensor are given, respectively, by 2 3 2 3 100 100 6 7 4 5 ij 6 1 7 g ¼ 0 r2 0 g ¼ 4 0 0 5 ij r2 001 001 Using the expressions deducted for the orthogonal coordinate systems: – i ¼ j ¼ k

1 ∂g Γ ¼ ii ii,i 2 ∂xi 1 1 1 Γ ¼ g ¼ 0 Γ ¼ g ¼ 0 Γ ¼ g ¼ 0 11,1 2 11, 1 22,2 2 22,2 33, 3 2 33,3

– i ¼ j 6¼ k

1 ∂g Γ ¼À ii ii, k 2 ∂xk 1 1 Γ ¼À g ¼ 0 Γ ¼À g ¼ 0 11,2 2 11,2 11,3 2 11,3 1 1 Γ ¼À g ¼Àr Γ ¼À g ¼ 0 22,1 2 22,1 22,3 2 22, 3 1 1 Γ ¼À g ¼ 0 Γ ¼À g ¼ 0 33,1 2 33,1 33,2 2 33,2

– i ¼ k 6¼ j

1 ∂g Γ ¼ ii ij,i 2 ∂xj 1 1 Γ ¼ g ¼ 0 Γ ¼ g ¼ 0 12, 1 2 11,2 13,1 2 11, 3 1 1 Γ ¼ g ¼ r Γ ¼ g ¼ 0 21,2 2 22, 1 23, 2 2 22,3 1 1 Γ ¼ g ¼ 0 Γ ¼ g ¼ 0 31,3 2 33, 1 ij,i 2 33,2

– i 6¼ j, j 6¼ k, i 6¼ k all the Christoffel symbols are null. Putting these symbols in matrix form, the result is 2.3 Derivatives of the Basis Vectors 97 2 3 2 3 000 0 r 0 4 5 4 5 Γij, 1 ¼ 0 Àr 0 Γij,2 ¼ r 00 Γij,3 ¼ ½0 000 000 For the Christoffel symbols of second kind it follows that

Γ p pkΓ ij ¼ g ij, k Γ p p1Γ Γ p p2Γ 22 ¼ g 22, 1 12 ¼ g 12, 2

– p ¼ 1

Γ1 11Γ Γ1 12Γ 22 ¼ g 22, 1 ¼ r 12 ¼ g 12,2 ¼ 0

– p ¼ 2

1 Γ2 ¼ g21Γ ¼ 0 Γ2 ¼ g22Γ ¼ 22 22,1 12 12, 2 r

– p ¼ 3

Γ3 31Γ Γ3 32Γ 22 ¼ g 22,1 ¼ 0 13 ¼ g 12, 2 ¼ 0

Putting these symbols in matrix form, the result is 2 3 2 3 1 6 0 0 7 000 6 r 7 1 6 7 2 6 7 3 Γ ¼ 4 0 Àr 0 5 Γ ¼ 6 1 7 Γ ¼ ½0 ij ij 4 005 ij 000 r 000

Exercise 2.6 Calculate the Christoffel symbols for the spherical coordinates system r x1, φ x2, θ x3, where À1 r 1,0 φ π,0 θ 2π, which metric tensor and its conjugated tensor are given, respectively, by 2 3 2 3 10 0 6 7 10 0 6 1 7 4 2 5 ij 6 0 2 0 7 gij ¼ 0 r 0 g ¼ 6 r 7 00r2 sin 2φ 4 1 5 00 r2 sin 2φ

Using the expressions deduced for the orthogonal coordinate systems the result is: – i ¼ j ¼ k 98 2 Covariant, Absolute, and Contravariant Derivatives

1 ∂g Γ ¼ ii ii,i 2 ∂xi 1 1 1 Γ ¼ g ¼ 0 Γ ¼ g ¼ 0 Γ ¼ g ¼ 0 11,1 2 11, 1 22,2 2 22,2 33, 3 2 33,3

– i ¼ j 6¼ k

1 ∂g Γ ¼À ii ii, k 2 ∂xk 1 1 Γ ¼À g ¼ 0 Γ ¼À g ¼ 0 11,2 2 11,2 11,3 2 11,3 1 1 Γ ¼À g ¼Àr Γ ¼À g ¼ 0 22,1 2 22,1 22,3 2 22, 3 1 1 Γ ¼À g ¼Àr sin 2φΓ ¼À g ¼Àr2 sin φ cos φ 33,1 2 33, 1 33, 2 2 33,2

– i ¼ k 6¼ j

1 ∂g Γ ¼ ii ij,i 2 ∂xj 1 1 Γ ¼ g ¼ 0 Γ ¼ g ¼ 0 12, 1 2 11,2 13,1 2 11, 3 1 1 Γ ¼ g ¼ r Γ ¼ g ¼ 0 21,2 2 22, 1 23, 2 2 22,3 1 1 Γ ¼ g ¼ r sin 2φΓ ¼ g ¼ r2 sin φ cos φ 31,3 2 33, 1 ij,i 2 33,2

– i 6¼ j, j 6¼ k, i 6¼ k all the Christoffel symbols are null. Putting these symbols in matrix form, the result is 2 3 2 3 00 0 0 r 0 4 5 4 5 Γij, 1 ¼ 0 Àr 0 Γij,2 ¼ r 00 00r sin 2φ 00Àr2 sin φ cos φ 2 3 00r sin 2φ 4 2 5 Γij, 3 ¼ 00r sin φ cos φ r sin 2φ r2 sin φ cos φ 0

For the Christoffel symbols of second kind it follows that

Γ p pkΓ ij ¼ g j,k

– p ¼ k ¼ 1 2.3 Derivatives of the Basis Vectors 99

Γ1 11Γ ij ¼ g ij, 1 Γ1 11Γ Γ1 11Γ 2φ 22 ¼ g 22,1 ¼Àr 33 ¼ g 33, 1 ¼Àr sin

– p ¼ k ¼ 2

Γ2 22Γ ij ¼ g ij, 2 1 Γ2 ¼ g22Γ ¼ Γ2 ¼ g22Γ ¼Àsin φ cos φ 12 12,2 r 33 33,2

– p ¼ k ¼ 3

Γ3 33Γ ij ¼ g ij, 3 1 Γ3 ¼ g33Γ ¼ Γ3 ¼ g33Γ ¼ cot φ 13 13, 3 r 23 23,3

Putting these symbols in matrix form, the result is 2 3 2 3 1 6 0 0 7 00 0 6 r 7 1 6 7 2 6 7 Γ ¼ 4 0 Àr 0 5 Γ ¼ 6 1 7 ij ij 4 005 00Àr sin 2φ r 0 0 sin φ cos φ 2 3 1 6 00 7 6 r 7 Γ3 6 φ 7 ij ¼ 4 0 0 cot 5 1 cot φ 0 r

ijk ijkΓ p ijkΓ p ijk Exercise 2.7 For the antisymmetric tensor A , show that A ij ¼ A jk ¼ A Γ p ik ¼ 0. The symmetry of the Christoffel symbol of second kind allows writing

ijkΓ p ijkΓ p A ij ¼ A ji ¼ 0 and replacing the indexes i ! j it follows that

ijkΓ p ijkΓ p A ij ¼ÀA ij the result is

ijkΓ p A ij ¼ 0 100 2 Covariant, Absolute, and Contravariant Derivatives

ijkΓp ijkΓp Proceeding in an analogous way for A jk and A ik the equalities of what was enunciated are veriﬁed.

Γ i Γ i δ i δ i Exercise 2.8 Given the expression jk ¼ jk þ j uk þ kuj, where ui is a covariant ij jkΓ i vector and A is an antisymmetric tensor, show that A jk ¼ 0. The symmetry of the Christoffel symbol of second kind allows writing

Γ i Γ i jk ¼ kj Γ i Γ i δ i δ i jk ¼ kj þ kuj þ j uk jkΓ i kjΓ i A jk ¼ A kj

jk and with the consideration of the anti-symmetry A ¼ÀAkj veriﬁes that

jkΓ i : : : A jk ¼ 0 Q E D

2.4 Covariant Derivative

The basic problem treated by the Tensorial Analysis is to research if the derivatives of tensors generate new tensors, which, in general, does not occur. For the case of Cartesian coordinate systems the variation rates of the tensors are expressed by partial derivatives. For instance, the variation rates of a vector’s components indicate the variation of this vector. However, for the curvilinear coordinate systems, the expressions for these variation rates are not expressed only by partial derivatives. Figure 2.3a shows this coordinate systems for the case in which the vector u has constant modulus and directions (Fig. 2.3a), but their components u1 vary. Figure 2.3b shows the behavior of the vectors u with constant modulus and different directions, whereby the three vectors are different, but their radial u1 and tangential u2 ¼ 0 components remain constant. This example indicates the need for

2 1 2 u X 1 X u u u u A A u B 1 B u u

C u C

θ θ

1 1 O X O X

Fig. 2.3 Polar coordinates: (a) vector u with constant modulus and direction and (b) vector u with constant modulus and variable direction 2.4 Covariant Derivative 101 researching the variation rates of vectors for the curvilinear coordinate systems, because the variation rates of their components do not represent the variation of these vectors. To exemplify this fact let the scalar function ϕ ¼Àmx, where m is a scalar, which generates the potential u ¼Àgradϕ, with Cartesian components u1 ¼ m, u2 ¼ 0. This scalar function in polar coordinates is deﬁned by ϕ ¼Àmr cos θ, which covariant components of its gradient are given by ∂ϕ ∂ϕ ¼Àm cos θ ¼ mr sin θ ∂r ∂θ * ∂ϕ θ * 1 ∂ϕ θ and its physical components areu1 ¼ ∂r ¼Àm cos andu2 ¼ r ∂θ ¼ m sin . These components are not constant. The interpretation of this variation is carried out admitting a polar coordinates point P(r; θ) being displaced to another point nearby 0 P ðÞr þ dr; θ þ dθ , so the covariant components of the vector u initially given by ∂ϕ ∂ϕ u ¼ ¼Àm cos θ u ¼ ¼ mr sin θ 1 ∂r 2 ∂θ stay for this new point

δu1 ¼Àm sin θdθδu2 ¼ m sin θdr þ mr cos θdθ The elemental variations of these new components are due to the change of coordinates, and not to the change of vector. This particular indicates the need of defining a kind of derivative that translates the vector’s variation in an invariant manner, and leads to the definition of the covariant derivative. The covariant derivative defines variation rate of parameters that are not dependent on the coordinate systems, and because of that it is of extreme importance in the expression of physical models, for it generates a new tensor. The denomination covariant derivative was adopted by Bruno Ricci-Curbastro when conceiving the Tensor Calculus. The term covariant denotes a kind of partial differentiation of tensors that generates new tensors with variance one order above the original tensors. The adjective covariant is used to indicate the tensorial characteristics of Γk the differentiation of tensors, in which the set of Christoffel symbols ij are the coefficients of connections of the tensorial space EN.

2.4.1 Contravariant Tensor

2.4.1.1 Contravariant Vector

Let the vector u deﬁned by its contravariant components uj:

j : : u ¼ u gj ð2 5 1Þ

j where the unit vectors gj ¼ gjðÞx of the curvilinear coordinate system are functions of the coordinates that deﬁne this referential system. 102 2 Covariant, Absolute, and Contravariant Derivatives

Differentiating the expression (2.5.1) with respect to an arbitrary coordinate xk results in

∂u ∂uj ∂g ¼ g þ uj j ∂xk ∂xk j ∂xk and using expression (2.4.1)

∂g j ¼ Γ mg ∂xk jk m then

∂u ∂uj ¼ g þ ujΓ mg ð2:5:2Þ ∂xk ∂xk j jk m

As j is a dummy index in the ﬁrst term to the right, it can be changed for the index m: ∂u ∂um ∂um ¼ g þ ujΓ mg ¼ þ ujΓ m g ∂xk ∂xk m jk m ∂xk jk m

This expression shows that the covariant derivative of a contravariant vector is given by the N2 functions

∂um ∂ um ¼ þ ujΓ m ð2:5:3Þ k ∂xk jk whereby

∂u ¼ ðÞ∂ um g ð2:5:4Þ ∂xk k m

For the Cartesian systems the Christoffel symbols are null, so the covariant ∂um derivative coincides with the partial derivative ∂xk . In expression (2.5.3) the result is the variation rate of the vector u along the axes ∂uj of the curvilinear coordinate system is given by ∂xk, and the variation of the unit ∂ gj vectors gj along the axes of this coordinate system is expressed by ∂xk. This physical Γm interpretation of the covariant derivative is associated to the Christoffel symbols jk that are the connection coefﬁcients of the tensorial space. Various notations are found in the literature for the term to the left of expression ∂ m m ∇ m m m m (2.5.3), the most usual being: ku ¼ Dku ¼ ku ¼ u jk ¼ u jk ¼ u ;k. 2.4 Covariant Derivative 103

m Expressions (2.5.1) and (2.5.4) are analogous, for ∂ku has the aspect of a vector. The transformation law of contravariant vectors is admitted to demonstrate that expression (2.5.4) is a tensor, thus

∂xi ui ¼ up ∂xp which differentiated with respect to the coordinate xj provides ! ∂ui ∂xi ∂up ∂xq ∂2xi ∂xq ¼ þ up ð2:5:5Þ ∂xj ∂xp ∂xq ∂xj ∂xq∂xp ∂xj and with expression (2.4.26)

2 i i ‘ m ∂ x ∂x i ∂x ∂x ¼ Γ m À Γ ð2:5:6Þ ∂xq∂xp pq ∂xn ‘m ∂xp ∂xq

Substituting expression (2.5.6) in expression (2.5.5) ∂ i ∂ i ∂ p ∂ q ∂ i ∂ q ∂ ‘ ∂ m ∂ q u x u x p n x x p i x x x ¼ þ u Γ À u Γ‘ ∂xj ∂xp ∂xq ∂xj pq ∂xn ∂xj m ∂xp ∂xq ∂xj

∂ i ∂ ‘ ∂ m ∂ q ∂ p ∂ i ∂ q ∂ i ∂ q u p i x x x u x x p n x x þ u Γ‘ ¼ þ u Γ ∂xj m ∂xp ∂xq ∂xj ∂xq ∂xp ∂xj pq ∂xn ∂xj

The dummy index p in the ﬁrst term to the right can be changed by the index n:

∂ i ∂ ‘ ∂ m ∂ q ∂ n ∂ i ∂ q ∂ i ∂ q u p i x x x u x x p n x x þ u Γ‘ ¼ þ u Γ ∂xj m ∂xp ∂xq ∂xj ∂xq ∂xn ∂xj pq ∂xn ∂xj and with expression

∂xm ¼ δ m ∂xj j results in ∂ i ∂ ‘ ∂ n ∂ i ∂ q u p i x m u p n x x þ u Γ‘ δ ¼ þ u Γ ∂xj m ∂xp j ∂xq pq ∂xn ∂xj

With the transformation law of contravariant vectors

‘ ‘ ∂x u ¼ up ∂xp 104 2 Covariant, Absolute, and Contravariant Derivatives the above expression becomes ∂ i ∂ n ∂ i ∂ q u ‘ i u p n x x þ u Γ‘ ¼ þ u Γ ð2:5:7Þ ∂xj j ∂xq pq ∂xn ∂xj

It is veriﬁed that in expression (2.5.7) the variety in parenthesis transforms as a mixed second-order tensor, then the covariant derivative of a contravariant vector is a mixed second-order tensor, i.e., of variance (1, 1).

2.4.2 Contravariant Tensor of the Second-Order

The transformation law of contravariant tensors of the second-order is given by

p q pq ∂x ∂x T ¼ Tij ∂xi ∂xj

The derivative of this expression with respect to coordinate x‘ is given by

pq ∂T ∂Tij ∂xk ∂xp ∂xq ∂2xp ∂xk ∂xq ∂xp ∂2xq ∂xk ¼ þ Tij þ Tij ∂x‘ ∂xk ∂x‘ ∂xi ∂xj ∂xk∂xi ∂x‘ ∂xj ∂xi ∂xk∂xj ∂x‘ and with expression (2.4.26)

2 p p ‘ m 2 q q ‘ m ∂ x ∂x p ∂x ∂x ∂ x ∂x q ∂x ∂x ¼ Γ r À Γ ¼ Γ r À Γ ∂xk∂xi ki ∂xr ‘m ∂xi ∂xk ∂xk∂xj kj ∂xr ‘m ∂xj ∂xk then ∂ pq ∂ ij ∂ k ∂ p ∂ q ∂ p ∂ ‘ ∂ m ∂ k ∂ q T T x x x ij Γ r x Γ p x x x x ‘ ¼ ‘ þ T ki À ‘m ‘ ∂x ∂xk ∂x ∂xi ∂xj ∂xr ∂xi ∂xk ∂x ∂xj ∂ q ∂ ‘ ∂ m ∂ k ∂ p ij r x q x x x x þ T Γ À Γ‘ kj ∂xr m ∂xj ∂xk ∂x‘ ∂xi

∂ pq ∂ ij ∂ k ∂ p ∂ q ∂ p ∂ k ∂ q ∂ ‘ ∂ m ∂ k ∂ q T T x x x ij r x x x ij p x x x x ¼ þ T Γ À T Γ‘ ∂x‘ ∂xk ∂x‘ ∂xi ∂xj ki ∂xr ∂x‘ ∂xj m ∂xi ∂xk ∂x‘ ∂xj ∂ q ∂ k ∂ p ∂ ‘ ∂ m ∂ k ∂ p ij r x x x ij q x x x x þ T Γ À T Γ‘ kj ∂xr ∂x‘ ∂xi m ∂xj ∂xk ∂x‘ ∂xi

With

∂ m ∂ k m x x δ‘ ¼ ∂xk ∂x‘ 2.4 Covariant Derivative 105 it follows that

∂ pq ∂ ij ∂ k ∂ p ∂ q ∂ p ∂ k ∂ q ∂ ‘ ∂ q T T x x x ij r x x x ij p x x ¼ þ T Γ À T Γ‘ ∂x‘ ∂xk ∂x‘ ∂xi ∂xj ki ∂xr ∂x‘ ∂xj m ∂xi ∂xj ∂ q ∂ k ∂ p ∂ ‘ ∂ p ij r x x x ij q x x þ T Γ À T Γ‘ kj ∂xr ∂x‘ ∂xi m ∂xj ∂xi

In the second term to the right interchanging the indexes i $ r and, likewise, in the same fourth term with the permutation of the indexes j $ r, it results in

∂ pq ∂ ij ∂ k ∂ p ∂ q ∂ p ∂ k ∂ q ∂ ‘ ∂ q T T x x x rj x x x i ij x x p ¼ þ T Γ À T Γ‘ ∂x‘ ∂xk ∂x‘ ∂xi ∂xj ∂xi ∂x‘ ∂xj kr ∂xi ∂xj m ∂ q ∂ k ∂ p ∂ ‘ ∂ p ir x x x j ij x x q þ T Γ À T Γ‘ ∂xj ∂x‘ ∂xi kr ∂xj ∂xi m and with the expressions

‘ q ‘ p ‘q ∂x ∂x ‘p ∂x ∂x T ¼ Tij T ¼ Tij ∂xi ∂xj ∂xj ∂xi it follows that ∂ pq ∂ ij ∂ k ∂ p ∂ q T T rj i ir j x x x ‘q p ‘p q ¼ þ T Γ þ T Γ À T Γ‘ À T Γ‘ ∂x‘ ∂xk kr kr ∂x‘ ∂xi ∂xj m m ∂ pq ∂ ij ∂ k ∂ p ∂ q T ‘q p ‘p q T rj i ir j x x x þ T Γ‘ þ T Γ‘ ¼ þ T Γ þ T Γ ∂x‘ m m ∂xk kr kr ∂x‘ ∂xi ∂xj

∂ k ∂ p ∂ q pq ij x x x ∂‘T ¼ ∂kT ð2:5:8Þ ∂x‘ ∂xi ∂xj where

∂Tij ∂ Tij ¼ þ TrjΓ i þ TirΓ j ð2:5:9Þ k ∂xk kr kr is the covariant derivative of the contravariant tensor of the second order. Expression (2.5.8) indicates that the covariant derivative of a contravariant tensor of the second order is a mixed tensor of the third order, twice contravariant and once covariant, i.e., variance (2, 1). For the Cartesian coordinates the Christoffel symbols are null, so the covariant derivative coincides with the partial ∂Tij derivative ∂xk . 106 2 Covariant, Absolute, and Contravariant Derivatives

2.4.2.1 Contravariant Tensor of Order Above Two

To generalize expression (2.5.9) for tensors of order above two, i.e., for instance, the covariant derivative of the contravariant tensor of the third order, which expression may be developed by means of the following steps: (a) The basic structure of its expression is written considering the expression obtained for the covariant derivative of a contravariant tensor of the second order

∂Tijk ∂ Tijk ¼ þ TΓ þ TΓ þ TΓ p ∂xp

(b) The indexes of the Christoffel symbols corresponding to the coordinate with respect to which the differentiation is being carried out are placed

∂Tijk ∂ Tijk ¼ þ TΓ þ TΓ þ TΓ p ∂xp p p p

∂Tijk ∂ Tijk ¼ þ TΓ i þ TΓ j þ TΓ k p ∂xp p p p

(d) The dummy index q is placed on the Christoffel symbol and in sequential form in the tensors

∂Tijk ∂ Tijk ¼ þ TqΓ i þ TqΓ j þ TqΓ k p ∂xp qp qp qp

(e) The remaining indexes are placed in the same sequence in which they appear on the tensor that is being differentiated

∂Tijk ∂ Tijk ¼ þ TqjkΓ i þ TiqkΓ j þ TijqΓ k p ∂xp qp qp qp

This tensor generated by the differentiation of a variance tensor (3, 0) has a variance (3, 1). Expression (2.5.9) can be generalized by adopting this indexes placement systematic for a contravariant tensor of order p > 3, then the variance of this new tensor will always be ( p, 1). Exercise 2.9 Calculate the covariant derivative of the contravariant components of vector u expressed in polar coordinates. In Exercise 2.4 the Christoffel symbols were calculated for the polar coordinates, given by 2.4 Covariant Derivative 107 2 3 1 1 0 00 0 x 00 6 1 7 Γ ¼ Γ ¼ Γ1 ¼ Γ2 ¼ 4 x 5 ij,1 0 Àx1 ij,2 x1 0 ij 0 Àx1 ij 1 0 x1

The expression for the derivative of the contravariant components of vector u is:

∂um ∂ um ¼ þ ujΓ m k ∂xk jk

– m ¼ 1

∂u1 ∂ u1 ¼ þ ujΓ1 k ∂xk jk ∂u1 ∂u1 k ¼ 1 ) ∂ u1 ¼ þ ujΓ1 ) ∂ u1 ¼ þ u1Γ1 þ u2Γ1 1 ∂x1 j1 1 ∂x1 11 21 ∂u1 ∂u1 ∂ u1 ¼ þ 0 þ 0 ¼ 1 ∂x1 ∂x1 ∂u1 ∂u1 k ¼ 2 ) ∂ u1 ¼ þ ujΓ1 ) ∂ u1 ¼ þ u1Γ1 þ u2Γ1 2 ∂x2 j1 2 ∂x2 12 22 ∂u1 u1 ∂u1 u1 ∂ u1 ¼ þ þ 0 ¼ þ 2 ∂x2 x1 ∂x2 x1

– m ¼ 2

∂u2 ∂ u2 ¼ þ ujΓ2 k ∂xk jk ∂u2 ∂u2 k ¼ 1 ) ∂ u2 ¼ þ ujΓ2 ) ∂ u2 ¼ þ u1Γ2 þ u2Γ2 1 ∂x1 j1 1 ∂x1 11 21 ∂u2 u2 ∂u2 u2 ∂ u2 ¼ þ 0 þ ¼ þ 1 ∂x1 x1 ∂x1 x1 ∂u2 ∂u2 k ¼ 2 ) ∂ u2 ¼ þ ujΓ2 ) ∂ u2 ¼ þ u1Γ2 þ u2Γ2 2 ∂x2 j2 2 ∂x2 12 22 ∂u2 u1 ∂u2 u1 ∂ u2 ¼ þ þ 0 ¼ þ 2 ∂x2 x1 ∂x2 x1

pffiffi ∂ðÞTij g ∂ ij p1ffiffi jpΓ i Exercise 2.10 Show that jT ¼ g ∂xj þ T jp. The expression of the covariant derivative of a contravariant tensor of the second order is given by 108 2 Covariant, Absolute, and Contravariant Derivatives

∂Tij ∂ Tij ¼ þ TmjΓ i þ TimΓ j j ∂xk km mk and assuming k ¼ j

∂Tij ∂ Tij ¼ þ TmjΓ i þ TimΓ j j ∂xk jm mj

In the study of the contraction of the Christoffel symbol, it was verified that ÀÁpffiffiffi ∂ ‘n g Γ i ¼ mj ∂xr

Substituting this expression in the previous expression ÀÁpffiffiffi ∂Tij ∂ ‘n g ∂ Tij ¼ þ TmjΓ i þ Tim j ∂xk jm ∂xm

As m is a dummy index, it can be changed by the index j in the third term to the right ÀÁpffiffiffi ÀÁ ∂Tij ∂ ‘n g ∂Tij ∂ 1 ‘ng ∂ Tij ¼ þ TmjΓ i þ Tij ) ∂ Tij ¼ þ Tij 2 þ TmjΓ i j ∂xk jm ∂xj j ∂xk ∂xj jm pffiffiffi and multiplying and dividing the two terms between brackets by g ffiffiffi ∂ ij ∂ ij 1 p T ij 1 g mj i ∂jT ¼ pffiffiffi g þ T pffiffiffi þ T Γ g ∂xk 2 g ∂xj jm

Changing the indexes j ! p and m ! j in the last term ÀÁffiffiffi ffiffiffi ∂ ij ∂ ∂ ijp ij 1 p T ij 1 g jp i ij 1 T g jp i ∂jT ¼ pffiffiffi g þ T pffiffiffi þ T Γ ) ∂jT ¼ pffiffiffi þ T Γ g ∂xk 2 g ∂xj pj g ∂xj pj

By means of the symmetry of the Christoffel symbol it results ÀÁffiffiffi ∂ ijp ij 1 T g jp i ∂jT ¼ pffiffiffi þ T Γ Q:E:D: g ∂xj pj 2.4 Covariant Derivative 109

2.4.3 Covariant Tensor

2.4.3.1 Covariant Vector

Let the vector u deﬁned by their covariant components uj:

j u ¼ uig ð2:5:10Þ ÀÁ j j where g ¼ g xj are the basis vectors of the curvilinear coordinate system, which are functions of the coordinates that deﬁne this referential system. Differentiating the expression (2.5.10) with respect to an arbitrary coordinate xk:

∂u ∂u ∂gi ¼ i gi þ u ð2:5:11Þ ∂xk ∂xk i ∂xk and substituting expression (2.4.4)

∂gi ¼ÀΓ i g j ∂xk kj in expression (2.5.11) the result is

∂u ∂u ¼ i gi À uiΓ i g j ð2:5:12Þ ∂xk ∂xk kj

As i is a dummy index in the ﬁrst term to the right of expression (2.5.12), it can be changed by j: ∂u ∂u ¼ j À uiΓ i g j ð2:5:13Þ ∂xk ∂xk kj thus the covariant derivative of a covariant vector is given by the N2 functions

∂u ∂ u ¼ j À uiΓ i ð2:5:14Þ k j ∂xk kj whereby

∂u ÀÁ ¼ ∂ u gj ð2:5:15Þ ∂xk k j

For the Cartesian coordinate systems the Christoffel symbols are null, so in these referential systems the covariant derivative of a covariant vector coincides with the ∂uj partial derivative ∂xk. 110 2 Covariant, Absolute, and Contravariant Derivatives

Expression (2.5.15) has the aspect of a vector, and to demonstrate that this expression is a tensor let the transformation law of covariant vectors

∂xi u ¼ u p ∂xp i that differentiated with respect to the coordinate xq provides

∂u ∂u ∂xk ∂xi ∂2xi p ¼ i þ u ð2:5:16Þ ∂xp ∂xk ∂xq ∂xq i ∂xq∂xp

Expression (2.4.25) can be written as

2 i i j k ∂ x ∂x s ∂x ∂x ¼ Γ À Γ i ∂xq∂xp ∂xs pq jk ∂xp ∂xq and substituting this expression in expression (2.5.15) k i i j k ∂u ∂u ∂x ∂x ∂x s ∂x ∂x p ¼ i þ u Γ À Γ i ∂xq ∂xk ∂xq ∂xp i ∂xs pq jk ∂xp ∂xq i k i j k ∂u ∂x s ∂u ∂x ∂x ∂x ∂x p À u Γ ¼ i À u Γ i ∂xq i ∂xs pq ∂xk ∂xq ∂xp i ∂xp ∂xq jk

Replacing the indexes i ! ‘, j ! i in the second term to the right of the expression, and with

∂xi u ¼ u s i ∂xs this expression becomes i k ∂up s ∂ui ‘ ∂x ∂x À u Γ ¼ À u‘Γ ∂xq s pq ∂xk ik ∂xp ∂xq

Putting

∂u s ∂ u ¼ p À u Γ q p ∂xq s pq the result is

∂xi ∂xk ∂ u ¼ ðÞ∂ u ð2:5:17Þ q p k i ∂xp ∂xq 2.4 Covariant Derivative 111

Then the covariant derivative of a covariant vector is a covariant tensor of the second order, i.e., of variance (0, 2). Various notations are found in the literature for the covariant derivative. For the ∂ ∇ covariant vector, the most usual ones are: kum ¼ Dkum ¼ kum ¼ umkj ¼ um;k.

2.4.3.2 Covariant Tensor of the Second Order

The transformation law of covariant tensors of the second order is given by

∂xi ∂xj T ¼ T pq ij ∂xp ∂xq and differentiating with respect to the coordinate xr

∂T ∂2xi ∂xj ∂xi ∂2xj ∂xi ∂xj ∂T ∂xk pq ¼ T þ T þ ij ∂xr ∂xr∂xp ∂xq ij ∂xp ∂xr∂xq ij ∂xp ∂xq ∂xk ∂xr

2 i i ‘ m ∂ x ∂x s ∂x ∂x ¼ Γ À Γ i ∂xr∂xp ∂xs rp ‘m ∂xp ∂xr

2 j j ‘ m ∂ x ∂x s ∂x ∂x ¼ Γ À Γ j ∂xr∂xp ∂xs rq ‘m ∂xr ∂xq

Substituting these two expressions in the expression of the covariant derivative i ‘ m j i j ‘ m ∂T ∂x s ∂x ∂x ∂x ∂x ∂x s ∂x ∂x pq ¼ Γ À Γ i T þ Γ À Γ j T ∂xr ∂xs rp ‘m ∂xp ∂xr ∂xq ij ∂xp ∂xs rq ‘m ∂xr ∂xq ij ∂xi ∂xj ∂T ∂xk þ ij ∂xp ∂xq ∂xk ∂xr i j j ‘ m j i ∂T ∂x ∂x s ∂x ∂x ∂x ∂x ∂x s pq ¼ T Γ À T Γ i þ T Γ ∂xr ij ∂xs ∂xq rp ij ∂xq ∂xp ∂xr ‘m ij ∂xs ∂xp rq ∂xi ∂x‘ ∂xm ∂T ∂xi ∂xj ∂xk À T Γ j þ ij ij ∂xp ∂xr ∂xq ‘m ∂xk ∂xp ∂xq ∂xr

In the second term to the right replacing the dummy index m ! k, and interchanging the indexes i $ ‘, and in the fourth term replacing the indexes ‘ ! k and interchanging the indexes j $ m results in

i j i j k i j ∂Tpq ∂x ∂x s ∂x ∂x ∂x ‘ ∂x ∂x s ¼ T Γ À T‘ Γ þ T Γ ∂xr ij ∂xs ∂xq rp j ∂xp ∂xq ∂xr ik ij ∂xp ∂xs rq ∂xi ∂xj ∂xk ∂T ∂xi ∂xj ∂xk À T Γ m þ ij im ∂xp ∂xq ∂xr kj ∂xk ∂xp ∂xq ∂xr 112 2 Covariant, Absolute, and Contravariant Derivatives and with the transformation law of covariant tensors of the second order

∂xi ∂xj ∂xi ∂xj T ¼ T T ¼ T sq ij ∂xs ∂xq ps ij ∂xp ∂xs ∂ ∂ ∂ i ∂ j ∂ k Tpq s s Tij ‘ m x x x À T Γ À T Γ ¼ À T‘ Γ À T Γ ∂xr sq rp ps rq ∂xk j ik im kj ∂xp ∂xq ∂xr

Replacing the dummy indexes m ! ‘: i j k ∂Tpq s s ∂Tij ‘ ‘ ∂x ∂x ∂x À T Γ À T Γ ¼ À T‘ Γ À T ‘Γ ∂xr sq rp ps rq ∂xk j ik i kj ∂xp ∂xq ∂xr whereby

ÀÁ∂xi ∂xj ∂xk ∂ T ¼ ∂ T k pq k ij ∂xp ∂xq ∂xr therefore the covariant derivative of a covariant tensor of the second order is a covariant tensor of the third order, i.e., of variance (0, 3). Whereby the covariant derivative of a covariant tensor of the second order is given by

∂Tij ‘ ‘ ∂ T ¼ À T‘ Γ À T ‘Γ ð2:5:18Þ k ij ∂xk j ik i kj

For the Cartesian coordinates the Christoffel symbols are null, so in these referential systems the covariant derivative of the tensor Tij coincides with the ∂Tij partial derivative ∂xk .

2.4.3.3 Covariant Tensor of Order Above Two

To generalize expression (2.5.18) for tensors of order above two, i.e., for instance, the covariant derivative of the covariant tensor of the third order, which expression may be developed by means of the following steps: (a) The basic structure of its expression is written considering the expression obtained for the covariant derivative of a covariant tensor of the second order

∂T ∂ T ¼ ijk þ T Γ þ T Γ þ T Γ p ijk ∂xp

(b) The indexes of the Christoffel symbols corresponding to the coordinate with respect to which the differentiation is being carried out are placed

∂T ∂ T ¼ ijk þ T Γ þ T Γ þ T Γ p ijk ∂xp p p p 2.4 Covariant Derivative 113

(c) The covariant indexes of the Christoffel symbols must be completed obeying the sequence of the indexes of the tensor that is being differentiated

∂T ∂ T ¼ ijk þ T Γ þ T Γ þ T Γ p ijk ∂xp ip jp kp

(d) The dummy index q is placed on the Christoffel symbols and in sequential form in the tensors

∂T ∂ T ¼ ijk þ T Γ q þ T Γ q þ T Γ q p ijk ∂xp q ip q jp q kp

(e) The remaining indexes are placed in the same sequence in which they appear on the tensor that is being differentiated

∂T ∂ T ¼ ijk þ T Γ q þ T Γ q þ T Γ q p ijk ∂xp qjk ip iqk jp ijq kp This tensor generated by the differentiation of a variance tensor (0, 4). Expres- sion (2.5.18) can be generalized by adopting this indexes placement systematic for a covariant tensor of order q > 3, and the variance of this new tensor will always be ðÞ0, q þ 1 .

2.4.4 Mixed Tensor

Consider the transformation law of the mixed tensors of the second

m j m ∂x ∂x T ¼ T i n j ∂xi ∂xn that can be written as

i j m ∂x ∂x T ¼ T i n ∂xm j ∂xn which derivative with respect to coordinate xr is given by

m i 2 i ∂ i k j 2 j ∂T ∂x m ∂ x T ∂x ∂x ∂ x n þ T ¼ j þ T i ∂xr ∂xm n ∂xr∂xm ∂xk ∂xr ∂xn j ∂xr∂xn and with the following expressions

2 i i ‘ j 2 j j ‘ p ∂ x ∂x s ∂x ∂x ∂ x ∂x s ∂x ∂x ¼ Γ À Γ i ¼ Γ À Γ j ∂xr∂xm ∂xs rm ‘j ∂xm ∂xr ∂xr∂xm ∂xs mr ‘p ∂xn ∂xr 114 2 Covariant, Absolute, and Contravariant Derivatives this expression becomes m i i ‘ j ∂T ∂x m ∂x s ∂x ∂x n þ T Γ À Γ i ∂xr ∂xm n ∂xs rm ‘j ∂xm ∂xr ∂ i k j j ‘ p T ∂x ∂x ∂x s ∂x ∂x ¼ j þ T i Γ À Γ j ∂xk ∂xr ∂xn j ∂xs mr ‘p ∂xn ∂xr

m q i e m ∂x ∂x m ∂x ∂x T ¼ T p T i ¼ T n q ∂xp ∂xn j e ∂xm ∂xj it follows that

m i i m q ‘ j ∂T ∂x m ∂x s ∂x ∂x ∂x ∂x n þ T Γ À T p Γ i ∂xr ∂xm n ∂xs rm q ∂xp ∂xn ∂xm ∂xr ‘j ∂ i k j i e j ‘ p T ∂x ∂x m ∂x ∂x ∂x s ∂x ∂x ¼ j þ T Γ À T i Γ j ∂xk ∂xr ∂xn e ∂xm ∂xj ∂xs mr j ∂xn ∂xr ‘p

m i i q j ∂T ∂x m ∂x s ‘ ∂x ∂x n þ T Γ À T pδ Γ i ∂xr ∂xm n ∂xs rm q p ∂xn ∂xr ‘j ∂ i k j i ‘ p T ∂x ∂x m ∂x s ∂x ∂x ¼ j þ T δ eΓ À T i Γ j ∂xk ∂xr ∂xn s ∂xm s mr j ∂xn ∂xr ‘p

m i i q j ∂T ∂x m ∂x s ‘ ∂x ∂x n þ T Γ À T Γ i ∂xr ∂xm n ∂xs rm q ∂xn ∂xr pj ∂ i k j i ‘ p T ∂x ∂x m ∂x s ∂x ∂x ¼ j þ T Γ À T i Γ j ∂xk ∂xr ∂xn s ∂xm mr j ∂xn ∂xr ‘p

Interchanging the indexes in the second term on the left m $ s, in the last term on the right, interchanging the indexes j $ ‘ and replacing the indexes p ! k results in

m i i q j ∂ i k j j ∂T ∂x s ∂x m ‘ ∂x ∂x T ∂x ∂x m ∂x s n þ T Γ À T Γ i ¼ j þ T Γ ∂xr ∂xm n ∂xm rs q ∂xn ∂xr ‘j ∂xk ∂xr ∂xn s ∂xm mr

j k ∂x ∂x ‘ À T i Γ ‘ ∂xn ∂xr jk and replacing the indexes j ! k and q ! j in the last term on the left 2.4 Covariant Derivative 115

m i i j k ∂ i k j ∂T ∂x s ∂x m ‘ ∂x ∂x T ∂x ∂x n þ T Γ À T Γ i ¼ j ∂xr ∂xm n ∂xm rs j ∂xn ∂xr ‘k ∂xk ∂xr ∂xn

j j k m ∂x s ∂x ∂x ‘ þ T Γ À T i Γ s ∂xm mr ‘ ∂xn ∂xr jk that can be written as ! m i ∂ i j k ∂T s m m s ∂x T ‘ ‘ ∂x ∂x n þ T Γ À T Γ ¼ j þ T Γ i À T iΓ ∂xr n rs s mr ∂xm ∂xk j ‘k ‘ jk ∂xn ∂xr then ! m ∂ i m j k ∂T s m m s T ‘ ‘ ∂x ∂x ∂x n þ T Γ À T Γ ¼ j þ T Γ i À T iΓ ð2:5:19Þ ∂xr n rs s mr ∂xk j ‘k ‘ jk ∂xi ∂xn ∂xr

Putting

m m ∂T s m m s ∂ T ¼ n þ T Γ À T Γ ð2:5:20Þ r n ∂xr n rs s mr

∂ i T ‘ ‘ ∂ T i ¼ j þ T Γ i À T iΓ ð2:5:21Þ r j ∂xr j ‘k ‘ jk the result is the expressions that represent the covariant derivative of the mixed m i tensors of the second-order Tn and Tj, whereby

m j k m ∂x ∂x ∂x ∂ T ¼ ∂ T i ð2:5:22Þ r n r j ∂xi ∂xn ∂xr

Expression (2.5.22) shows that the derivative of a mixed tensor of the second order is a mixed tensor of the third order, once contravariant and twice covariant, i.e., of variance (1, 2). The covariant derivative of a mixed tensor of variance ( p, q) generates a variance tensor ðÞp, q þ 1 . To generalize expression (2.5.22) for mixed tensors of order above two, assume as an example the covariant derivatives of a mixed tensor of the third order of variance (1, 2) and of a mixed tensor of ﬁfth order of variance (3, 2), which are given, respectively, by the expressions

j ∂T ‘ ∂ T j ¼ p À T j Γ q À T j Γ q þ T q Γ j k p‘ ∂xk q‘ pk pq ‘k p‘ kq j‘m ‘ ∂T ‘ ‘ ‘ ‘ ‘ ∂ Tj m ¼ rs À Tj mΓ q À Tj mΓ q þ Tq mΓ j þ TjqmΓ þ Tj qΓ m k rs ∂xk qs rk rq sk rs kq rs kq rs kq 116 2 Covariant, Absolute, and Contravariant Derivatives

2.4.5 Covariant Derivative of the Addition, Subtraction, and Product of Tensors

Expression (2.5.21) shows that the covariant derivative of a mixed tensor comprises a partial derivative of this tensor and the terms containing Christoffel symbols, which are always linear in the components of the original tensor. This characteristic indicates that the covariant differentiation follows the same rules of the ordinary differentiation of Differential Calculus. To stress the properties of the covariant derivative let the scalar ϕ(xi) which ordinary derivative is equal to its covariant derivative, that can be written as the dot i product of the vectors u and vi expressed in Cartesian coordinates ÀÁ i i ϕ x ¼ u vi and differentiating

ÀÁ ÀÁduðÞiv dui dv ∂ ϕ xi ¼ ∂ uiv ¼ i ¼ v þ ui i k k i dxk dxk i dxk

As the covariant and ordinary derivatives are equal, it results in ÀÁ ÀÁ i i i ∂kϕ x ¼ ∂k u vi þ u ∂kðÞvi

Substituting the expressions of the covariant derivatives of contravariant and covariant vectors ÀÁ ∂ui ∂v ∂ui ∂v ∂ ϕ xi ¼ þ uiΓ i v þ ui i À v Γ i ¼ v þ ui i k ∂xk kj i ∂xk i kj ∂xk i ∂xk

This expression suggests that the covariant derivative of an inner product of tensors behaves in a manner that is similar to the ordinary derivative. To prove this assumption, let, for instance, the tensors Aij and Bij for which the following properties of the covariant derivative are admitted a priori as valid: ÀÁ ∂ ∂ ∂ (a) kÀÁAij þ Bij ¼ kAij þ kBij; ∂ ∂ ∂ (b) kÀÁAij À Bij ÀÁ¼ kAij À kBijÀÁ; (c) ∂k AijBij ¼ ∂kAij Bij þ Aij ∂kBij .

To demonstrate property (a) let the tensor Cij ¼ Aij þ Bij,so 2.4 Covariant Derivative 117

ÀÁ ∂Cij ‘ ‘ ∂ A þ B ¼ ∂ C ¼ À C‘ Γ À C ‘Γ k ij ij k ij ∂xk j ik i kj ÀÁ ÀÁ ∂ Aij þ Bij ‘ ‘ ¼ À A‘ þ B‘ Γ À ðÞA ‘ þ B ‘ Γ ∂xk j j ik i i kj ∂Aij ‘ ‘ ∂Bij ‘ ‘ ¼ À A‘ Γ À A ‘Γ þ À B‘ Γ À B ‘Γ ∂xk j ik i kj ∂xk j ik i kj

¼ ∂kAij þ ∂kBij

In an analogous way, it is possible to prove property (b), replacing only the addition sign for the subtraction sign in the previous demonstration. To demonstrate property (c) let the inner product AijB‘m ¼ Cij‘m that generates a covariant tensor of the fourth order

ÀÁ ∂ Cij‘m p p p p ∂ A B‘ ¼ ∂ C ‘ ¼ À C ‘ Γ À C ‘ Γ À C Γ À C ‘ Γ k ij m k ij m ∂xk pj m ki ip m kj ijpm k‘ ij p km

Substituting the expressions of the tensor of the fourth order in terms of the inner product ÀÁ ÀÁ∂ AijB‘m p p p p ∂ A B‘ ¼ À A B‘ Γ À A B‘ Γ À A B Γ À A B‘ Γ k ij m ∂xk pj m ki ip m k‘ij pm k‘ ij p km ∂ ∂ Aij p p B‘m p p ¼ À A Γ À A Γ B‘ þ A À B Γ À B‘ Γ ∂xk pj ki ip kj m ij ∂xk pm k‘ p km

The terms in parenthesis are the covariant derivatives of the covariant tensors of the second-order, whereby ÀÁÀÁ ∂k AijB‘m ¼ ∂kAij B‘m þ AijðÞ∂kB‘m thus the covariant derivative of an inner product of tensors follows the same rule as the derivative of the product of functions in Differential Calculus.

ij δi 2.4.6 Covariant Derivative of Tensors gij, g , j

Ricci’s Lemma

The metric tensor behaves as a constant when calculating the covariant derivative. 118 2 Covariant, Absolute, and Contravariant Derivatives

The covariant derivative of the metric tensor gij is calculated to demonstrate this lemma, thus

∂g ∂ g ¼ ij À g Γ p À g Γ p k ij ∂xk pj ik ip kj ∂g ÀÁ ∂ g ¼ ij À Γ þ Γ k ij ∂xk ik,j kj, i and with the Ricci identity

∂g ij ¼ Γ þ Γ ∂xk ik,j jk, i and by the symmetry Γjk,i ¼ Γkj,i

∂g ∂g ∂ g ¼ ij À ij ¼ 0 k ij ∂xk ∂xk

In an analogous way the conjugate metric tensor gij is given by

∂gij ∂ gij ¼ þ gpjΓ i þ gipΓ j ð2:5:23Þ k ∂xk kp kp

Since ∂ jp gijg ∂g ∂gjp g gjp ¼ δ p ) ¼ 0 ) ij gjp þ g ¼ 0 ij i ∂xk ∂xk ij ∂xk and multiplying by giq

∂g ∂gjp ∂g ∂gjp ∂gqp ∂g giqgjp ij þ giqg ¼ 0 ) giqgjp ij þ δ q ¼ 0 ) ¼Àgiqgjp ij ∂xk ij ∂xk ∂xk j ∂xk ∂xk ∂xk it follows that

∂gqp ÀÁ ¼Àgiqgjp Γ þ Γ ¼ÀgiqgjpΓ þÀgiqgjpΓ ∂xk ik,j jk, i ik,j jk, i iqΓ p iqΓ q ¼Àg ik À g jk

Replacing the indexes i ! p, q ! i, and p ! j:

∂gqp ¼ÀgpiΓ j À gpjΓ i ∂xk pk pk 2.4 Covariant Derivative 119 and substituting this expression in expression (2.5.23) ∂ ij piΓ j pjΓ i pjΓ i ipΓ j kg ¼Àg pk À g pk þ g kp þ g kp and with the symmetry of gij and the Christoffel symbol of second kind ∂ ij ipΓ j pjΓ i pjΓ i ipΓ j kg ¼Àg kp À g kp þ g kp þ g kp ¼ 0

Following the same systematic it implies for the covariant derivative of the Kronecker delta

∂δ i ∂ δ i ¼ j þ δ pΓ i À δ i Γ p ¼ 0 þ Γ i À Γ i ¼ 0 k j ∂xk j pk p jk jk jk

These deductions show that the conjugate metric tensor gij and the Kronecker δi delta j also behave as constants in calculating the covariant derivative. ∂ i im ∂ i Exercise 2.11 Show that kTj ¼ g kTj . Expressing the mixed tensor by

i im Tj ¼ g Tmj the result for its covariant derivative is ÀÁ ∂ i ∂ im i im ∂ i kTj ¼ kg Tj þ g kTj

im As ∂kg ¼ 0, it results in ∂ i im ∂ i : : : kTj ¼ g kTj Q E D ∂ ∂ui uj Exercise 2.12 Show that ∂xj À ∂xi is a covariant tensor of the second order, being ui a covariant vector. The covariant derivative of a covariant vector is given by

∂u ∂u ∂ u ¼ i À u Γ p ) i ¼ ∂ u þ u Γ p j i ∂xj p ij ∂xj j i p ij and replacing the indexes i ! j results in

∂u j ¼ ∂ u þ u Γ p ∂xi i j p ji 120 2 Covariant, Absolute, and Contravariant Derivatives

Carrying out the subtraction presented in the enunciation ∂u ∂u i À j ¼ ∂ u þ u Γ p À ∂ u þ u Γ p ∂xj ∂xi j i p ji j i p ij

Γ p Γ p and with the symmetry ij ¼ ji ∂u ∂u i À j ¼ ∂ u À ∂ u ∂xj ∂xi j i i j

As the covariant derivative of a covariant vector is a tensor of the second order, then this expression represents a tensor of variance (0, 2). ∂ ∂T ∂T Γ p 1 pq Tik jk ij Exercise 2.13 Show that ij ¼ 2 T ∂xj þ ∂xi À ∂xk , being Tij a symmetric tensor and detTij 6¼ 0, and with covariant derivative ∂kTij ¼ 0. The tensor Tpk can be written under the form

pk ip jk T ¼ g g Tij

For the tensor Tij the covariant derivative is given by

∂T ∂T ∂ T ¼ ij À T Γ p À T Γ p ¼ 0 ) ij ¼ T Γ p þ T Γ p k ij ∂xk pj ik ip jk ∂xk pj ik ip jk

Interchanging the indexes i, j, k cyclically

∂T ∂T jk ¼ T Γ p þ T Γ p ki ¼ T Γ p þ T Γ p ∂xi pk ji jp ki ∂xj pi kj kp ij and adding these two expressions and subtracting the one that comes before them, and considering the tensor’s symmetry ∂T ∂T ∂T jk þ ki À ij ¼ T Γ p þ T Γ p þ T Γ p þ T Γ p À T Γ p þ T Γ p ∂xi ∂xj ∂xk pk ji jp ki pi kj kp ij pj ik ip jk Γ p ¼ 2Tkp ij

The dummy index p can be changed by the index q, so ∂T ∂T ∂T 1 ∂T ∂T ∂T jk þ ki À ij ¼ 2T Γ q ) jk þ ki À ij ¼ T Γ q ∂xi ∂xj ∂xk kq ij 2 ∂xi ∂xj ∂xk kq ij and multiplying by Tpk 1 ∂T ∂T ∂T Tpq jk þ ki À ij ¼ TpkT Γ q 2 ∂xi ∂xj ∂xk kq ij 2.4 Covariant Derivative 121 and with the contraction

pk δ p T Tkq ¼ q it follows that 1 ∂T ∂T ∂T Tpq jk þ ki À ij ¼ δ pΓ q ¼ Γ p Q:E:D: 2 ∂xi ∂xj ∂xk q ij ij

2.4.7 Particularities of the Covariant Derivative

To exemplify a particularity of the covariant derivative let the vector u deﬁned by i its covariant components uj ¼ giju , then ∂ ∂ i ∂ i ∂ i kuj ¼ k giju ¼ kgij u þ gij ku and with Ricci’s lemma ∂ i ∂ i k giju ¼ gij ku

The covariant derivative of the contravariant vector is given by

i ∂u ‘ ∂ ui ¼ þ u Γ i k ∂xk ‘k so by substitution i ∂u ‘ ∂ g ui ¼ g þ u Γ i k ij ij ∂xk ‘k

The contravariant components of the vector can be expressed in terms of their covariant components ÀÁ ÀÁ i‘ i‘ ∂ ‘ ∂ ‘ i g u ‘ i g u ‘ ∂ g u ¼ g þ g u Γ ¼ g þ u Γ‘ k ij ij ∂xk ij ‘k ij ∂xk k,j ∂ i‘ ∂ g i‘ u‘ ‘ ¼ g u‘ þ g g þ u Γ‘ ij ∂xk ij ∂xk k, j

Rewriting expression (2.4.31)

i‘ ∂g ‘ ‘ ¼Àg mΓ i À gimΓ ∂xk mk mk 122 2 Covariant, Absolute, and Contravariant Derivatives which substituted in the previous expression provides

ÀÁ∂ i ‘m i im ‘ i‘ u‘ ‘ ∂ g u ¼ g Àg Γ À g Γ u‘ þ g g þ u Γ‘ k ij ij mk mk ij ∂xk k, j ÀÁ∂ ‘m i im ‘ ‘ u‘ ‘ ¼ g Àg u‘Γ À g u‘Γ þ δ þ u Γ‘ ij mk mk j ∂xk k,j ∂ m i m ‘ uj ‘ ¼Àg u Γ À δ u‘Γ þ þ u Γ‘ ij mk j mk ∂xk k,j ∂ m ‘ uj ‘ ¼ u Γ À u‘Γ þ þ u Γ‘ mk, j mk ∂xk k, j

Replacing the dummy indexes ‘ ! m:

∂ i m ‘ uj m i ∂ g u ¼Àu Γ À u‘Γ þ þ u Γ ) ∂ u ¼ ∂ g u k ij mk, j mk ∂xk mk, j k j k ij

∂uj ‘ ¼ À u‘Γ ∂xk mk then the covariant derivative of a covariant vector is equal to the covariant derivative of the product of the metric tensor by the contravariant components of this vector. This characteristic of the covariant derivative can be generalized for tensors of order above one, for instance, for a contravariant tensor of the second order the result is ∂ pq ∂ k gipgjqT ¼ kTij

Another particularity of the covariant derivative is its successive differentiation of a scalar function. Let a scalar function ϕ that represents an invariant, so its derivative with respect to its coordinate xi is a covariant vector given by

∂ϕ ϕ ¼ ¼ ∂ ϕ , i ∂xi i Taking the derivative of this function again, now with respect to the coordinate xj:

∂2ϕ ∂2ϕ ∂ϕ ϕ ¼ ¼ ∂ ðÞ¼∂ ϕ À Γ m , ij ∂xj∂xi j i ∂xj∂xi ∂xm ij

The dummy index m can be changed, and as the Christoffel symbol is symmetric, it results in ÀÁ ∂jðÞ¼∂iϕ ∂i ∂jϕ

Then the covariant derivative of an invariant is commutative. 2.5 Covariant Derivative of Relative Tensors 123

2.5 Covariant Derivative of Relative Tensors

The covariant derivative of relative tensors has characteristics that differ from the covariant derivative of absolute tensors. For studying the derivatives of these varieties in a progressive manner, a scalar density of weight W with respect to the i coordinate system X is admitted, given by JWϕðÞxi . Taking the derivative of this function ÀÁ ∂ JWϕ ∂ϕ ∂xk ∂J ¼ JW þ WJWÀ1 ϕ ð2:6:1Þ ∂xj ∂xk ∂xj ∂xj

The second parcel on the right shows that the gradient of a scalar density is not a vector. It is veriﬁed that for W ¼ 0 the result is a scalar function and

∂ϕ ∂ϕ ∂xk ¼ ∂xj ∂xk ∂xj is the transformation law of the vectors. Let the Jacobian cofactor

∂xk C m ¼ k ∂xm or

∂xr ∂xm C m ¼ Jδ r ) C m ¼ J ∂xj r r r ∂xk it follows that ∂J ∂ ∂xk ∂J ∂2xk ∂xm ¼ C m ) ¼ J ∂xj ∂xj ∂xm k ∂xj ∂xj∂xm ∂xk

The substitution of this expression in expression (2.6.1) provides ÀÁ ! ∂ JWϕ ∂ϕ ∂xk ∂2xk ∂xm ¼ JW þ W ϕ ð2:6:2Þ ∂xj ∂xk ∂xj ∂xj∂xm ∂xk that is the transformation law of the pseudoscalar JWϕ(xi). Using expression (2.4.25) the second term in parenthesis can be written as

2 k m m k q ∂ x ∂x m ∂x ∂x ∂x p ¼ Γ ‘ À Γ ∂xj∂xm ∂xk j ∂xp ∂xj ∂x‘ kq 124 2 Covariant, Absolute, and Contravariant Derivatives

The contraction in the indexes m and ‘ provides

2 k m ‘ k q ∂ x ∂x m ∂x ∂x ∂x p ¼ Γ ‘ À Γ ∂xj∂xm ∂xk j ∂xp ∂xj ∂x‘ kq and with

∂x‘ ∂xq δ q ¼ p ∂xp ∂x‘ the result is

2 k m k ∂ x ∂x m ∂x q ¼ Γ ‘ À Γ ∂xj∂xm ∂xk j ∂xj kq

The substitution of this expression in expression (2.6.2) provides ÀÁ ∂ Wϕ ∂ϕ ∂ k ∂ k J W x W m W x q ¼ J þ WJ Γ ‘ ϕ À WJ Γ ϕ ∂xj ∂xk ∂xj j ∂xj kq

Let a scalar density which transformation law is given by

ϕ ¼ JWϕ it results in ÀÁ ∂ Wϕ ∂ k ∂ϕ J m W x q À WΓ ‘ ϕ ¼ J À WΓ ϕ ∂xj j ∂xj ∂xk kq

The term in parenthesis to the right represents a covariant pseudovector of weight W. This expression shows that the covariant derivative of a scalar density presents an additional term in its expression, in which the factor multiplies the contracted Christoffel symbol. For W ¼ 0 this expression is reduced to the gradient expression of the scalar function ϕ(xi)

∂ϕ ∂ϕ ∂xk ¼ ∂xj ∂xk ∂xj

For a contravariant pseudovector of weight W it follows by means of this expression that is analogous to the one shown for a scalar density, the next expression

∂u j ∂ u j ¼ þ uqΓ j À Wu jΓ q ð2:6:3Þ k ∂xk kq kq 2.5 Covariant Derivative of Relative Tensors 125 and the contraction of the indexes j and k provides

∂u j ∂ uj ¼ þ uqΓ j À Wu jΓ q j ∂xj jq jq The dummy index j in the third term to the right can be changed by the index q:

∂u j ∂ uj ¼ þ ðÞ1 À W uqΓ q j ∂x j jq ∂ j ∂u j If the pseudovector has weightW ¼ 1this expression is simpliﬁed for ju ¼ ∂x j, which represents the divergence of vector u j. The generalization of expression (2.6.3) for a relative tensor of weight W and variance (1, 1) is given by

∂ i T ‘ ‘ ∂ T i ¼ j þ T Γ j À T iΓ À WT iΓ q ð2:6:4Þ r j ∂xr j ‘k ‘ jk j rq

iÁÁÁ For a relative tensor TjÁÁÁ of weight W and variance ( p, q) it results in

∂ iÁÁÁ T ‘ ‘ ∂ TiÁÁÁ ¼ jÁÁÁ þ T Γ j þ ÁÁÁÁÁÁÁÁÁÁÁÁ ÀT iΓ À ÁÁÁÁÁÁÁÁÁÁÁÁ ÀWT iΓ q r jÁÁÁ ∂xr j ‘k |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} ‘ jk |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} j rq terms relative to terms relative to the contravariance the covariance ð2:6:5Þ

By means of the considerations presented in the ﬁrst paragraph of item (2.5.4), iΓq and adding that the parcel WTj rq in expression (2.6.5) is linear in terms of the original tensor, it implies that the rules of ordinary differentiation of Differential Calculus are applicable to the covariant differentiation of relative tensors.

2.5.1 Covariant Derivative of the Ricci Pseudotensor

The covariant derivative of the Ricci pseudotensor in its contravariant form is given by eijk jk‘ ∂ pffiffi pk‘ jp‘ jkp jk‘ e g j e k e ‘ e ∂iε ¼ ∂i pffiffiffi ¼ þ Γ pffiffiffi þ Γ pffiffiffi þ Γ pffiffiffi g ∂xi ip g ip g ip g ejk‘ 1 1 ∂ pffiffi jk‘ ∂ pffiffi ∂ pffiffi jk‘ g 1 ∂e g g e ∂g ffiffiffi jk‘ jk‘ i ¼ p i þ e i ¼ e i ¼À 2 i ∂x g ∂x ∂x ∂x 2g3 ∂x 126 2 Covariant, Absolute, and Contravariant Derivatives

The contraction of the Christoffel symbol provides

∂g ¼ 2gΓ p ∂xi pi whereby ejk‘ ∂ pffiffi jk‘ g e ¼Àpffiffiffi Γ p ∂xi g pi

Substituting this expression in the expression of covariant derivative jk‘ jk‘ pk‘ jp‘ jkp jk‘ e e p j e k e ‘ e ∂iε ¼ ∂i pffiffiffi ¼Àpffiffiffi Γ þ Γ pffiffiffi þ Γ pffiffiffi þ Γ pffiffiffi g g pi ip g ip g ip g

The conditions for which this pseudotensor is non-null are that the first three indexes be different, i.e., j 6¼ k 6¼ ‘, then for j ¼ 1, k ¼ 2, ‘ ¼ 3: jk‘ 123 p23 1p3 12p jk‘ e e p j e k e ‘ e ∂iε ¼ ∂i pffiffiffi ¼Àpffiffiffi Γ þ Γ pffiffiffi þ Γ pffiffiffi þ Γ pffiffiffi g g pi ip g ip g ip g

With p ¼ 1, 2, 3: jk‘ 123 ÀÁ123 123 123 jk‘ e e 1 2 3 j e k e ‘ e ∂iε ¼ ∂i pffiffiffi ¼Àpffiffiffi Γ þ Γ þ Γ þ Γ pffiffiffi þ Γ pffiffiffi þ Γ pffiffiffi g g 1i 2i 3i i1 g i2 g i3 g and with the symmetry of the Christoffel symbol it results in jk‘ jk‘ e ∂iε ¼ ∂i pffiffiffi ¼ 0 g

With an analogous expression for the covariant form of the Ricci pseudotensor pffiffiffi εijk ¼ geijk it results for its covariant derivative ÀÁffiffiffi ÀÁp pffiffiffi ∂ geijk pffiffiffi pffiffiffi pffiffiffi ∂ ε ¼ ∂ ge ¼ À ge Γ p À ge Γ p À ge Γ p i ijk i ijk ∂xi pjk i‘ ipk j‘ ijp k‘

The partial derivative referent to the first term to the right is given by ÀÁpffiffiffi ÀÁpffiffiffi ÀÁ ∂ geijk ∂ g pffiffiffi ∂ eijk ¼ e þ g ∂x‘ ∂x‘ ijk ∂x‘ but ÀÁ ∂ eijk ¼ 0 ∂x‘ 2.5 Covariant Derivative of Relative Tensors 127 it results in ÀÁpffiffiffi ÀÁpffiffiffi ∂ geijk ∂ g ¼ e ∂x‘ ∂x‘ ijk

Expression (2.4.23) can be written as ÀÁpffiffiffi ÀÁpffiffiffi ∂ g pffiffiffi ∂ geijk pffiffiffi ¼ gΓ p ) ¼ ge Γ p ∂x‘ p‘ ∂x‘ ijk p‘

Substituting this expression in the expression of the covariant derivative ÀÁffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi ∂ ε ∂ p p Γ p p Γ p p Γ p p Γ p i ijk ¼ i geijk ¼ geijk p‘ À gepjk i‘ À geipk j‘ À geijp k‘

The conditions for which this pseudotensor is non-null are that the first three indexes be different, i.e., j 6¼ k 6¼ ‘, then for j ¼ 1, k ¼ 2, ‘ ¼ 3: ÀÁffiffiffi ffiffiffi ffiffiffi ffiffiffi ffiffiffi ∂ ε ∂ p p Γ p p Γ p p Γ p p Γ p i ijk ¼ i geijk ¼ ge123 p‘ À gep23 1‘ À ge1p3 2‘ À ge12p 3‘

With p ¼ 1, 2, 3: ÀÁpffiffiffi ∂iεijk ¼ ∂i geijk ffiffiffi ÀÁffiffiffi ffiffiffi ffiffiffi p Γ1 Γ2 Γ3 p Γ1 p Γ2 p Γ3 ¼ ge123 1‘ þ 2‘ þ 3‘ À ge123 1‘ À ge123 2‘ À ge123 3‘ whereby ÀÁpffiffiffi ∂iεijk ¼ ∂i geijk ¼ 0

These derivatives show that ÀÁÀÁ ÀÁ ∂ δijk ∂ εijkε ∂ ε ijk ε εijk∂ ε i pqr ¼ i pqr ¼ i pqr þ i pqr ¼ 0

ijk The covariant derivatives of the Ricci pseudotensors ε , εpqr and the generalized δijk Kronecker delta pqr being null, it implies that these varieties behave as constants in the calculation of the covariant derivative. As an example of an application of this characteristic, let the tensorial expression ijk ε ∂juk, which covariant derivative is given by ÀÁ ÀÁ ÀÁ ijk ijk ijk ∂i ε ∂juk ¼ ε ∂i ∂juk þ ∂juk∂i ∂jε but with

ijk ∂iε ¼ 0 this expression becomes 128 2 Covariant, Absolute, and Contravariant Derivatives ÀÁ ÀÁ ijk ijk ∂i ε ∂juk ¼ ε ∂i ∂juk

2.6 Intrinsic or Absolute Derivative

The absolute derivative of a variety is calculated when the coordinates xi vary as a function of time, i.e., xi ¼ xiðÞt . A covariant derivative of an invariant ϕ(xk) is given by ÀÁ ÀÁ ∂ϕ xk ∂ ϕ xk ¼ k ∂xk which is equal to its partial derivative. For the absolute derivative ÀÁ ÀÁ ÀÁ ÀÁ ÀÁ δϕ xk ∂ϕ xk ÀÁdxk ∂ϕ xk ∂ϕ xk dxk dϕ xk ¼ þ ∂ ϕ xk ¼ þ ¼ δt ∂t k dt ∂t ∂xk dt dt then this derivative is equal to its total derivative. For the vector u(xi) where xi varies as a function of time, which is expressed by means of its contravariant coordinates, or ÂÃÂÃ k i i u ¼ u x ðÞt , t gk x ðÞt

The derivative with respect to time is given by ÀÁ du dukg duk ∂g dxi ¼ k ¼ g þ uk k ð2:7:1Þ dt dt dt k ∂t dt and with

duk ∂uk ∂uk dxi ¼ þ ð2:7:2Þ dt ∂t ∂xi dt

The following expression (item 2.3)

∂g k ¼ g Γ m ∂xi m ki substituted in expression (2.7.1) provides

du duk dxi ¼ g þ ukΓ m g dt dt k ki dt m Replacing the indexes k ! m in the ﬁrst term to the right 2.6 Intrinsic or Absolute Derivative 129 du dum dxi ¼ þ ukΓ m g dt dt ki dt m thus the absolute derivative of a vector generates a vector. The covariant derivative of the contravariant vector is written as

δum dum dxi ¼ þ ukΓ m ð2:7:3Þ δt dt ki dt and substituting expression (2.7.2) in expression (2.7.3) δum ∂uk ∂uk dxi dxi δum ∂uk ∂uk dxi ¼ þ þ ukΓ m ) ¼ þ þ ukΓ m δt ∂t ∂xi dt ki dt δt ∂t ∂xi ki dt

The covariant derivative of the contravariant vector is given by

∂uk ∂ uk ¼ þ ukΓ m i ∂xi ki or in vectorial form

∂u ÀÁ ¼ ∂ uk g ∂xi i k whereby for the absolute derivative of vector u it results that

δum ∂uk dxi ¼ þ ∂ uk δt ∂t i dt or in vectorial form du δum ¼ g dt δt m

The vector u in terms of their covariant components is given by ÂÃÂÃ i k i u ¼ uk x ðÞt , t g x ðÞt and with an analogous analysis to the one shown for the contravariant vectors, and with

k Γ k m g, i ¼À img it results for the absolute derivative of vector u 130 2 Covariant, Absolute, and Contravariant Derivatives

δu ∂u dxi k ¼ m þ ∂ u δt ∂t i k dt where ∂iuk is the covariant derivative of the covariant vector. These expressions can be generalized for the tensors

δTij ∂Tij dxk ¼ þ ∂ Tij ð2:7:4Þ δt ∂t k dt

δT ∂T dxk ij ¼ ij þ ∂ T ð2:7:5Þ δt ∂t k ij dt

δTij ∂Tij dxk m ¼ m þ ∂ Tij ð2:7:6Þ δt ∂t k m dt

The differentiation rules of Differential Calculus are applicable to absolute differentiation, which can be proven, for instance, for two tensors Aij and Bij, which algebraic addition generates the tensors Cij ¼ Aij Æ Bij, and which product results in AijBij. Calculating the absolute derivative of this sum

δC dxk ÀÁdxk dxk dxk δA δB ij ¼ ∂ C ¼ ∂ A þ B ¼ ∂ A þ ∂ B ¼ ij þ ij δt k ij dt k ij ij dt k ij dt k ij dt δt δt

Calculating the absolute derivative of the product of the tensors ÀÁ ÀÁk ÀÁ k ÀÁk δ AijBij dx dx dx ¼ ∂ A B ¼ ∂ A B þ A ∂ B δt k ij ij dt k ij ij dt ij k ij dt dxk dxk δA δB ¼ ∂ A B þ A ∂ B ¼ ij B þ A ij k ij dt ij ij k ij dt δt ij ij δt

The absolute derivative of vector u calculated along the curve xi ¼ xiðÞt can be deﬁned by means of the inner product of its covariant derivative by the tangent dxi vector to this curve dt . For a tensor of order above the unit, and with an analogous way, the absolute derivative is the inner product of this tensor by the vector tangent to a curve, then

δTij dxk pqr ¼ ∂ Tij δt k pqr dt

This deﬁnition in conjunction with the considerations made in the ﬁrst paragraph of item 2.5.4 indicates that the absolute derivative follows the rules of Differential Calculus, such as shown for the addition and product of two tensors. The derivative of the metric tensor gij is given by 2.6 Intrinsic or Absolute Derivative 131

δg ∂g dxk ij ¼ ij þ ∂ g δt ∂t k ij dt ’ ∂ Ricci s lemma shows that kgij ¼ 0, then

δg ∂g ij ¼ ij δt ∂t

∂ gij As the metric tensor is independent of time it implies that ∂t ¼ 0, whereby it δ gij results that δ t ¼ 0, i.e., its absolute derivative is null. ij δi For the tensors g and j, which have the same characteristics of the metric tensor, developing an analysis analogous to the one shown for this tensor it results in

δgij ∂gij dxk ∂gij ¼ þ ∂ gij ¼ ¼ 0 δt ∂t k dt ∂t

δδ i ∂δ i dxk ∂δ i j ¼ j þ ∂ δ i ¼ j ¼ 0 δt ∂t k j dt ∂t

2.6.1 Uniqueness of the Absolute Derivative

The covariant derivative of a Cartesian tensor coincides with its partial derivative, then the absolute derivative of this variety, calculated along a curve xi ¼ xiðÞt , can be deﬁned by means of the scalar product of this derivative by the vector tangent to dxi this curve dt . For instance, for a Cartesian tensor of variance (2, 3) it results in

δTij ∂Tij dxk pqr ¼ pqr δt ∂t dt

As the partial derivative of a Cartesian tensor is unique, and the scalar product that deﬁnes the absolute derivative generates an invariant, it is possible to conclude that this derivative is also unique. This analysis can be generalized for arbitrary tensors. i j i j Exercise 2.14 Calculate the absolute derivative of: (a) giju v ; (b) giju u ; (c) vector δui ui knowing that δ t ¼ 0. i j (a) The expression giju v represents a scalar, and taking the derivative 132 2 Covariant, Absolute, and Contravariant Derivatives δ i j i j giju v dgiju v δ ¼ t dt δ i j δ giju v gij δðÞui δðÞvj δðÞui δðÞvj ¼ uivj þ g vj þ g ui ¼ g vj þ g ui δt δt ij δt ij δt ij δt ij δt

(b) The change of vector v j by vector u j in the expression calculated in the previous item provides δ i j giju u δðÞui δðÞuj ¼ g uj þ g ui δt ij δt ij δt

Interchanging the indexes i $ j in the ﬁrst term to the right, and with the symmetry of the metric tensor results in δ i j giju u δðÞuj δðÞuj δðÞuj ¼ g ui þ g ui ¼ 2g ui δt ji δt ij δt ij δt

δ i j i j 2 ðÞgiju u δuj As giju u ¼ kku , it implies that δt ¼ 0, which indicates that δt ¼ 0. (c) The covariant components of the vector are given by

i uj ¼ giju

whereby differentiating δ i δu giju δg δui δui j ¼ ¼ ij ui þ g ¼ g ¼ 0 δt δt δt ij δt ij δt δ dxi d2xi dxj dxk Exercise 2.15 Show that ¼ þ Γ i . δt dt dt2 jk dt dt Putting

dxi ui ¼ dt results for the absolute derivative of this vector δui dxk ∂ui dxk ∂ui dxk dxk ¼ ∂ u ¼ þ ujΓ i ¼ þ ujΓ i δt k i dt ∂xk jk dt ∂xk dt jk dt and with 2.7 Contravariant Derivative 133

dxj uj ¼ dt it implies

δui ∂ui dxk dxj dxk ¼ þ Γ i δt ∂xk dt jk dt dt

It follows that δ dxi d dxi dxj dxk ¼ þ Γ i δt dt dt dt jk dt dt δ dxi d2xi dxj dxk ¼ þ Γ i Q:E:D: δt dt dt2 jk dt dt

2.7 Contravariant Derivative

The contravariant derivative is deﬁned considering the tensorial nature of the covariant derivative, for the raising of the index of tensor ∂k ...the result is

‘ k‘ ∂ ...¼ g ∂k ... ð2:8:1Þ

’ ∂k ∂k ij It is promptly veriﬁed with Ricci s lemma that gij ¼ 0, as well as g ¼ 0 and ∂kδ i ij δi j ¼ 0. These relations show that the tensors gij, g , j behave as constants in the calculation of the contravariant derivative. For the variance tensors ( p, q) the result by means of the expression (2.8.1)is ÀÁ ∂k ÁÁÁ kj∂ ÁÁÁ ∂ kj ÁÁÁ : : TÁÁÁ ¼ g jTÁÁÁ ¼ j g TÁÁÁ ð2 8 2Þ

Then the contravariant derivative is equivalent to the raising of the indexes of ∂ ... kj ÁÁÁ tensor k , or the covariant derivative of tensor g TÁÁÁ. For instance, for the covariant vector uk: ÀÁ k kj kj j ∂ uk ¼ g ∂juk ¼ ∂j g uk ¼ ∂ju

Problems 2.1 Calculate the Christoffel symbols for the coordinates Xi which metric tensor is given by 134 2 Covariant, Absolute, and Contravariant Derivatives 2 3 10 g ¼ 4 1 5 ij 0 ðÞx2 2

Answer: 2 3 00 Γ Γ 4 1 5 ij, 1 ¼ 0fori, j ¼ 1, 2 ij,2 ¼ 0 À ðÞx2 2 00 Γ1 ¼ 0fori, j ¼ 1, 2 Γ2 ¼ ij ij 0 À1

2.2 Calculate the Christoffel symbols for the coordinate system Xi which metric tensor and its conjugated metric tensor are given by 2 3 2 3 10 0 6 7 10 0 6 1 7 6 2 7 6 0 0 7 4 0 ðÞx1 0 5 ij 6 ðÞx1 2 7 gij ¼ g ¼ 6 7 1 2 2 4 1 5 00ðÞx sin x 00 ðÞx1 sin x2 2

Answer: 2 3 2 3 00 0 0 x1 0 6 7 6 7 1 1 Γij,1 ¼ 4 0 Àx 0 5 Γij,2 ¼ 4 x 005 2 2 00Àx1ðÞsin x2 00ÀðÞx1 sin x2 cos x2 2 3 2 00x1ðÞsin x2 6 7 6 1 2 2 2 7 Γij, 3 ¼ 4 00ðÞx sin x cos x 5 2 2 x1ðÞsin x2 ðÞx1 sin x2 cos x2 0 2 3 00 0 6 7 Γ1 4 1 5 ij ¼ 0 Àx 0 2 00Àx1ðÞsin x2 2 3 2 3 1 1 6 0 0 7 00 6 x1 7 6 1 7 6 7 6 x 7 Γ2 6 1 7 Γ3 6 2 7 ij ¼ 6 007 ij ¼ 6 0 0 cot x 7 4 x1 5 4 5 1 2 2 cot x 0 00ÀðÞx1 sin x2 cos x2 x1 2.7 Contravariant Derivative 135

2.3 Calculate the Christoffel symbols of the second kind, where F(x1; x2)isa function of the coordinates, for the referential system which metric tensor is 10 g ¼ ij 0 FxðÞ1; x2

Answer: 2 3 2 3 1 ∂F 10 1 6 ∂ 1 7 Γ1 4 5 Γ2 6 2F x 7 ij ¼ 1 ∂F ij ¼ 4 5 0 À 1 ∂F 1 ∂F 2 ∂x1 2F ∂x1 2F ∂x2

2.4 Calculate the Christoffel symbols for the space deﬁned by the metric tensor 2 3 À10 0 0 6 0 À10 07 g ¼ 6 7 ij 4 00À105 000eÀx4

Γ 1 Àx4 ; Γ4 1 Answer: 44,4 ¼À2 e 44 ¼À2. j ‘m 2.5 Calculate the covariant derivative of the inner product of the tensors Ak and Bn with respect to coordinate xp. Answer: ÀÁ ∂ j ‘m j ∂ ‘m pAk Bn þ Ak pBn ffiffi ∂ p ij ffiffiffi ðÞgg Γ j p pq 2.6 Show that ∂xi þ pq gg ¼ 0. Chapter 3 Integral Theorems

3.1 Basic Concepts

The integral theorems and the concepts presented in this chapter are treated in Differential and Integral Calculus of multiple variables. The approach of this subject is carried in a concise and direct manner, and seeks solely to provide theoretical subsides so that the gradient, divergence, and curl differential operators can be physically interpreted.

3.1.1 Smooth Surface

The surface S, open or closed, with upward normal n unique in each point, which direction is a continuous function of its points, is classiﬁed as a smooth surface. For instance, the surface of a sphere is closed smooth, and the surface of a cube is closed smooth by parts, for it can be decomposed into six smooth surfaces.

3.1.2 Simply Connected Domain

For every closed curve C defined in the domain D, the region formed by C and its interior is fully contained in D. This curve defines a region R D, and D is called simply connected domain (Fig. 3.1a). The interior of a circle and the interior of a sphere are simply connected regions. Two concentric spheres define a simply connected region.

Fig. 3.1 Domain: ab (a) simply connected and (b) multiply connected

R R C C 1

C2 C D

3.1.3 Multiply Connected Domain

Multiply Connected Domain is the domain D that contains a region R with N “holes” (Fig. 3.1b). A circle excluded its center deﬁnes a simply connected domain, and the “hole” is reduced to a point, but the region between two coaxial cylinders is multiply connected.

3.1.4 Oriented Curve

The closed smooth curve C that limits a region R is counterclockwise oriented if this region stays to its left, i.e., this curve is positively oriented.

3.1.5 Surface Integral

Consider S a smooth surface by parts with upward unit normal vector n, and ϕ(xi)a function that represents a smooth curve C over this surface (Fig. 3.2). Dividing this ﬁnite area surface, deﬁned by the function ϕ(xi)inN elementary i areas dSi, i ¼ 1, 2, ..., N, where the elementary area contains the point P(x ), and XN ÀÁ i carrying out the sum ϕ x dSi and for N !1, thus dSi ! 0, implies the limit ðð i¼1 ÀÁ ϕ xi dS that represents the integral of surface S.

S This limit exists and is independent of theðð number of divisions made. For a vectorial function, it results in a similar way udS.

S 3.1 Basic Concepts 139

Fig. 3.2 Smooth surface u

α n

P ()x i φ ()x i S

ab Flow of u u ⋅ n dS

S dS S

Fig. 3.3 Flow: (a) through the surface S and (b) component of the vectorial function u in the direction normal to the surface S

3.1.6 Flow

Let the vectorial function u dependent on point P(xi) located on the surface S. The component of u in the direction of the unit normal vector to the surface in this point is given by the scalar product u Á n. With this dot product for all the points located in XN the surface elements dS, and carrying out the sum u Á ndS, and for N !1, and i¼1 dS ! 0 implies the integral ðð F ¼ u Á ndS ð3:1:1Þ S that deﬁnes the ﬂow of the vectorial function u on the surface S (Fig. 3.3). The surface area element dS is associated to the area vector dS, with modulus dS and same direction of n, then 140 3 Integral Theorems

dS ¼ ndS ð3:1:2Þ

Expression (3.1.1) is written as ðð ðð F ¼ u Á ndS ¼ u Á dS ð3:1:3Þ S S and the integration shown in this expression is independent of the coordinate system, because the dot product u Á n is invariant. In terms of the components of u, it follows that ðð

F ¼ uinidS ð3:1:4Þ S where ni are the direction cosines of the unit normal vector n.

3.2 Oriented Surface

Let S a surface oriented by means of its upward unit normal vector n, then its outline C is oriented positively if S stays to its left, thus this curve is anticlockwise oriented. Figure 3.4 shows a smooth surface S with upward unit normal vector n, deﬁned in a Cartesian coordinate system. This surface is expressed by the function z ¼ ϕðÞx; y , 3 which orthogonal projection in plane OX determines the region R ¼ S12.Theunit

Fig. 3.4 Smooth surface

S with upward unit normal 3 vector n which outline is a X curve closed smooth C dS P xii S C

O X 2

11 S dx 12 dx2

1 X C12 3.2 Oriented Surface 141 normal vector n forms an angle α with the axis OX3, being cos α its direction cosine. The orthogonal projection of the area element dS is given by

dx1dx2 dS ¼ cos α

The dot product of the unit vectors n and k is given by

n Á k ¼ kkn kkk cos α so

kkn Á k ¼ cos α and therefore dx1dx2 dS ¼ kkn Á k

Substituting this expression in expression (3.1.3) results in ðð ðð dx1dx2 F ¼ u Á ndS ¼ u Á n ð3:1:5Þ kkn Á k S S then the surface integral can be calculated as a double integral defined in the region R. The algebraic value of the flow depends on the field’s orientation. If π π α < then F > 0, i.e., the flow “is outward,” and if α > then F < 0, i.e., “the flow 2 2 is inward.”

3.2.1 Volume Integral

Consider the closed smooth surface S that contains a volume V, and ϕ(xi) a function of position deﬁned on this volume. Dividing V into elementary volumes dVi, then for the point P(xi) situated over S implies ϕ½¼PxðÞi ϕðÞxi . Carrying out the sum of XN ÀÁ i elementary volumes ϕ x dVi and for N !1, thus dVi ! 0, results the limit ððð i¼1 ÀÁ ϕ xi dV that represents the volume integral. This limit exists and is indepen-

V dentððð on the number of divisions. If the function is vectorial, it results in a similar way udV.

V 142 3 Integral Theorems

3.3 Green’s Theorem

Consider R a region in the plane OX1X2 involved by the closed smooth curve 1 2 1 2 C with R to its left. Let the real continuous functions F1(x ; x ) and F2(x ; x ), with continuous partial derivatives in R [ C. Then ðð þ ∂F ∂F ÀÁ 2 À 1 dx1dx2 ¼ F dx1 þ F dx2 ð3:2:1Þ ∂x1 ∂x2 1 2 R C

This theorem is due to George Green (1793–1841) and deals with a generalization of the fundamental theorem of Integral Calculus for two dimensions. Figure 3.5 shows the region R involved by the closed smooth curve C, in which there are lines parallel to the coordinate axes that are tangent to this curve. It is assumed as a premise that C is intersect by straight lines parallel to the coordinate axes in a maximum of two points. The region R is deﬁned by a x1 b, fxðÞ1 x2 gxðÞ1 c x2 d, pxðÞ2 x1 qxðÞ2

Let C ¼ AEB [ BFA, with AEB given by x2 ¼ fxðÞ1 , and BFA by x2 ¼ gxðÞ1 .In an analogous way results C ¼ FAE [ EBF, with FAE given by x1 ¼ pxðÞ2 , and EBF by x1 ¼ qxðÞ2 . With 2 3 1 ðð ðb gxðÞð ðb

∂F 6 ∂F 7 ÀÁgx1 1 1 2 6 1 27 1 1; 2 ðÞ dx dx ¼ 4 dx 5dx ¼ F1 x x 1 ∂x2 ∂x2 fxðÞ R a fxðÞ1 a ðb ðb ÂÃÀÁ ÂÃÀÁ 1 1 1 1 1 1 1 dx ¼À F1 x ; fx dx À F1 x ; gx dx a a Fig. 3.5 Simply connected X 2 region F d

A R B

C E

1 O b X 3.3 Green’s Theorem 143

Fig. 3.6 Region simply X 2 connects with segments parallel to one of the C coordinate axes P H

Q G

O X 1

The two right members are the line integrals, then ðð ð ð þ ∂F ÀÁ ÀÁ ÀÁ 1dx1dx2 ¼À F x1; x2 dx1 À F x1; x2 dx1 ¼ F x1; x2 dx1 ∂x2 1 1 1 R BFC AEB C

2 If the segment of curve C is parallelð to axis OX , the results of the integrals 1 are not modiﬁed (Fig. 3.6). The integral F1dx is cancelled in segment GH, for x1 ¼ constant then dx1 ¼ 0. The same occurs for segment PQ. With the segment QG given byx2 ¼ fxðÞ1 , and the segment HP given byx2 ¼ gxðÞ1 : ðð þ ∂F ÀÁ À 1dx1dx2 ¼ F x1; x2 dx1 ð3:2:2Þ ∂x2 1 R C and in the same way ðð þ ∂F ÀÁ À 2dx1dx2 ¼ F x1; x2 dx2 ð3:2:3Þ ∂x1 2 R C

Adding expressions (3.2.2) and (3.2.3) results in ðð þ ∂F ∂F ÀÁ 2 À 1 dx1dx2 ¼ F dx1 þ F dx2 Q:E:D: ∂x1 ∂x2 1 2 R C

To prove the validity of this theorem for the more general cases being the region R ¼ R1 [ R2, in which the integrals are calculated for each subregion (Fig. 3.7). 144 3 Integral Theorems

Fig. 3.7 Division of the X 2 simply connected region into two simply connected regions S R1

C T R2

O X 1

In the segment ST the line integrals are calculated twice, but as they are of different direction they cancel each other when they are added, hence þ þ ÀÁÀÁ 1 2 1 2 F1dx þ F2dx þ F1dx þ F2dx ¼ 0 TS ST

Therefore, the expression of Green’s theorem is valid for the subdivision of region R (Fig. 3.7). This ascertaining is generalized for a ﬁnite region R ¼ R1 [ R2 ÁÁÁRN comprising N simple regions, with the outline curves Ci, i ¼ 1, 2, ..., N, then ðð þ ∂F ∂F XN ÀÁ 2 À 1 dx1dx2 ¼ F dx1 þ F dx2 ∂ 1 ∂ 2 1 2 x x i¼1 R Ci

The consequence of this division of region R into parts is that this theorem can be applicable to multiply connected regions (Fig. 3.8). The region involved by the curve TSBSTAT is simply connected, so Green’s theorem is valid for this region, hence ðð þ ∂F ∂F ÀÁ 2 À 1 dx1dx2 ¼ F dx1 þ F dx2 ∂x1 ∂x2 1 2 R TSBSTAT

To demonstrate the validity of Green’s theorem for this kind of region, let the line integrals written in a symbolic way ð ð ð ð ð ð þ þ þ þ ¼ þ ¼

TS C2 ST C1 C2 C1 C 3.3 Green’s Theorem 145

Fig. 3.8 Multiply X 2 connected regions T

B A

R S 1 C 2

R C 1

O X 1 for ð ð ¼À TS ST therefore ðð þ ∂F ∂F ÀÁ 2 À 1 dx1dx2 ¼ F dx1 þ F dx2 ∂x1 ∂x2 1 2 R C proves the previous statement. ∂F ∂F With the condition 2 ¼ 1 in the region R it follows by Green’s theorem ∂x1 ∂x2 þ ÀÁ 1 2 F1dx þ F2dx ¼ 0 C thus the line integral is independent of the path on the closed curve C. To demonstrate that the admitted condition is necessary and sufﬁcient being the segments C1 and C2 of the curve C shown in Fig. 3.9, for the line integral it follows that þ ÀÁ 1 2 F1dx þ F2dx ¼ 0 ADBEA

Writing the line integrals of the various segments of curve C under symbolic form 146 3 Integral Theorems

Fig. 3.9 Segments C1 and B C2 of the closed curve C

D C 1

C R 2

ð ð þ ¼ 0 ADB BEA ð ð ð ¼À ¼

ADB BEA AEB then þ þ ÀÁÀÁ 1 2 1 2 F1dx þ F2dx ¼ F1dx þ F2dx

C1 C2

∂F ∂F where by 2 ¼ 1 is the necessary and sufﬁcient condition for this ∂x1 ∂x2 independence. To admit that a parallel straight line of a coordinated axis intersects the region R in only two points is not essential, because R can be divided into a number of subregions which separately fulﬁll this property. In vectorial notation with the function F ¼ F1i þ F2 j and the position vector r ¼ x1i þ x2j, and in differential form dr ¼ dx1i þ dx2j, the line integral along the curve C is given by þ þ ÀÁ 1 2 F1dx þ F2dx ¼ F Á dr C C 3.4 Stokes’ Theorem 147

3.4 Stokes’ Theorem

Consider the surface S with upward unit normal vector n involved by a closed smooth curve C with S to its left, which direction cosines are ni > 0. Let the 1 2 3 1 2 3 1 2 3 continuous real functions F1(x ; x ; x ), F2(x ; x ; x ), F3(x ; x ; x ) with continuous partial derivatives in S [ C.Then ðð ∂F ∂F ∂F ∂F ∂F ∂F 3 À 2 n þ 1 À 3 n þ 2 À 1 n dS ∂x2 ∂x3 1 ∂x3 ∂x1 2 ∂x1 ∂x2 3 S þ ÀÁ 1 2 3 ¼ F1dx þ F2dx þ F3dx ð3:3:1Þ C

To demonstrate this theorem admit that a line parallel to axis OX3 intersects S only 1 2 in a point, then the projection of S on the plane OX X will be the region S12 involved by the closed smooth curve C12 oriented positively (Fig. 3.10), then

dS12 ¼ n3dS ð3:3:2Þ and n3 > 0. The equation of surface S is given explicitly by x3 ¼ ϕðÞx1; x2 , which allows substituting the line integral along the curve C by the line integral along curve C12: þ þ ÀÁ ÂÃÀÁ 1 2 3 1 1 2 1 2 1 F1 x ; x ; x dx ¼ F1 x ; x ; ϕ x ; x dx

C C12

In the term to the right the coordinate x2 appears twice, in a direct way and in the function that represents the surface S. Applying Green’s theorem it follows that þ ðð ÂÃÀÁ ∂F ∂F ∂ϕ F x1; x2; ϕ x1; x2 dx1 ¼À 1 þ 1 dx1dx2 1 ∂x2 ∂ϕ ∂x2 C S12 where

1 2 dS12 ¼ dx dx and using expression (3.3.2) 148 3 Integral Theorems

3 X

S C

O S 2 X

S12

1 X C12

B C2

Fig. 3.10 Stokes theorem: (a) projection of the smooth surface S with upward unit normal vector n on the plane OX1X2;(b) surface delimited by the closed smooth curve C; and (c) surface with outline delimited by more than one curve

þ ðð ÀÁ ∂F ½x1; x2; ϕðÞx1; x2 ∂F ½x1; x2; ϕðÞx1; x2 ∂ϕðÞx1; x2 F x1; x2; x3 dx1 ¼À 1 þ 1 n dS 1 ∂x2 ∂ϕ ∂x2 3 C S ð3:3:3Þ

In Integral Calculus of Multiple Variables when studying the surface integrals of x3 ¼ ϕðÞx1; x2 the following expressions are deducted for the direction cosines of its upward unit normal vector n:

∂ϕ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∂x1 : : n1 ¼ ð3 3 4Þ ∂ϕ 2 ∂ϕ 2 Æ 1 þ ∂x1 þ ∂x2 3.4 Stokes’ Theorem 149

∂ϕ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∂x2 : : n2 ¼ ð3 3 5Þ ∂ϕ 2 ∂ϕ 2 Æ 1 þ ∂x1 þ ∂x2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 : : n3 ¼ ð3 3 6Þ ∂ϕ 2 ∂ϕ 2 Ç 1 þ ∂x1 þ ∂x2

As the direction cosines are positive, expression (3.3.5) provides

∂ϕðÞx1; x2 n3 ¼Àn3 ∂x2 and substituting this expression in expression (3.3.3) þ ðð ∂F ∂F F dx1 ¼À n 1 À n 1 dS 1 2 ∂x3 3 ∂x2 C S

In an analogous way for the projections of S on the planes OX2X3 and OX3X1 it follows that þ ðð ∂F ∂F F dx2 ¼À n 2 À n 2 dS 2 3 ∂x1 1 ∂x3 C S þ ðð ∂F ∂F F dx3 ¼À n 3 À n 3 dS 3 1 ∂x2 2 ∂x1 C S

Adding these three expressions results in ðð ∂F ∂F ∂F ∂F ∂F ∂F 3 À 2 n þ 1 À 3 n þ 2 À 1 n ∂x2 ∂x3 1 ∂x3 ∂x1 2 ∂x1 ∂x2 3 S þ ÀÁ 1 2 3 dS ¼ F1dx þ F2dx þ F3dx Q:E:D: C

Admit that a line parallel to one of the coordinate axis cuts the surface S only in a point is not an essential premise. Figure 3.10b, c shows two kinds of surface that do not fulfill this condition. In this case the surfaces must be divided into a finite number of subsurfaces, which separately fulfills this hypothesis, allowing Stokes’ theorem to be applied to these subsurfaces, and add the partial results obtained. Then the line integrals referent to the outlines common to the projections of these surfaces on a plane of the coordinate system cancel each other, for they are integrated twice, but with the signs changed. 150 3 Integral Theorems

For a surface formed by several closed curves it is also possible to apply Stokes’ theorem. Figure 3.10c shows a surface S limited by the closed and smooth curves C1 and C2. The section S along the curve AB generates a new surface, which outlines are the curves C1, C2 and AB, considered in opposite directions. Then the line integral referent to curve AB is calculated twice, but with opposite signs, whereby it cancels itself, leaving only the results referent to the line integrals of the curves C1 and C2. The Stokes theorem is a generalization of Green’s theorem for the tridimensional space. In vectorial notation with the function F ¼ F1i þ F2 j þ F3k and the vector r ¼ x1i þ x2j þ x3k, which differential is dr ¼ dx1i þ dx2j þ dx3k, the line integral along the curve C is given by þ þ ÀÁ 1 2 3 F1dx þ F2dx þ F3dx ¼ F Á dr ð3:3:7Þ C C

The surface integrals that are present in Stokes’ theorem also have a vectorial interpretation (item 4.4).

3.5 Gauß–Ostrogradsky Theorem

Consider the volume V with upward unit normal vector n involved by a closed and smooth surface S, which direction cosines are ni > 0. Let the continuous 1 2 3 1 2 3 1 2 3 real functions F1(x ; x ; x ),F2(x ; x ; x ),F3(x ; x ; x ) with continuous partial derivatives in V [ S. Then ððð ðð ∂F ∂F ∂F 1 þ 2 þ 3 dx1dx2dx3 ¼ ðÞF n þ F n þ F n dS ∂x1 ∂x2 ∂x3 1 1 2 2 3 3 V S ð3:4:1Þ

Consider a line parallel to axis OX2 that intersects the surface S in a maximum of two points P and P0, with upward unit normal vector n(P) and n(P0), respectively 3 1 (Fig. 3.11). Then the projection of S on OX X will be S31, it follows that ððð ðð ð ðð hi ∂F ∂F 0 2dx1dx2dx3 ¼ 2 dx2 dS ¼ F P À F ðÞP dS ∂x2 ∂x2 31 2 2 31 V S31 S31 3.5 Gauß–Ostrogradsky Theorem 151

n P n P '

dS 31

j V S 31 C S

O X2

Fig. 3.11 Volume V with upward unit normal vector n, which outline is a closed and smooth surface S

For the area element in this plane and with the direction cosines of the upward normal n(P) and n(P0): 0 0 dS31 ¼ dSðÞ P n2ðÞ¼ÀP dS P n2 P

Substituting it results for the point P on S: ððð ðð ∂F 2dx1dx2dx3 ¼ F ðÞP n ðÞP dS ∂x2 2 2 V S

In an analogous way, for the projections of S on the planes OX1X2 and OX2X3: ððð ðð ∂F 3dx1dx2dx3 ¼ F ðÞP n ðÞP dS ∂x3 3 3 V S ððð ðð ∂F 1dx1dx2dx3 ¼ F ðÞP n ðÞP dS ∂x1 1 1 V S 152 3 Integral Theorems ab X 3 S S2 S1 V V2 V 2 V1 n1 V n2 V * 1 S S O X 2 * S * S1 2 S * X 1

Fig. 3.12 Gauß–Ostrogradsky theorem: (a) volume cut in more than two points by a straight line parallel to a coordinated axis and (b) volume V with voids V1 and V2

The addition of these three expressions results ððð ðð ∂F ∂F ∂F 1 þ 2 þ 3 dx1dx2dx3 ¼ ðÞF n þ F n þ F n dS Q:E:D: ∂x1 ∂x2 ∂x3 1 1 2 2 3 3 V S

One of the premises adopted in the proof of the theorem of Carl Friedrich Gauß and Mikhail Vasilievich Ostrogradsky (1801–1861) is that the surface S has two sides, with a single upward and inward normal in each point. To admit that a straight line parallel to a coordinate axis intersects the volume V in only two points is not an essential hypothesis, for V can be divided into a number of subvolumes that separately fulﬁll the property admitted initially, allowing the Gauß–Ostrogradsky theorem to be applied to these subvolumes and adding the partial results obtained. Figure 3.12a shows the volume V cut in more than two points by a straight line 2 parallel to axis OX . The division of V into two volumes V1 and V2, separated by surface S*, with opposite unit normal vector n1 and n2, being V1 involved by * * S1 [ S , and V2 by S2 [ S . Then the surface integrals referent to this part common to the two volumes cancel each other, remaining the integrals on the surfaces S1 and S2. This makes the applying of this theorem valid to volume V. If the closed surface S that involves volume V is not smooth, it can be divided into a ﬁnite number of smooth surfaces represented by the functions ϕ(xi), which have continuous partial derivatives, each one involving a subvolume. This proce- dure allows applying the Gauß–Ostrogradsky theorem to these subvolumes and adding the results obtained. 3.5 Gauß–Ostrogradsky Theorem 153

Figure 3.12b shows the volume V involved by the closed surface S with empty volumes V1 and V2, with which are involved, respectively, by the smooth closed surfaces S1 and S2. In this case it is necessary to cut the total volume and the volumes of the voids by a plane π and the surfaces of their outlines to project in this * Ã Ã plane, originating the surfaces S , S1 and S2, and then apply the Gauß–Ostrogradsky theorem considering these surfaces. In vectorial notation with F ¼ F1i þ F2 j þ F3k results ððð ðð ∂F ∂F ∂F 1 þ 2 þ 3 dx1dx2dx3 ¼ F Á ndS ð3:4:2Þ ∂x1 ∂x2 ∂x3 V S

The volume integral that is present in Gauß–Ostrogradsky theorem also has a vectorial interpretation (item 4.3). Chapter 4 Differential Operators

4.1 Scalar, Vectorial, and Tensorial Fields

4.1.1 Initial Notes

The study of the scalar, vectorial, and tensorial fields is strictly related with the differential operators which are applied to the analytic functions that represent these fields. In this chapter the differential operators gradient, divergence, and curl will be defined, and their physical interpretations, as well as various fundamental relations with these operators, will be presented. These expressions form the mathematical backbone for the practical applications of the Field Theory. The conception of fields is of fundamental importance to the formulation of Tensor Calculus, and allows defining various concepts and deducing several expressions which form the framework for the study of the tensors contained in the tensorial space that defines the field. The scalar, vectorial, and tensorial fields are formulations carried out on a point i x 2D, the domain D EN being an open subset and embedded in the ordinary geometric space. In these three kinds of fields the formulations are the functions smooth, continuous, and derivable. By defining an arbitrary origin in the space EN a biunivocal correspondence is determined for each domain with a variety, scalar, vector, or tensor that defines the kind of field. The scalar and vectorial fields are particular cases of the tensorial fields. The behavior of a tensorial field is measured by the variation rate of the tensor in the points contained in the field. In the literature it is usual to call this variation rate as tensor derivative, which is incorrect, for what exists is the variation rate of the field defined by this variety, so the proper denomination is variation of the tensorial field. However, on account of being customary by use, the denomination tensor derivative will be used in this text to express this variation.

4.1.2 Scalar Field

Let a scalar be associated to a point in the Euclidian space E3 given by a function of the coordinates xi, which is defined as ϕ ¼ ϕðÞxi; t , i ¼ 1, 2, 3, where t is the time in the instant in which the scalar is measured. A scalar field is defined by the function of field ϕ(xi; t), and if the time variable t is constant, the level surface of the field ϕðÞ¼xi C is defined, where C is a constant. For several values of C there is a family of level surfaces, which characterize the field geometrically. These surfaces do not intersect, for if they did the function ϕ(xi)would have to assume various values, which is impossible, for this function has only one value for each xi. As an example of scalar field there is a point in the interior of a reservoir containing liquid, where each particle of this fluid is submitted to a pressure proportional to the distance of this particle up to the top of the free surface. Another example is the field of temperatures due to a heat source, where the isotherms are spherical surfaces, with the temperature decreasing to the points farthest from this source.

4.1.3 Pseudoscalar Field

If the field function defines a pseudoscalar then the field is pseudoscalar. The specific mass ρ(xi) of the points of a solid body is an example of this sort of field.

4.1.4 Vectorial Field

i i If the vector u(x , t) is associated with the point P(x ) of the space EN, then a vectorial field is defined, and if t ¼ constant the field is homogeneous. For the space E3 which points are referenced to a Cartesian coordinate system there are i i i three scalar functions of these points, f 1ðÞx , f 2ðÞx , f 3ðÞx , i ¼ 1, 2, 3, which express the field vector ÀÁ ÀÁ ÀÁ ÀÁ i i i i u x ¼ f 1 x i þ f 2 x j þ f 3 x k

Field lines are defined for a vectorial field determined by the vectorial function u(xi), in which for each point P(xi) the field vectors are collinear with the vectors tangents to these lines (Fig. 4.1). The condition of collinearity between the vector u(xi) and the tangent vector t(xi) is given by the nullity of cross product

εijkujdxk ¼ 0 4.1 Scalar, Vectorial, and Tensorial Fields 157

a b t u

Field line s

Fig. 4.1 Vectorial field: (a) field lines and field vector and (b) field vectors

Developing provides: – i ¼ 1

ε1jkujdxk ¼ 0 ) ε12ku2dxk ¼ 0 ) ε123u2dx3 ¼ 0 ) u2dx3 ¼ 0

ε13ku3dx2 ¼ 0 ) ε132u3dx2 ¼ 0 )Àu3dx2 ¼ 0

– i ¼ 2

ε2jkujdxk ¼ 0 ) ε213u1dx3 ¼ 0 )Àu1dx3 ¼ 0

ε2jkujdxk ¼ 0 ) ε231u3dx1 ¼ 0 ) u3dx1 ¼ 0

– i ¼ 3

ε3jkujdxk ¼ 0 ) ε312u1dx2 ¼ 0 ) u1dx2 ¼ 0

ε3jkujdxk ¼ 0 ) ε321u2dx1 ¼ 0 )Àu2dx1 ¼ 0

Thus the following system results 8 > u3 dx3 8 > ¼ > u2 dx2 <> u3dx2 ¼ u2dx3 <> u dx u dx ¼ u dx ) 1 ¼ 1 :> 1 2 2 1 > u dx > 2 2 u3dx1 ¼ u1dx3 > u dx : 3 ¼ 3 u1 dx1 158 4 Differential Operators

For a flat vectorial field the condition of collinearity between the field vector and the vector tangent to the field lines is given by

u2dx1 À u1dx2 ¼ 0

The gravitational, the electric, and the magnetic are examples of vectorial ﬁelds.

4.1.5 Tensorial Field

The fundamental problem of Tensor Calculus is associated to the concept of tensorial field. If the tensorial field is fixed the tensor T(xi) is a function of the i coordinates of a point P(x ) situated in the tensorial space EN. When this tensor is function of xi and other parameters then the tensorial field is variable. For tensor TðÞxi which components are defined with respect to a curvilinear i coordinates X which origin is the point P(xi), a few difficulties arise in the calculation of its derivatives, because the local coordinate system varies as a function of the point. The study of the tensorial fields in a tensorial space EN, considering curvilinear local coordinate systems, is associated to the basis of this space. Exercise 4.1 Calculate the parametric equation of the lines of the vectorial field u ¼Àx2i þ x1 j þ mk that contains the point of coordinates (1; 0; 0) where m is a scalar. The differential equations of the field lines are

dx dx dx 1 ¼ 2 ¼ 3 Àx2 x1 m

Following with the first two differential relations ð ð 2 2 x1dx1 þ x2dx2 ¼ 0 ) x1dx1 þ x2dx2 ¼ C0 ) ðÞx1 þ ðÞx2 ¼ C1; C1 > 0 and introducing a parameter t pffiffiffiffiffiffi pffiffiffiffiffiffi x1 ¼ C1 cos tx2 ¼ C1 sin t so pffiffiffiffiffiffi dx2 ¼ C1 cos t dt and with the differential relations 4.1 Scalar, Vectorial, and Tensorial Fields 159

dx dx 2 ¼ 3 x1 m it follows that ÀÁpffiffiffiffiffiffi C1 cos t dt dx3 pffiffiffiffiffiffi ¼ ) dx3 ¼ mdt ) x3 ¼ mt þ C2 C1 cos t m

As the field line contains the point of coordinates (1; 0; 0), then pffiffiffiffiffiffi 1 ¼ C1 cos t ) t ¼ 2kπ; k ¼ 0, Æ 1, ...

– k ¼ 0 ! C1 ¼ 1; – k ¼ 0 ! t ¼ 0soC2 ¼ 0; – verifying that 0 ¼ mt þ C2 for t ¼ 0. The parametric equations of the ﬁeld lines represent a circular helix given by

x1 ¼ cos t; x2 ¼ sin t; x3 ¼ mt

4.1.6 Circulation

Consider the ﬁeld deﬁned by the vectorial function u and the point P(xi) located on an open curve C, continuous by parts, smooth, and derivable, which is the hodograph of the position vector r(s), where s is the curvilinear abscissa, and admitting that this point varies in the interval a xi b, then the line integral of this curve is given by

ðb I ¼ u Á dr ð4:1:1Þ a where line integral deﬁnes the circulation of the vectorial function u on the curve C. Let u Á dr be the differential total of the function ϕ(xi), thus

ðb ÀÁ ÀÁ ϕ i ϕ i b ϕ ϕ : : I ¼ d x ¼ x a ¼ ðÞÀb ðÞa ð4 1 2Þ a

The value of this integral depends only on the extreme points of the interval for which the function ϕ(xi) is deﬁned, regardless of the integration path. Expression (4.1.2) is a generalization of the fundamental theorem of the Integral Calculus. 160 4 Differential Operators

ab Path

C2 C Path b C 2 C b Path C Path C1 1 D D

Fig. 4.2 Closed curve paths: (a) with no self-intersection and (b) with a ﬁnite number of self- intersections

For a closed curve the extreme points of this interval are coincident, which allows concluding that þ u Á dr ¼ 0 ð4:1:3Þ

This expression defines the circulation of vector u along the closed curve C. The line integral of an open curve C defined in a certain interval will be independent of the path adopted in this calculation, and will be null if the curve C is closed. Figure 4.2 shows two types of closed paths of curves defined in domain D—the single closed path in which there are no self-intersection points and the closed path with self-intersection points. For the closed spatial curves defined in the domain D self-intersecting in a finite number of points, the line integral is calculated dividing the path in a finite number of single closed paths. For an infinite number of intersections, a reasonable approx- imation is obtained with the integrals on paths which are polygonal segments, using a limit process to achieve a finite number of intersections.

4.2 Gradient

In item 2.2 the gradient of a scalar ﬁeld was deﬁned, by a function ϕ(xi) which differential is given by

∂ϕ dϕ ¼ dxi ð4:2:1Þ ∂xi that is called differential parameter of the ﬁrst order of Beltrami. Expression (4.2.1) shows that there is no difference between the total differential dϕ and the absolute differential, which allows adopting the notation ϕ,i for the partial derivatives of this function. It was also shown that the gradient is a vector 4.2 Gradient 161 obtained by means of applying a vectorial operator to the scalar function ϕ(xi), that with respect to a coordinate system Xi is given by

∂ÁÁÁ ∇ÁÁÁ¼ei ∂xi

j For a curvilinear coordinate system X by the chain rule

∂ÁÁÁ ∂xj ∂ÁÁÁ ¼ ∂xi ∂xi ∂xj and with the transformation law of unit vectors

∂xi ei ¼ gk ∂xk it follows that

∂ÁÁÁ ∂xi ∂ÁÁÁ ∂xi ∂xj ∂ÁÁÁ ∂ÁÁÁ ∂ÁÁÁ ∇ÁÁÁ¼ei ¼ gk ¼ gk ¼ gkδ j ¼ gk ∂xi ∂xk ∂xi ∂xk ∂xi ∂xj k ∂xj ∂xk

The several notations for the gradient vector are

ÀÁ ∂ϕðÞxi gradϕ xi ¼ GðÞ¼ϕ ∇ðÞ¼ϕ gk ¼ gkϕ ð4:2:2Þ ∂xk , k

This comma notation will hereafter be used in some special case for derivatives with respect to coordinates. The classic notation for the operator that defines the gradient of a scalar function is grad ϕ(xi), and was introduced by Maxwell, Rie- mann, and Weber. The other notation is an inverted delta, called nabla operator να0 βλα ∇ k ∂ÁÁÁ (in Greek ¼ harp), del, atled (inverted delta), expressed as ÁÁÁ¼g ∂xk. This notation was designed by Hamilton in 1837, initially was not used to represent ⊲ the gradient of a function, but was written with the rotated delta , and represented ÀÁ 2 ÀÁ d 2 d d 2 symbolically the Laplace operator dx þ dy þ dz that was already well used at the time, thereby the denomination Hamilton operator, or Hamiltonian operator. Another interpretation for the name nabla is due to Maxwell, who remarks that the rotated delta calls to cuneiform writing, which name in Hebrew would be this one. The use of the nabla operator has many advantages with respect to the spelling grad, especially in the development of expressions, for it reinforces the tensorial characteristics of the gradient. This symbolic vector enables making the spelling for the differential operators uniform, and complies with the Vectorial Algebra rules. Figure 4.3a shows schematically a scalar field defined by a function ϕ(xi), where in a field line contained in the level surface the ϕðÞ¼xi C, being C ¼ constant, a point P is defined, and with an arbitrary origin O for the coordinate system Xi, 162 4 Differential Operators

ab ∇φ u(P) dr dr P dt φ xi = C b P element of curve line C

Fig. 4.3 Scalar ﬁeld: (a) gradient and (b) line element

k results in the position vector r, which derivative is the vector dr ¼ dx gk tangent to the ﬁeld line, and denotes the line element. The differential of the scalar function that represents this ﬁeld is given by the dot product

‘ ∂ϕ ‘ ∂ϕ ∂ϕ dϕ ¼ dr Á ∇ϕ ¼ dxkg Á g ¼ δ dxk ¼ dxk k ∂x‘ k ∂x‘ ∂xk

The ﬁeld represented by the gradient of a function is conservative, thus this function is called potential, or ﬁeld gradient. As the operator nabla is a vector, it is invariant for a change in the coordinate j system, which can be proven admitting ∇ for a coordinate system X , and ∇ for a coordinate system Xi, so by means of the vectors transformation law ∂ÁÁÁ ∂ÁÁÁ ∂x‘ ¼ ∂xk ∂x‘ ∂xk k ‘ ∂ÁÁÁ ∂x ∂x ∂ÁÁÁ ‘ ∂ÁÁÁ ‘ ∂ÁÁÁ ∇ÁÁÁ¼g ¼ gm ¼ δ gm ¼ g ¼ ∇ÁÁÁ k ∂xk ∂xm ∂xk ∂x‘ m ∂x‘ ∂x‘ therefore the operator ∇ is a vector. The gradient for the function ϕ ¼ xk, where xk represents a coordinate of the ∇ϕ ∂xk i referential system, is given by ¼ ∂xi g , then the gradient is the unit vector for the coordinate axis. For the product of two scalar functions ϕ(xi) and ψ(xi) the result is 4.2 Gradient 163

∂ðÞϕψ ∂ϕ ∂ψ gradðÞ¼ϕψ ∇ðÞ¼ϕψ gk ¼ gk ψ þ ϕgk ¼ ðÞ∇ϕ ψ þ ϕðÞ∇ψ ∂xk ∂xk ∂xk

In this demonstration it is observed that the nabla operator acts on each parcel of the expression in a distinct way, maintaining a parcel variable and the other constant. If it comes before the parcel it acts with a variable, if it comes after, it acts as a constant. The gradient can be deﬁned by means of the Gauß-Ostrogradsky theorem. Let the ﬁeld be determined by the vectorial function u ¼ vϕðÞxi , v being a constant vector, then

∂u1 ∂ϕðÞxi ¼ v Á ∂ i ∂ i x x ∂u1 ∂u2 ∂u3 ∂ϕ ∂ϕ ∂ϕ þ þ ¼ v Á þ þ ¼ v Á ∇ϕ ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 it follows that ððð ðð ∂u1 ∂u2 ∂u3 þ þ dV ¼ u Á n dS ∂x1 ∂x2 ∂x3 V ððð ðð S v Á ∇ϕdV ¼ v Á u Á n dS 0 V S 1 ððð ðð v Á @ ∇ϕdV À u Á n dSA ¼ 0 V S

For the point P in the scalar ﬁeld ϕ(xi) contained in an elementary volume, and ∂ϕ ∇ϕ with the component ∂x1 of by the mean value theorem of the Integral Calculus ððð ∂ϕ ∂ϕ dV ¼ V ∂ 1 ∂ 1 x x P* V where P* is the midpoint of volume V. Applying the Gauß-Ostrogradsky theorem ðð ∂ϕ 1 ¼ ϕn dS ∂ 1 1 x P* V S where n1 is the direction cosine of the angle between the upward normal unit vector n and the coordinate axis OX1. 164 4 Differential Operators

When the point P approaches the point P*, the volume V and the surface S also come close to P, and with continuous function ϕ(xi) and its partial derivatives it results in ðð ∂ϕ 1 ¼ lim ϕn dS ∂ 1 1 x P V!0 V S

For the coordinates x2 and x3 the result with analogous formulations is ðð ðð ∂ϕ 1 ∂ϕ 1 ¼ lim ϕn dS ¼ lim ϕn dS ∂ 2 2 ∂ 3 3 x P V!0 V x P V!0 V S S

If these limits exist, the gradient of the scalar function ϕ(xi) in point P is determined by ðð ÀÁ 1 ÀÁ ∇ϕ xi ¼ lim ϕ xi ndS ð4:2:3Þ V!0 V S that is valid for any coordinate system, which shows that the gradient is independent of the coordinate system.

4.2.1 Norm of the Gradient

The norm of the gradient is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ϕ ∂ϕ kk∇ϕ ¼ kk∇ϕ Á ∇ϕ ¼ g ð4:2:4Þ ij ∂xi ∂xj

Δ ϕ ∂ϕ ∂ϕ A few authors use the spelling 1 ¼ gij ∂xi ∂xj to designate the first differential parameter of Beltrami. δ For the orthogonal coordinate systems gij ¼ ij: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ϕ 2 kk∇ϕ ¼ ð4:2:5Þ ∂xi 4.2 Gradient 165

4.2.2 Orthogonal Coordinate Systems

Consider the point P(xi) be coincident with the origin of the curvilinear orthogonal j coordinate X , and r its position vector with respect to the Cartesian coordinate Xj. Rewriting expression (2.3.4) the result for this vector’s differential is

∂r dr ¼ dxi ∂xi with

∂r ¼ h g ∂xi i i then

∂r hi ¼ ∂xi

j pffiffiffiffiffiffiffiffi where gi are the unit vectors of the coordinate system X , and hi ¼ gðÞii are the metric tensor coefficients that represent scale factors of the magnitudes of the vectors tangents to the curves of this coordinate system, where the indexes in parenthesis do not indicate summation. They are called Lame´ coefficients for orthogonal coordinate systems. The differential of the scalar function ϕ(xi) is given by

∂ϕ dϕ ¼ dxi ∂xi but

dϕ ¼ ∇ϕ Á dr i dr ¼ hidx gi then ∂ϕ dϕ ¼ dxi gi Á dr ∂xi whereby ∂ϕ ∇ϕ 1 i : : ¼ i g ð4 2 6Þ hi ∂x 166 4 Differential Operators that provides the components of the gradient vector in a curvilinear orthogonal coordinate system. The physical components of the vector ∇ϕ are given by

∂ϕ ∇ϕ * 1 : : ðÞ¼ i ð4 2 7Þ hi ∂x

4.2.3 Directional Derivative of the Gradient

Figure 4.3b shows the differential element of the line contained in the level surface ϕðÞ¼xi C and its tangent unit vector t, collinear with vector dr, which allows writing for the line element dr ¼ tds. The geometric interpretation of the gradient of a scalar ﬁeld is given by the dot product

dϕ ¼ t Á ∇ϕ ð4:2:8Þ ds that defines the field directional derivative. The symbol ∇ϕ characterizes the field, and unit vector t being independent of the function ϕ(xi), this indicates the direction in which the derivative is calculated. If ∇ϕðÞxi exists in the point P, defined by the field ϕ(xi), it will be possible to calculate the directional derivative of this function in all the directions of the field. Then the field ϕ(xi) is non-homogeneous. Let α be the angle between the two vectors from expression (4.2.8), the dot product provides

dϕ ¼ kkt kk∇ϕ cos α ¼ kk∇ϕ cos α ds

As ϕ ¼ constant, it results in dϕ ¼ 0, so dr Á ∇ϕ ¼ 0, then the vector ∇ϕ is perpendicular to the vector dr. The variation rate of the ﬁeld deﬁned by the function ϕ is maximum in the direction of ∇ϕ, for α ¼ 0 results in cos α ¼ 1, then dϕ ¼ kk∇ϕ > 0 dn max

The directional derivative is calculated in the direction of the unit normal vector n to the level surface ϕðÞ¼xi C (Fig. 4.4), thus

dϕ ∇ϕ ¼ n ð4:2:9Þ dn 4.2 Gradient 167

Fig. 4.4 Interpretation of the gradient as a vector normal to surface

t P

φ xi = C

4.2.4 Dyadic Product

k The nabla operator applied to a vectorial function u ¼ u gk results in the dyadic product

∇ u ¼ ∇u ¼ T ÀÁ ∂u ∂ ukg ∂uk ∂g T ¼ gi ¼ gi k ¼ gi g þ uk k ∂xi ∂xi ∂xi k ∂xi and with expression (2.3.10) ∂g k ¼ Γ mg ∂xi ki m it follows that ∂uk T ¼ gi g þ ukΓ mg ∂xi k ki m

Interchanging the indexes m $ k in the second member to the right ∂uk T ¼ þ umΓ k gi g ∂xi mi k

The covariant derivative of a contravariant vector results in a variance tensor (1, 1) 168 4 Differential Operators

∂uk T k ¼ ∂ uk ¼ þ umΓ k i i ∂xi mi then

k i : : T ¼ Ti g gk ð4 2 10Þ

This analysis shows that the gradient and the covariant derivative represent a same concept, i.e., they represent the derivative of a scalar, vectorial, or tensorial function, increasing their variance from one unit. Formulating an analogous analysis for a covariant vector

∂u T ¼ ∇ u ¼ ∇u ¼ gi ∂xi ÀÁ k k ∂ ukg ∂u ∂g ∂u T ¼ gi ¼ gi k gk þ u ¼ gi k gk À u Γ mgm ∂xi ∂xi k ∂xi ∂xi m ki ∂u ¼ gi k À u Γ k gi gk ∂xi k mi then

i k i k ∇ u ¼ ∇u ¼ ðÞ∂iuk g g ¼ Tikg g ð4:2:11Þ

The differential of a vectorial ﬁeld is a vector, for the differential of vector u: ÀÁÀÁ ∇ i ∂ i k i∂ i k du ¼ dr Á u ¼ dx gi iukg g ¼ dx iukgi Á g g whereby

i k du ¼ dx ∂iukg ð4:2:12Þ

For the ﬁelds vectorial there is the directional derivative

du ¼ t Á ∇ u ð4:2:13Þ ds

The same considerations formulated for the directional derivative of a scalar ﬁeld ϕ(xi) are applicable to the vectorial ﬁelds. The physical components for the gradient of vector u* are obtained considering the physical components of the second-order tensor. 4.2 Gradient 169

4.2.5 Gradient of a Second-Order Tensor

The generalization of the concept of gradient for an arbitrary tensorial field is immediate. For a coordinate system Xi with unit vector g‘, and with the tensor T defining the tensorial field

‘ ∂T ∇ T ¼ gradT ¼ g ð4:2:14Þ ∂x‘

∇ ‘ ∂ÁÁÁ Then the gradient of a tensor T is calculated by nabla operator ¼ g ∂x‘ applying to this tensor. This operator is deﬁned for a contravariant base. The tensorial product of the nabla operator by the second-order tensor T is given by ÀÁ ∂ Tkig g ∂Tki ∂g ∂g ∇ T ¼ ∇T ¼ gm k i ¼ gm g g þ Tki k g þ Tkig i ∂xm ∂xm k i ∂xm i k ∂xm and with the expressions

∂g ∂g k ¼ Γ p g i ¼ Γ p g ∂xm km p ∂xm im p it follows that ∂Tki ∇ T ¼ gm g g þ TkiΓ p g g þ TkiΓ p g g ∂xm k i km p i im k p

Interchanging the indexes p $ k in the second term to the right, and the indexes p $ i in the third term ∂Tki ∇ T ¼ þ TkiΓ p þ TkiΓ p gm g g ∂xm km im k i this expression becomes

∇ ∂ ki m : : T ¼ mT g gk gp ð4 2 15Þ and shows that the gradient of a second-order tensor is a variance tensor (1, 2). The other components of the gradient of tensor T are given by expressions (2.5.18) and (2.5.21). The generalization of the deﬁnition of the gradient of a third-order tensor T is immediate. The components of the fourth-order tensor that result from applying this operator to tensor T being given by their covariant derivatives, for instance, for tensor Tijk: 170 4 Differential Operators ∂T ‘ ∇ T ¼ ijk À T Γ m À T Γ m À T Γ m g gi g j gk ∂x‘ mjk i‘ imk j‘ ijm k‘

For a tensor T of order p the variety ∇ T is a tensor of order ðÞp þ 1 . The differential of a tensorial ﬁeld is a tensor, is gives by then the differential of the second-order tensor T thus:

∇ j ∂ ki m j∂ ki m dT ¼ dr Á T ¼ dx gj Á mT g gk gp ¼ dx mT gj Á g gk gp j∂ kiδ m ¼ dx mT j gk gp whereby j∂ kj : : dT ¼ dx jT gk gp ð4 2 16Þ

The physical components for the gradient of tensor T* are obtained considering the physical components of the tensor of the third order. The same considerations formulated for the directional derivative of a scalar field ϕ(xi) and of a vectorial field are applicable to the tensorial fields, where

dT ¼ t Á ∇ T ð4:2:17Þ ds

4.2.6 Gradient Properties

The ascertaining achieved in the previous paragraphs allow establishing the condi- i tions so that a vector is gradient of a scalar function, for if theð vector u(x ) deﬁned in a single or multiply connected region, and if the line integral u Á dr is independent

C of the path, then a scalar function ϕ(xi) exists and fulﬁlls the condition u ¼ ∇ϕðÞxi in all of this region of the space. The gradient operator applied to the addition of two tensors provides

‘ ∂ðÞT þ A ‘ ∂T ‘ ∂A ∇ ðÞ¼T þ A g ¼ g þ g ¼ ∇ T þ ∇ A ∂x‘ ∂x‘ ∂x‘

The applying of this operator to the multiplication of the scalar m by the tensor T provides

‘ ∂ðÞmT ‘ ∂T ∇ ðÞ¼mT g ¼ mg ¼ m∇ T ∂x‘ ∂x‘

These two demonstrations prove that the gradient is a linear operator, which is already implicit, because it is a vector. 4.2 Gradient 171

Exercise 4.2 Calculate: (a) v Á ∇u; (b) ∇ðÞu Á v . (a) The gradient for the ﬁeld deﬁned by a vectorial function is given by

i k ∇u ¼ ∂iukg g

With

‘ v ¼ v g‘

it follows that

‘ i k ‘ i k ‘ i k i k v Á ∇u ¼ v g‘ Á ∂iukg g ¼ v ∂iukg‘ Á g g ¼ v ∂iukδ‘g ¼ v ðÞ∂iuk g

Thus for the Cartesian coordinates

ðÞ∂u v Á ∇u ¼ v k g i ∂xi k

(b) The gradient of the scalar ﬁeld represented by the dot product of the vectorial functions u and v is given by

∂ðÞuiv ÂÃÀÁ ∇ðÞ¼u Á v gk i ¼ ∂ ui v þ uiðÞ∂ v gk ∂xk k i k i

and with the expressions

∂ui ∂v ∂ ui ¼ þ umΓ i ∂ v ¼ i À v Γ m k ∂xk mk k i ∂xk m ik

it follows that ∂ui ∂v ∇ðÞ¼u Á v v þ umΓ i v þ ui i À uiv Γ m gk ∂xk i mk i ∂xk m ik

In the last term in parenthesis interchanging the indexes i $ m: ∂ui ∂v ∇ðÞ¼u Á v v þ umΓ i v þ ui i À umv Γ i gk ∂xk i mk i ∂xk i mk

then ∂ui ∂v ∇ðÞ¼u Á v v þ ui i gk ∂xk i ∂xk 172 4 Differential Operators

For the Cartesian coordinates ∂u ∂v ∇ðÞ¼u Á v i v þ u i g ∂xk i i ∂xk k

Another way of expressing ∇ðÞu Á v is to use the expression between the covariant derivative of a covariant vector and the covariant derivative of a contravariant vector, which is given by

∂ ∂ i kum ¼ gim ku

The multiplying of this expression by gin provides

in∂ in ∂ i δ n ∂ i ∂ i im∂ m g kum ¼ g gim ku ¼ m ku ) ku ¼ g kum

whereby

i im m vi∂ku ¼ vig ∂kum ¼ v ∂kum

Replacing the dummy index m ! i:

i i vi∂ku ¼ v ∂kui and by substitution ÀÁ i i k ∇ðÞ¼u Á v v ∂kui þ u ∂kvi g

i i Adding and subtracting the terms v ∂iuk and u ∂ivk ÂÃ i i i i k ∇ðÞ¼u Á v v ðÞþ∂kui À ∂iuk v ∂iuk þ u ðÞþ∂kvi À ∂ivk u ∂ivk g

and with the expressions

i k i k v Â ðÞ¼∇ Â u v ðÞ∂kui À ∂iuk g u Â ðÞ¼∇ Â v u ðÞ∂kvi À ∂ivk g i k i k v Á ∇u ¼ v ðÞ∂iuk g u Á ∇v ¼ u ðÞ∂ivk g

it results in

∇ðÞ¼u Á v v Á ∇u þ v Â ðÞþ∇ Â u u Á ∇v þ u Â ðÞ∇ Â v

For the particular case in which u ¼ v:

1 v Á ∇v ¼ ∇v2 À v Â ðÞ∇ Â v 2 4.2 Gradient 173

Exercise 4.3 Calculate the gradient of the scalar function ϕ(xi) expressed in cylindrical coordinates. pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi For the cylindrical coordinates g11 ¼ g33 ¼ 1, g22 ¼ r, then

1 ∂ϕ ∇ϕ ¼ ffiffiffiffiffi g p ∂ i k gii x it follows that

∂ϕ 1 ∂ϕ ∂ϕ ∇ϕ ¼ g þ gθ þ g ∂r r r ∂θ ∂z z

Exercise 4.4 Calculate the gradient of the scalar function ϕ(x1) expressed in spherical coordinates. ffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi p p p ϕ For the spherical coordinates g11 ¼ 1, g22 ¼ r, g33 ¼ r sin , then

∂ϕ 1 ∂ϕ 1 ∂ϕ ∇ϕ ¼ g þ gϕ þ gθ ∂r r r ∂ϕ r sin ϕ ∂xθ

∂2ϕðÞxi Exercise 4.5 Show that is a second-order tensor, where ϕ(xi) is a scalar ∂xi∂xj function. Putting ∂2ϕ T ¼ ¼ ϕ ij ∂xi∂xj , ij

i for the coordinate system X ∂2ϕ ∂ϕ ∂ϕ ∂xk ∂ϕ ∂ϕ ∂xk ∂xm ∂xk ∂xm ∂2ϕ ¼ ¼ ¼ ∂xi∂xj ∂xi ∂xk ∂xk ∂xm ∂xk ∂xj ∂xi ∂xj ∂xi ∂xm∂xk

∂2ϕðÞxi This transformation law proves that is a second-order tensor. ∂xi∂xj Exercise 4.6 Calculate the directional derivative of the function ϕðÞ¼x, y x2 þ y2 À 3xy3, at the point P(1; 2) in the direction of vector pffiffi 1 3 u ¼ 2 e1 þ 2 e2, being e1(1; 0), e2(0; 1). The gradient of the scalar field is given by ÀÁÀÁ 3 2 ∇ϕ ¼ 2x À 3y e1 þ 2y À 9xy e2 174 4 Differential Operators and in point P(1; 2)

∇ϕ ¼À22e1 À 32e2

pffiffi u 1 3 The vector u is a unit vector, for kku ¼ 2 e1 þ 2 e2, whereby it follows that for the directional derivative pffiffiffi 1 3 pffiffiffi ∇ϕ Á u ¼À22 Â À 32 Â ¼À11 À 16 3 2 2

4.3 Divergence

The analysis of the flow magnitude for the vectorial field u that passes through the volume V involved by the closed surface S with respect to this volume leads to the conception of a differential operator (Fig. 4.5a). In volume V let the elementary parallelepiped with sides dx1, dx2, dx3, and the vectorial function u continuous with continuous partial derivatives (Fig. 4.5b). The study of the flow of u that passes through the volume V with respect to this volume is carried out considering the point P(x1; x2; x3) at the center of an elementary parallelepiped (Fig. 4.5). For the face dS1, with upward normal unit vector n (1; 0; 0), the component of u in the direction of axis OX1 is given by

u Á n ¼ u1

abn

X 3 u * S u S 12 12 dx3 P n n 2 dx1 dx O P X 2 S V X 1

Fig. 4.5 Flow of the vectorial function u:(a) that passes through the volume V and (b) in the elementary parallelepiped 4.3 Divergence 175 1 2 1 dx1 ; The center of the elementary area dS1 ¼ dx dx has coordinates x þ 2 dx2; dx3Þ, whereby it follows that for the surface integral in this face of the elementary parallelepiped dV ðð dx1 u Á ndS ﬃ u1 x1 þ ; dx2; dx3 dx2dx3 2 dS1 for the area considered is elementary, which allows calculating approximately the surface integral as the dot product

u:ndS ¼ u1dx2dx3

* 2 3 For the elementary face dS1 ¼ dxdx with upward normal unit vector ; ; 1 dx1 ; 2; 3 nðÞÀ1 0 0 centered in the midpoint x À 2 dx dx it follows that in an analogous way ðð dx1 u:n dS ﬃÀu1 x1 À ; dx2; dx3 dx2dx3 2 * dS1

Adding these contributions ðð dx1 dx1 u:n dS ﬃ u1 x1 þ ; dx2; dx3 À u1 x1 À ; dx2; dx3 dx2dx3 2 2 * dS1þdS 1 ÀÁ ﬃ u1 x1 dx2dx3

The component u1 in the point of coordinate dx1 varies according to the rate

∂u1 du1 ¼ dx1 ∂x1 then ðð ∂u1 ∂u1 u Á n dS ffi dx1dx2dx3 ffi dV ∂x1 ∂x1 * dS1þdS1 and the same way for the components u2 and u3 in the other faces of the parallelepiped the result is, respectively 176 4 Differential Operators ðð ðð ∂u2 ∂u3 u Á n dS ffi dV u Á n dS ffi dV ∂x2 ∂x3 * * dS2þdS2 dS3þdS3

Adding these three expressions results for the six faces of the elementary parallelepiped ðð ððð ∂u1 ∂u2 ∂u3 u Á n dS ¼ þ þ dV ∂x1 ∂x2 ∂x3 S V

Putting

∂u1 ∂u2 ∂u3 divu ¼ þ þ ð4:3:1Þ ∂x1 ∂x2 ∂x3 this analysis leads to the following deﬁnition for the divergence of the vectorial function u at point P(xi) ðð 1 divu ¼ lim u Á n dS ð4:3:2Þ V!0 V S that can be interpreted as the dot product between the nabla operator and the vectorial function u, thus ∂ÁÁÁ divu ¼ ∇ Á u ¼ gj Á uig ∂xj i

To demonstrate that expression (4.3.2) represents the divergence of the vectorial function u, consider the sphere of radius R > 0, of surface S(R) and volume V(R), centered at point P located in the vectorial space E3. For the vectorial ﬁeld u acting in the space ðð 1 divuðÞ¼P lim u Á n dS ð4:3:3Þ R!0 VRðÞ SRðÞ

Let gPðÞ¼divu, and admitting that g(xi) is a continuous function that can be written as ÀÁ ÀÁ gxi ¼ gPðÞþhxi where ÀÁ i hx xi!P ¼ 0 4.3 Divergence 177

Applying the divergence theorem to the vectorial ﬁeld ðð ððð ððð ððð 1 1 ÀÁ 1 1 ÀÁ u:n dS ¼ hxi dV ¼ hPðÞdV þ hxi dV VRðÞ VRðÞ VRðÞ VRðÞ SRðÞ VRðÞ VRðÞ VRðÞ

As gPðÞ¼divu: ððð ððð 1 1 1 gPðÞdV ¼ gPðÞ dV ¼ gPðÞVRðÞ¼gPðÞ VRðÞ VRðÞ VRðÞ VRðÞ VRðÞ

For the function h(xi) the result when R ! 0is

ððð ððð ÀÁ ÀÁ ÀÁ 1 1 hxi dV ¼ Max hxi dV Max hxi i i VRðÞ kkx ÀP R VRðÞ kkx ÀP R VRðÞ VRðÞ

The maximum value of this function fulfills the condition kkhxðÞi ! 0 when kkxi À P ! 0, then the expression (4.3.2) represents the divergence of the vectorial function u at the point P. This expression is valid for any kind of coordinate system, which shows that the divergence is independent of the referential system. This analysis was formulated for a Cartesian coordinate system for a question of simplicity, being that for the case of curvilinear coordinate systems it was enough to adopt the local trihedron with unit vectors (g1; g2; g3), and one elementary parallelepiped of volume dV. The scalar field generated by the applying of the divergence to the vectorial field defined by the vectorial function u is called solenoidal or vorticular field, when divu ¼ 0, where u is a solenoidal vector, and this field is called field without source.

4.3.1 Divergence Theorem

The divergence allows writing the Gauß-Ostrogradsky theorem as ðð ððð u Á n dS ¼ ∇ Á u dV ð4:3:4Þ S V which is called the divergence theorem. The symbology adopted in expression (4.3.4) does not change the characteristics and properties shown in item 3.4. Let a solenoidal ﬁeld acting in a region R be located between the two closed surfaces S1 and S2 (Fig. 4.6), then 178 4 Differential Operators

Fig. 4.6 Solenoidal ﬁeld in a region R between two volumes S1

V2 V1

R = V1 −V2

ðð ðð u Á n dS ¼ u Á n dS

S1 S2

To demonstrate this equality consider the closed surface S1 with upward normal unit vector n, with R to the left of the outline of this surface that involves the volume V1, and the closed surface S2 with unit downward unit normal vector n, involving the volume V2. Applying the divergence theorem ðð ðð ðð ðð divu ¼ u Á n dS À u Á n dS ¼ 0 ) u Á n dS ¼ u Á n dS

S1 S2 S1 S2 then it is enough to calculate only the integral of a surface. For a ﬁeld represented by the vectorial function u ¼ ϕ∇ψ, where ϕ and ψ are scalar functions

∇ Á u ¼ ∇ðÞ¼ϕ∇ψ ϕ∇ Á ∇ψ þ ∇ϕ Á ∇ψ ¼ ϕ∇2ψ þ ∇ϕ Á ∇ψ

The component of u in the direction of the normal unit vector n is given by

∂ψ u Á n ¼ ϕn Á ∇ψ ¼ ϕ ∂n and applying the divergence theorem ðð ððð u Á n dS ¼ ∇ Á u dV S V 4.3 Divergence 179 results in ðð ððð ∂ψ ÀÁ ϕ dS ¼ ϕ∇2ψ þ ∇ϕ Á ∇ψ dV ∂n S V that is called Green’s ﬁrst formula. If the vectorial function is given by u ¼ ϕ∇ψ þ ψ∇ϕ, in an analogous way

∇ Á u ¼ ϕ∇2ψ þ ψ ∇2ϕ ∂ψ ∂ϕ u Á n ¼ ϕ À ψ ∂n ∂n whereby ðð ððð ∂ψ ∂ϕ ÀÁ ϕ À ψ dS ¼ ϕ∇2ψ þ ψ ∇2ϕ dV ∂n ∂n S V that is called Green’s second formula.

4.3.2 Contravariant and Covariant Components

The vectorial function u can be expressed by means of their contravariant or covariant components, so it is necessary to calculate this function’s divergence for these components. For the vector’s contravariant coordinates ∂uig divu ¼ ∇ Á u ¼ g j Á i ð4:3:5Þ ∂xj and for its covariant coordinates ∂u gi divu ¼ ∇ Á u ¼ g Á i ð4:3:6Þ j ∂xj

The terms in parenthesis in these expressions indicate that this deﬁnition can be ampliﬁed considering the vector’s covariant derivatives, expressed in their contravariant and covariant coordinates. Let the covariant derivative of the contravariant vector ui be:

∂ui ∂ ui ¼ þ u jΓ i k ∂xk jk 180 4 Differential Operators that generates a tensor which contraction for i ¼ k provides

∂ui ∂ ui ¼ þ ujΓ i i ∂xi ji and rewriting the expression (2.4.23) ÀÁpffiffiffi ÀÁpffiffiffi ∂ ‘n g 1 ∂ g Γ i ¼ ¼ pffiffiffi ji ∂xj g ∂xj

The use of this expression is more adequate, for it abbreviates the calculation of the Christoffel symbol. Substituting this expression in the previous expression ÀÁffiffiffi ÀÁ ∂ i ∂ p ffiffiffi ∂ i j ∂ 1‘ i u j 1 g 1 p u u 2 ng ∂iu ¼ þ u pffiffiffi ¼ pffiffiffi g þ pffiffiffi ∂xi g ∂xj g ∂xi g ∂xj and replacing the indexes i ! j of the first term to the right ÀÁ ffiffiffi ∂ j j ∂ 1‘ i 1 p u u 2 ng ∂iu ¼ pffiffiffi g þ pffiffiffi g ∂xj 2 g ∂xj or in a compact form ÀÁffiffiffi ∂ p j i 1 gu ∂iu ¼ pffiffiffi ð4:3:7Þ g ∂xj

It is veriﬁed that expression (4.3.7), deducted by means of the contravariant vector ui, represents a scalar, for it was obtained by means of contraction of the second-order tensor. The other way of formulating this analysis is by means of the covariant derivative of their covariant components. Let the covariant derivative of the covariant vector ui be: ÀÁ ij ∂iui ¼ ∂i g uj that developed leads to the following expression ÀÁ ÀÁ ij ij ∂iui ¼ ∂i g uj þ g ∂i uj

ij Ricci’s lemma shows that ∂iðÞ¼g 0 whereby ÀÁ ij j ∂iui ¼ g ∂i uj ¼ ∂iu 4.3 Divergence 181 and the contraction of this tensor for i ¼ j provides ÀÁffiffiffi ∂ p i i i 1 gu ∂iui ¼ ∂iu ¼ divu ¼ pffiffiffi ð4:3:8Þ g ∂xi

i Expressions (4.3.7) and (4.3.8) provide the same result, i.e., ∂iui ¼ ∂iu . Then the covariant derivative of a vector is independent of the type of the component. The divergence deﬁned by expressions (4.3.5) and (4.3.6) is the dot product of the nabla operator by the vector to which it is applied. The development of the derivatives indicated in these expressions leads to the same results of expressions (4.3.7) and (4.3.8), whereby these last expressions represent the divergence of a vectorial function. For the Cartesian coordinates

∂u ∇ Á u ¼ i ð4:3:9Þ ∂xi

4.3.3 Orthogonal Coordinate Systems

Consider the elementary parallelepiped with sides ds1, ds2, ds3, defined in the curvilinear orthogonal coordinates OXj, by means of which the flow of the field is represented by the vectorial function u (Fig. 4.7). The divergence of this field is given by ðð 1 ∇ Á u ¼ lim u Á ndS V!0 V S

Let

i dsi ¼ hidx

X 3 u ()x 1 ; x 2 ; x 3

n = − g 2 u ()x 1 ; x 2 + ∂x 2 ; x 3 ds3

ds O 1 ds2 n = g 2 X 1 X 2

Fig. 4.7 Divergence of the vectorial function u in the curvilinear orthogonal coordinates 182 4 Differential Operators

1 2 3 dV ¼ ds1ds2ds3 ¼ h1h2h3dx dx dx there is, respectively, for the face with upward normal unit vector n ¼Àg2 and n ¼ g2 ∂ðÞu2h h Àu Á g ds ds ¼Àu2h h dx1dx3 ¼Àu2 h h þ 1 3 dx2 dx1dx3 2 1 3 1 3 1 3 ∂x2

In an analogous way, for the other faces ∂ðÞu1h h Àu Á g ds ds ¼Àu1h h dx2dx3 ¼Àu1 h h þ 2 3 dx2 dx2dx3 1 2 3 2 3 2 3 ∂x1 ∂ðÞu2h h Àu Á g ds ds ¼Àu2h h dx1dx3 ¼Àu2 h h þ 1 3 dx1 dx1dx3 2 1 3 1 3 1 3 ∂x2 ∂ðÞu3h h Àu Á g ds ds ¼Àu3h h dx1dx2 ¼Àu3 h h þ 1 2 dx3 dx1dx2 3 1 2 1 2 1 2 ∂x3

Adding the expressions relative to the six faces of the parallelepiped ðð ∂ðÞu1h h ∂ðÞu2h h ∂ðÞu3h h u Á n dS ¼ 2 3 þ 1 3 þ 1 2 dx1dx2dx3 ∂x1 ∂x2 ∂x3 S but dV dx1dx2dx ¼ h1h2h3 then ðð ∂ 1 ∂ 2 ∂ 3 ∇ 1 ðÞu h2h3 ðÞu h1h3 ðÞu h1h2 Á u ¼ lim u Á n dS ¼ 1 þ 2 þ 3 V!0 h1h2h3 ∂x ∂x ∂x S

The result for the orthogonal coordinate system is ∂ i ∇ 1 h1h2h3u : : Á u ¼ divu ¼ i ð4 3 10Þ h1h2h3 ∂x hi pffiffiffiffiffiffiffiffi where hi ¼ gðÞii are the components of the metric tensor, and the indexes in parenthesis do not indicate summation. 4.3 Divergence 183

4.3.4 Physical Components

With expression (4.3.6) the physical components of the divergence of a vector takes the form ∂ *i ∇ * * 1 h1h2h3u : : Á u ¼ divu ¼ i ð4 3 11Þ h1h2h3 ∂x hi where u* i are the vector’s physical components.

4.3.5 Properties

As the divergence is the dot product of the nabla operator for a vectorial function, the distributive property of the dot product is valid. For the sum of two vectorial functions u and v:

divðÞ¼u þ v ∇ Á ðÞ¼u þ v ∇ Á u þ ∇ Á v ð4:3:12Þ and in terms of the covariant derivative

i i ∇ Á ðÞ¼u þ v ∂iu þ ∂iv ð4:3:13Þ and for the Cartesian coordinates the result is

∂ui ∂vi ∇ Á ðÞ¼u þ v þ ð4:3:14Þ ∂xi ∂xi

Considering the vectorial function mu, where m is a scalar, the result of the dot product of vectors is

divðÞ¼ mu ∇ Á ðÞ¼mu m∇ Á u ð4:3:15Þ

These two demonstrations prove that the divergence, for these cases, is a linear operator. In general, the divergence is not a linear operator, as it will be shown in Exercise 4.7.

4.3.6 Divergence of a Second-Order Tensor

The generalization of the divergence theorem for tensorial ﬁelds is immediate. Consider, for example, the ﬁeld represented by the tensorial function of the second 184 4 Differential Operators

order T(r) in space E3, which components depend on the position vector, i.e., Tij ¼ TijðÞr . For the surface S smooth and continuous by parts in its two faces, with normal unit vector n(n1; n2; n3) varying on each point of the surface, the ﬂow of this tensorial function through S is given by the vector v of components ðð

vi ¼ TijnjdS; i, j ¼ 1, 2, 3 S or ðð

vi ¼ TjinjdS; i, j ¼ 1, 2, 3 S

In absolute notation for the ﬂow v the result is ðð v ¼ T ndS ð4:3:16Þ S

The ﬂow of the unit tensor δij through the closed surface S is given by the components of vector n: ðð ðð

vi ¼ δijnjdS ¼ nidS S S or in absolute notation ðð

vi ¼ ndS S

The comparison with expression (4.2.3) ðð ÀÁ 1 ÀÁ ∇ϕ xi ¼ lim ϕ xi ndS V!0 V S shows that ϕðÞ¼xi 1, i.e., ϕðÞ¼xi constant, so ∇ϕðÞ¼xi 0, which indicates that v ¼ 0. Concluding that for a unitary tensorial field the flow through the closed surface S is null. The concept of a field’s divergence can be extended to the tensorial fields, for it is enough that the tensors be contravariant. In the case of covariant tensors their indexes must be raised by means of the metric tensor, next they must be derived and contracted. 4.3 Divergence 185

There are distinct divergences, depending on the index to be contracted. For Tij ij ij ij ji there are two divergences: ∂iT and ∂jT . If the tensor is symmetrical T ¼ T then ij ij ∂iT ¼ ∂jT , i.e., the divergence is unique. The divergence components of a contravariant second-order tensor are given by

ij ij divT ¼ ∂jT ð4:3:17Þ and the covariant derivative of the components for this tensor is ∂Tij ∂ Tij ¼ þ TmkΓ i þ TimΓ j k ∂xk mk mk

With k ¼ j: ∂Tij ∂ Tij ¼ þ TmjΓ i þ TimΓ j j ∂xj mj mj being ÀÁpffiffiffi ∂ ‘n g Γ j ¼ mj ∂xm it follows that ÀÁpffiffiffi ∂Tij ∂ ‘n g ∂ Tij ¼ þ TmjΓ i þ Tim j ∂xj mj ∂xm

The change of the indexes m ! j in the last term in brackets provides ÀÁpffiffiffi ∂Tij ∂ ‘n g ∂ Tij ¼ þ TmjΓ i þ Tij k ∂xj mj ∂xj or ÀÁffiffiffi ffiffiffi ∂ ij ∂ p ij mj i 1 p T ij g ∂kT ¼ T Γ þ pffiffiffi g þ T mj g ∂xj ∂xj then ÀÁpffiffiffi 1 ∂ gTij divTij ¼ TmjΓ i þ pffiffiffi ð4:3:18Þ mj g ∂xk that shows that the divergence of a second-order tensor is a vector. 186 4 Differential Operators

For a mixed second-order tensor the result is

i ∂ i divTj ¼ iTj and rewriting expression (2.5.21)

∂T i ∂ T i ¼ j þ T mΓ i À T i Γ m k j ∂xk j mk m jk

Assuming i ¼ k:

∂T i ∂ T i ¼ j þ T mΓ i À T i Γ m i j ∂xk j mi m ji and with ÀÁpffiffiffi ∂ ‘n g Γ i ¼ mi ∂xm then "# ÀÁpffiffiffi ∂T i ∂ ‘n g ∂ T i ¼ j þ T m À T i Γ m i j ∂xk j ∂xm m ji

The change of indexes m ! i in the second term in brackets provides "# ÀÁpffiffiffi ∂T i ∂ ‘n g ∂ T i ¼ j þ T i À T i Γ m i j ∂xk j ∂xi m ji or ÀÁffiffiffi ffiffiffi ∂ i ∂ p i p Tj 1 i g i m ∂iT ¼ g þ pffiffiffi T À T Γ j ∂xk g j ∂xi m ji then pffiffiffi ∂ T i g i 1 j i m ∂iT ¼ pffiffiffi À T Γ ð4:3:19Þ j g ∂xi m ji

The generalization of the deﬁnition of the divergence for a third-order tensor is immediate 4.3 Divergence 187 ÀÁ ∇ j ∂ j ∂ kim ∂ kim j Á T ¼ g Á jT ¼ g Á jT gk gi gm ¼ jT g Á gk gi gm ÀÁ ∂ kim δ j ¼ jT kgi gm thus

∇ ∂ jim : : Á T ¼ jT gi gm ð4 3 20Þ

This expression shows that ∇ Á T is a second-order tensor. For a tensor T of order p then ∇ Á T is a tensor of order ðÞp À 1 . In absolute or invariant notation the result for the divergence of tensor T is

∇ Á T ¼ ðÞ∇ T G ð4:3:21Þ where G is the metric tensor. In the particular case in which divT ¼ 0 the tensor T defines a tensorial solenoidal field. The divergence theorem also applies to a tensorial field. Let the field be defined by u ¼ Tv, which in terms of the components of vectors and tensor is given by ui ¼ Tikvk, being v an arbitrary and constant vector. Applying the divergence theorem to this field ðð ððð u Á ndS ¼ ∇ Á udV S V where

∇ Á u ¼ ∇ðÞ¼Tv ∇T Á v

∂T This vector has components ik v , and the component of vector u in the ∂xk i direction of the normal unit vector n is given by the dot product

u Á n ¼ ðÞTikvi nk then ðð ððð ∂T ðÞT v n dS ¼ ik v dV ik i k ∂xk i S V whereby in terms of the tensor components the result is ðð ððð ∂T T n dS ¼ ikdV ð4:3:22Þ ik k ∂xk S V 188 4 Differential Operators and in absolute notation this expression becomes ðð ððð T ndS ¼ ∇ Á TdV ð4:3:23Þ S V

Exercise 4.7 Calculate: (a) ∇ Á ðÞϕu ; (b) ∇ Á ðÞu Â v . (a) The ﬁeld divergence deﬁned by the product of a scalar function ϕ(xi)bya vector u is given by ÀÁ ∂ ϕukg ∂ϕ ∂uk ∂g divðÞ¼ϕu ∇ Á ðÞ¼ϕu gm k ¼ gm ukg þ ϕ g þ ϕuk k ∂xm ∂xm k ∂xm k ∂xm

and substituting (2.3.10)

∂g k ¼ Γ p g ∂xm km p

in the previous expression ∂ϕ ∂uk divðÞ¼ϕu gm ukg þ ϕ g þ ϕukΓ p g ∂xm k ∂xm k km p

The permutation of the indexes p $ k in the third member in parenthesis provides ∂ϕ ∂uk ∂ϕ ∂uk divðÞ¼ϕu uk þ ϕ þ ϕupΓ k gm Á g ¼ uk þ ϕ þ ϕupΓ k ∂xm ∂xm pm k ∂xm ∂xm pm

and with

∂ϕ ∂uk ∂ϕ ∂ uk ¼ uk þ ϕ ) divðÞ¼ϕu uk þ ϕ∂ uk m ∂xm ∂xm ∂xm m

Putting

∂ϕ ∂ uk ¼ ∇uk ¼ ∇ϕ m ∂xm

thus

∇ Á ðÞ¼ϕu ðÞÁ∇ϕ u þ ϕðÞ∇u 4.3 Divergence 189 or

divðÞ¼ϕu gradϕ Á u þ ϕgradu

In this case the divergence is not a linear operator, but for ϕ ¼ m, where m is a constant, the result is expression (4.3.15), verifying the linearity of this operator. (b) The ﬁeld represented by the vectorial function generated by the cross product w ¼ u Â v is given by

p εpqr w ¼ w gp ¼ uqvrgp

The divergence of this function is given by ÀÁ ∂ wpg ∂wp ∂g ∇ Á ðÞ¼u Â v gi Á p ¼ g þ wp p Á gi ∂xi ∂xi p ∂xi

and the expression ∂g p ¼ Γ j g ∂xi pi j

substituted in the previous expression provides ∂wp ∇ Á ðÞ¼u Â v g þ wpΓ j g Á gi ∂xi p pi j

Interchanging indexes p $ j in the second term in parenthesis it follows that ∂wp ∇ Á ðÞ¼u Â v þ wjΓ p gi Á g ¼ δ i ∂ wp ∂xi ji p p i

or ÀÁÀÁ ÀÁ p pqr pqr pqr ∂pw ¼ ∂p ε uqvr ¼ ∂pε uqvr þ ε ∂p uqvr

and with ÀÁ p pqr ∂pw ¼ ε ∂p uqvr

thus ÀÁ pqr ∇ Á ðÞ¼u Â v ε ∂p uqvr 190 4 Differential Operators whereby ÂÃÀÁ pqr ∇ Á ðÞ¼u Â v ε ∂puq vr þ uq∂pvr

With the εpqr ¼ εqpr and εpqr ¼Àεrpq the results for the terms to the right are ÀÁ ÀÁ pqr rpq ε ∂puq vr ¼ ε ∂puq vr ¼ v Á ∇u ÀÁ ÀÁ pqr qpr ε uq ∂pvr ¼Àε uq ∂pvr ¼Àu Á ∇v

whereby

∇ Á ðÞ¼u Â v v Á ∇ Â u À u Á ∇ Â v ) divðÞ¼u Â v v Á rotu À u Á rotv

For the Cartesian coordinates ÀÁ ∂ ujvk ∇ Á ðÞ¼u Â v ε ijk ∂xi

ij i i Exercise 4.8 Let T and Tj be associated tensors, write div Tj in terms of the symmetrical tensor Tij. The divergence of a second-order tensor is given by pffiffiffi ∂ T i g i i 1 j i m divT ¼ ∂iT ¼ pffiffiffi À T Γ j j g ∂xi m ji and with Γ m mkΓ ij ¼ g ij, k thus pffiffiffi ∂ T i g i i 1 j i mk divT ¼ ∂iT ¼ pffiffiffi À T g Γij k j j g ∂xi m ,

Let 1 ∂g ∂g ∂g Γ ¼ jk þ ik þ ij ij,k 2 ∂xi ∂xj ∂xk i mk ik Tmg ¼ T then pffiffiffi ∂ T i g ∂ ∂ ∂ i 1 j ik1 gjk gik gij ∂iT ¼ pffiffiffi À T þ þ j g ∂xi 2 ∂xi ∂xj ∂xk 4.3 Divergence 191 or pffiffiffi ∂ T i g ∂ ∂ ∂ i 1 j 1 ik gjk 1 ik gik 1 ik gij ∂iT ¼ pffiffiffi À T À T À T j g ∂xi 2 ∂xi 2 ∂xj 2 ∂xk

Interchanging the indexes i $ j in the last term to the right pffiffiffi ∂ T i g ∂ ∂ ∂ i 1 j 1 ik gjk 1 ik gik 1 ki gkj ∂iT ¼ pffiffiffi À T À T À T j g ∂xi 2 ∂xi 2 ∂xj 2 ∂xk

ik ki gjk ¼ gkj T ¼ T thus pffiffiffi ∂ T i g ∂ i 1 j 1 ik gik ∂iT ¼ pffiffiffi À T j g ∂xi 2 ∂xj

Exercise 4.9 Calculate the divergence of vector ui expressed in cylindrical coordinates. pffiffiffi For the cylindrical coordinates g ¼ r, and with the contravariant components θ of vector (ur, u , uz) it follows that ÀÁffiffiffi ÀÁ ∂ p i r ∂ θ z 1 gu 1 ∂ðÞrui 1 ∂ðÞru ru ∂ðÞru divui ¼ pffiffiffi ¼ ¼ þ þ g ∂xi r ∂xi r ∂r ∂θ ∂z ∂ur ∂uθ ∂uz ¼ þ þ þ ur ∂r ∂θ ∂z

In an analogous way in terms of the vector’s covariant components

∂u 1 ∂uθ ∂u divu ¼ r þ þ z þ u j ∂r r2 ∂θ ∂z r

Exercise 4.10 Calculate the divergence of vector ui expressed in spherical coordinates. pffiffiffi For the spherical coordinates g ¼ r2 sin ϕ, and with the contravariant components of vector (ur, uϕ, uθ) it follows that 192 4 Differential Operators

ÀÁpffiffiffi 1 ∂ gui 1 ∂ðÞr2 sin ϕui divui ¼ pffiffiffi ¼ g ∂xi r2 sin ϕ ∂xi ÀÁÀÁ ϕ θ 1 ∂ðÞr2 sin ϕur ∂ r2 sin ϕu ∂ r2 sin ϕu ¼ þ þ r2 sin ϕ ∂r ∂ϕ ∂θ r ϕ θ r ∂u ∂u ∂u 2u ϕ ¼ þ þ þ þ ðÞcot ϕ u ∂r ∂ϕ ∂θ r

For the vector’s covariant components the result is

∂ur 1 ∂uϕ 1 ∂uθ 2ur ðÞcot ϕ divu ¼ þ þ þ þ uϕ j ∂r r2 ∂ϕ r2 sin 2ϕ ∂θ r r2

Exercise 4.11 Let r be the position vector of theÀÁ points in the spaceÀÁ E3, show that: n n r r 2 (a) divr ¼ 3; (b) divðÞ¼r r ðÞn þ 3 r ; (c) div r3 ¼ 0; (d) div r ¼ r. (a) With the deﬁnition of divergence ∂ ∂r ∇ Á r ¼ g Á r ¼ g Á i ∂xi i ∂xi

but ∂r ¼ g ∂xi i

then ∇ Á r ¼ gi Á gi

For i ¼ 1, 2, 3 the result is

divr ¼ 3Q:E:D:

(b) Let

divðÞ¼ϕu ϕdivu þ ugradϕ

and putting

u ¼ r ϕ ¼ rn thus 1 ÀÁ divðÞ¼rnr rn divr þ rgradrn ¼ 3rn þ r Á nrnÀ1 r ¼ 3rn þ nrnÀ2r2 r 4.3 Divergence 193

then

divðÞ¼rn r ðÞn þ 3 rn Q:E:D:

it follows that r divðÞ¼rÀ3r rÀ3divr þ r Á gradrÀ3 ¼ 3rÀ3 þ r ÁÀðÞ¼3rÀ4gradr 3rÀ3 þ r ÁÀ3rÀ4 r 1 ¼ 3rÀ3 þ r2 À3rÀ4 r

whereby ÀÁ div rÀ3r ¼ 0Q:E:D:

This conclusion shows that rÀ3r is a solenoidal vectorial function. (d) Putting r 1 div ¼ div r r r r ¼ xi þ yj þ zk

it follows that 1 x y z ∂ x ∂ y ∂ z div r ¼ div i þ j þ k ¼ þ þ r r r r ∂x r ∂y r ∂z r 1 x ∂r 1 y ∂r 1 z ∂r ¼ À þ À þ À r r2 ∂x r r2 ∂y r r2 ∂z

and with

r2 ¼ x2 þ y2 þ z2 ∂r x ∂r y ∂r z ¼ ¼ ¼ ∂x r ∂y r ∂z r

thus 1 3 x x y y z z div r ¼ À þ þ r r r2r r2r r2r 194 4 Differential Operators whereby r 2 div ¼ Q:E:D: r r

4.4 Curl

The vector product of the nabla operator by a vector generates a differential operator linked to the direction of rotation of the coordinate system deﬁning the curl, also called rotation or whirl. In absolute notation this operator is written as

∇ Â u ¼ rot u ¼ v ð4:4:1Þ

In English literature the notation curl u is used to designate rotational of vector u, which was adopted ﬁrstly by Maxwell. The term curl literally means ring, and it designates the pseudovector ∇ Â u. With ÀÁ j j ∂ ujg ∂u ∂g v ¼ gi Â ¼ gi Â j g j þ gi Â u ∂xi ∂xi j ∂xi and rewriting expression (2.4.4)

∂g j ¼ÀΓ j gm ∂xi ki it follows for the second term of the member to the right of expression (4.4.1)

∂gm u gi Â ¼Àu Γ j gi Â gm j ∂xi j ki

The cross product of these vectors is given by 8 > > þ1 for an even number of > <> permutations of the indexes eimk i m ffiffiffi g Â g ¼ p gk ¼ > À1 for an odd number of g > > permutations of the indexes :> 0 when there are repeated indexes and substituting results in

∂ m i g uj j j ujg Â ¼Àpffiffiffi Γ À Γ g ¼ 0 ∂xi g ki ik k 4.4 Curl 195 whereby

∂u ∇ Â u ¼ v ¼ j gi Â gm ∂xi

eijk gi Â g j ¼ pffiffiffi g g k in tensorial terms

ijk e ∂uj ∇ Â u ¼ pffiffiffi g ð4:4:2Þ g ∂xi k

As a function of Ricci’s pseudotensor

∂u ∇ Â u ¼ εijk j g ð4:4:3Þ ∂xi k and with

eijk εijk ¼ pffiffiffi g the expression (4.4.3) after a cyclic permutation of the indexes i, j, k ¼ 1, 2, 3 takes the form 1 ∂uj ∂ui ∇ Â u ¼ pffiffiffi À g ð4:4:4Þ g ∂xi ∂xj k

In a space provided with metric, the curl of a vectorial function can also be deﬁned by means of its contravariant components, for these relate with its covariant components by means of the metric tensor. In an analogous way, the results for the contravariant coordinates are ÀÁ k k ‘ ∂ u g ‘ ∂u ∂g ∇ Â u ¼ g Â k ¼ g Â g þ uk k ∂x‘ ∂x‘ k ∂x‘ ∂g k Γ mg ∂ ‘ ¼ k‘ m x k k ‘ ∂u ∂u ‘ ∇ Â u ¼ g Â g þ ukΓ mg ¼ þ umΓ k g Â g ∂x‘ k k‘ m ∂x‘ m‘ k 196 4 Differential Operators

∇ ∂ k ‘ Â u ¼ ‘u g Â gk ‘ ‘j g ¼ g gj ∇ ∂ k ‘j Â u ¼ ‘u g gj Â gk ε i gj Â gk ¼ ijkg ÀÁ k ‘j i ∇ Â u ¼ εijk ∂‘u g g ð4:4:5Þ

A vectorial ﬁeld is called an irrotational ﬁeld when ∇ Â u ¼ 0, then

∂u ∂u j ¼ i ∂xi ∂xj

In space E3 the curl∇ Â uis an axial vector (vectorial density), so it is associated to an antisymmetric second-order tensor, which components are 2 3 ∂u2 ∂u1 ∂u3 ∂u1 6 0 À À 7 6 ∂x1 ∂x2 ∂x1 ∂x3 7 6 1 2 3 2 7 ij 6 ∂u ∂u ∂u ∂u 7 A ¼ 6 À 0 À 7 ð4:4:6Þ 6 ∂ 2 ∂ 1 ∂ 2 ∂ 3 7 4 x x x x 5 ∂u1 ∂u3 ∂u2 ∂u3 À À 0 ∂x3 ∂x1 ∂x3 ∂x2

∇ 1 For the space EN the curl Â u has 2 NNðÞÀ 1 independent components. In space E2 the curl is a pseudoscalar. For the Cartesian coordinates

∂u ∇ Â u ¼ e j g ð4:4:7Þ ijk ∂xi k or in a determinant form

ijk

∇ ∂ ∂ ∂ : : Â u ¼ ð4 4 8Þ ∂x1 ∂x2 ∂x3 u1 u2 u3

4.4.1 Stokes Theorem

In expression (3.3.7) with F ¼ u, and with the expression (4.4.8) Stokes theorem in vectorial notation is given by ðð þ n Á ∇ Â u dS ¼ u Á dr ð4:4:9Þ

S C 4.4 Curl 197 ab

n X 3 C3 S C P 3 i C C = h dx P 4 2 3 g 3 S i 2 C 1 = h 2 dx O g 2 X 2 g1

X 1

Fig. 4.8 Concept of curl: (a) circulation in a closed surface and (b) elementary rectangle

A more consistent deﬁnition of the curl can be formulated analyzing the circulation of the vectorial ﬁeld u in a closed surface S with upward unit normal vector n (Fig. 4.8a). Consider the elementary rectangle dS determined in the orthogonal curvilinear j 1 2 2 3 coordinate systemÀÁX , with sides h1dx and h2dx located in the plane OX X ,with j the point P x1; x2; x3 located in its center. Locally, the coordinate system X is considered as a Cartesian orthogonal system (Fig. 4.8b), with the scale factors hi, i ¼ 1, 2, 3. þ The line integral u Á dr along the perimeter of this rectangle is carried out

C dividing this perimeter into segments C1, C2, C3, C4. The center of segment C1 of 1; 2; 3 dx3 perimeter of the rectangle is given by the coordinates x x x À 2 then u Á dr ¼ u2dx2. As this length is elementary its contribution to the line integral is given by þ dx3 u Á dr ﬃ u2 x1; x2; x3 À h dx2 2 2 C1 1; 2; 3 dx3 For segment C3 with center x x x þ 2 : þ dx3 u Á dr ﬃÀu2 x1; x2; x3 þ h dx2 2 2 C3 where the negative sign indicates that the direction of the path is contrary to the coordinate axis. 198 4 Differential Operators

Adding the contributions of these two segments þ dx3 dx3 u Á dr ﬃ u2 x1; x2; x3 À À u2 x1; x2; x3 þ h dx2 2 2 2 C1þC3

The component u2 varies according to the rate

∂u2 du2 ¼À dx3 ∂x3 where the negative sign indicates that this variation decreases in the positive 1 direction of axis OX , it follows that þ ∂u2 u Á dr ﬃÀ dx3h dx2 ∂x3 2 C1þC3

2 3 and dividing by dS ¼ h2h3dx dx þ 2 1 1 ∂ðÞu h2 u Á dr ﬃÀ 3 dS h2h3 ∂x C1þC3

Adopting analogous formulations for the other segments þ dx3 dx3 ∂u3 u Á dr ffi u3 x1; x2; x3 þ À u3 x1; x2; x3 À h dx3 ffi dx2h dx3 2 2 3 ∂x2 3 C þC 2 4 þ 3 1 1 ∂ðÞu h3 u Á dr ffi 2 dS h2h3 ∂x C2þC4

Adding these contributions the result when dS ! 0is þ ∂ðÞh u3 ∂ðÞh u2 1 3 À 2 ¼ lim u Á dr ∂x2 ∂x3 dS!0 dS C or ∂ðÞh u3 ∂ðÞh u2 e Á ∇ Â u ¼ 3 À 2 1 ∂x2 ∂x3 4.4 Curl 199

For the components u1 and u2 the result is, respectively,

∂ðÞh u1 ∂ðÞh u3 e Á ∇ Â u ¼ 1 À 3 2 ∂x3 ∂x1 ∂ðÞh u2 ∂ðÞh u1 e Á ∇ Â u ¼ 2 À 1 3 ∂x1 ∂x2

Concluding that the curl components of the vectorial ﬁeld u in the direction of the upward unit normal vector to the closed surface S are given by þ 1 n Á ∇ Â u ¼ lim u Á dr ð4:4:10Þ S!0 S C

This expression is valid for any type of referential system, which shows that the curl is independent of the coordinate system. For demonstrating that expression (4.4.9) represents the Stokes theorem, let the surface S which outline is curve C, and the ﬁeld represented by the vectorial function u(r), continuous and with continuous partial derivatives in S [ C. Dividing S in N cells Si, i ¼ 1, 2, ...N, which components of the upward normal unit vectors are ni, with closed outline curves Ci (Fig. 4.8a), and with expression (4.4.10) the result for each cell of S is þ 1 ni Á ∇ Â u ¼ lim u Á dr Si!0 Si Ci

Applying this expression to point P contained in cell Si with boundary Ci, the result when the area of this cell is reduced approaching the outline P is þ ÀÁ i ðÞni Á ∇ Â u Si ¼ u Á dr þ h x Si

Ci where kkhðÞxi > 0 is a function with very small value, which decreases with the reduction of size of Si. i With the division of the surface S into N parts the result is that Nh½iðÞx > 0, then i i i Max hiðÞx < hðÞx . For N !1the result is hðÞ!x 0, so 1iN

þ XN ÀÁXN ÀÁ ∇ < i i ðÞni Á Â u Si À u Á dr h x Si ¼ h x S i¼1 i¼1 Ci 200 4 Differential Operators and with þ þ XN u Á dr ¼ u Á dr i¼1 Ci C for the outlines of the cells Si are calculated twice, but in opposite directions, whereby these parcels cancel each other, leaving only the parcel of boundary C of S. Then

þ ÀÁ i ðÞni Á ∇ Â u Si À u Á dr < h x S

As N !1the result is þ

lim ðÞni Á ∇ Â u Si ¼ u Á dr N!1 C whereby the result of the expression of Stokes theorem is ðð þ n Á ∇ Â u dS ¼ u Á dr

S C

This theorem is a particular case of the divergence theorem. To demonstrate this assertion let the vectorial function u ¼ v Â w, and an arbitrary and constant vector, then

∇ Á ðÞ¼v Â w w Á ∇ Â v and

n Á ðÞ¼v Â w w Á n Â v

Applying the divergence theorem to the function u it is written as ðð ððð u:n dS ¼ ∇ Á u dV S V

The substitution of the previous expressions in this expression shows that ððð ðð ðð ∇ Á u dV ¼ ðÞÁv Â w n dS ¼ w Á ðÞn Â v dS

V S S 4.4 Curl 201 whereby ððð ðð w Á ∇Âv ¼ w Á n Â v dS

V S

As w is arbitrary it results in ððð ðð ∇Âv ¼ n Â v dS

V S

The concept of curl of a vector u can be generalized for a space EN, in which the vector is associated to an antisymmetric tensor A, and its order depends on the dimension of the space. This tensor is generated by means of the dot product between the Ricci pseudotensor and the vector’s covariant derivative

i1 i2ÁÁÁipÀ2 i1 i2ÁÁÁipÀ2 jk A ¼ ε ∂juk ð4:4:11Þ

4.4.2 Orthogonal Curvilinear Coordinate Systems

With the expressions used in the previous item to demonstrate expression (4.4.10), there is in index notation for the curl coordinates of vector u in a curvilinear orthogonal coordinate system ∂ j ∂ i ∇ hk hju hiu : : Â u ¼ i À j gk ð4 4 12Þ h1h2h3 ∂x ∂x pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi where hi ¼ gðÞii , hj ¼ gðÞjj , hk ¼ gðÞkk are the components of the metric tensor, and the indexes in parenthesis indicate no summation. In a determinant form the result is

h1g1 h2g2 h3g3 1 ∂ ∂ ∂ ∇ Â u ¼ ð4:4:13Þ 1 2 3 h1h2h3 ∂x ∂x ∂x 1 2 3 h1u h2u h3u and with the physical components of vector u*i it follows that ∂ *j ∂ *i ∇ * hk hju hiu : : Â u ¼ i À j gk ð4 4 14Þ h1h2h3 ∂~x ∂x 202 4 Differential Operators

4.4.3 Properties

As the curl is the cross product of the nabla operator by a vectorial function, the properties of this vector product are valid. For the sum of two vectorial functions u and v:

∇ Â ðÞ¼u þ u ∇ Â u þ ∇ Â v ð4:4:15Þ and the successive applying of the curl to this sum provides

∇ Â ∇ Â ðÞ¼u þ u ∇ Â ∇ Â u þ ∇ Â ∇ Â v ð4:4:16Þ

Considering the vectorial function mu, where m is a scalar, the result of the cross product

∇ Â ðÞ¼mu m∇ Â u ð4:4:17Þ

Expressions (4.4.16) and (4.4.17) show that the curl, for these cases, is a linear operator, which is valid for the general case as will be shown in item 4.5.

4.4.4 Curl of a Tensor

The concept of curl of a vector in space EN is developed in an analogous way. For instance, for the second-order tensor Tk1k2 in space EN exists the curl of order ðÞp À 3 , given by the cross product between the Ricci pseudotensor of order p and the tensor’s covariant derivative, then

i1 i2ÁÁÁipÀ3 εi1 i2ÁÁÁipÀ3 jk1k2 ∂ : : A ¼ jTk1k2 ð4 4 18Þ

Expression (4.4.18) shows that the Ricci pseudotensor is the generator of the antisymmetric tensor that represents the rotational of the tensor. In absolute notation the result is

∇ Â T ¼ ðÞ∇ T E ð4:4:19Þ where E is the Ricci pseudotensor. In the particular case of the space E4 the curl for the second-order tensor Tk‘ is given by the components of vector Ai:

i ijk‘ A ¼ ε ∂jTk‘ 4.4 Curl 203

Assuming that the second-order tensor is decomposed into two tensors, one symmetric and the other antisymmetric

T ¼ S þ A then

rotT ¼ rotS þ rotA

The components of the curl of the symmetric tensor are given by εi1 i2ÁÁÁipÀ3 jk1k2 ∂ jSk1k2 , i.e., are obtained by means of the dot product of the Ricci pseudotensor (antisymmetric) by the symmetric tensor which is null, whereby rotS ¼ 0. Concluding that the curl of a symmetric tensor is null, and that only the antisymmetric tensor A generates the rotational of tensor T. In the particular case in which rotT ¼ 0 the tensor T defines an irrotational tensorial field. The definition of curl of a second-order tensor can be applied to a tensor of order p > 2, whereby

i1 i2ÁÁÁiqÀpÀ1 εi1 i2ÁÁÁiqÀpÀ1 jk1ÁÁÁkp ∂ : : A ¼ jTk1ÁÁÁkp ð4 4 20Þ being ðÞq À 1 the order of the Ricci pseudotensor, and ðÞq À p À 1 the order of the antisymmetric tensor that represents the curl of the tensor. Exercise 4.12 Calculate: (a) ∇ Â ϕu; (b) u Â ðÞ∇ Â v ; (c) ∇ Â ðÞu Â v . (a) The curl of the ﬁeld deﬁned by the product of a scalar function ϕ(xi) for a vectorial function u is given by ÀÁ k k ∂ ϕukg ∂ϕ ∂u ∂g ∇ Â ϕu ¼ g j Â ¼ g j Â u gk þ ϕ k gk þ ϕu ∂xj ∂xj k ∂xj k ∂xj

Substituting expression (2.4.4)

∂gk ¼ÀΓ k gm ∂xj mj

in this expression ∂ϕ ∂u ∇ Â ϕu ¼ g j Â u gk þ ϕ k gk À ϕu Γ k gm ∂xj k ∂xj k mj

The permutation of the indexes k $ m in the last term provides ∂ϕ ∂u ∇ Â ϕu ¼ u þ ϕ k À ϕu Γ m g j Â gk ∂xj k ∂xj m kj 204 4 Differential Operators

and with expressions

∂u g j Â gk ¼ εijkg ∂ u ¼ k À u Γ m i j k ∂xj m kj

it follows that ∂ϕ ∂ϕ ÀÁ ∇ Â ϕu ¼ u þ ϕ∂ u εijkg ¼ u εijkg þ ϕ ∂ u εijkg ∂xj k j k i ∂xj k i j k i

Putting ∂juk ¼ ∇uk

∇ Â ϕu ¼ ∇ϕ Â u þ ϕ∇ Â u ) rotϕu ¼ gradϕ Â u þ ϕrotu

For the Cartesian coordinates ∂ϕ ∂u ∇ Â ϕu ¼ ε u þ ε ϕ k g ijk k ∂xj ijk ∂xj i

It is veriﬁed for this case that the curl is not a linear operator. For ϕ ¼ m, where m is a constant, the result with expression (4.4.17) is that this operator’s linearity is valid by this particular case. (b) The curl ∇ Â v is given by

kmn k ∇ Â v ¼ w ¼ ε ∂mvng

then

ε j k u Â w ¼ ijku w gi

whereby substituting

∇ ε jεkmn∂ ε εkmn j∂ u Â ðÞ¼Â v ijku mvngi ¼ ijk u mvngi

and with

ε εkmn δmn δmn δ mδ n δ mδ n ijk ¼ ij ij ¼ i j À j i

is it follows that hiÀÁ ∇ δ mδ n j∂ δ mδ n j∂ j∂ j∂ u Â ðÞ¼Â v i j u mvn À j i u mvn gi ¼ u ivj À u jvi gi 4.4 Curl 205

For the Cartesian coordinates the result is ∂v ∂v u Â ðÞ¼∇ Â v u j À u i g j ∂xi j ∂xj i

‘ m n u Â v ¼ w ¼ w g‘ ¼ ε‘mnu v g‘

thereby

ij‘ ‘ ij‘ m n ij m n ∇ Â ðÞ¼u Â v ε ∂ w g ¼ ε ∂ ðÞε‘ u v g ¼ δ ∂ ðÞu v g ÀÁj i j mn ÂÃi mn j i δ i δ j δ i δ j ∂ m n ∂ i j ∂ j i ¼ m n À n m jðÞu v gi ¼ jðÞÀu v jðÞu v gi

For the Cartesian coordinates ∂ui ∂vj ∂u j ∂vi ∇ Â ðÞ¼u Â v vj þ ui À vi À u j g ¼ v Á ∇ Â u À u Á ∇ Â v ∂xj ∂xj ∂xj ∂xj i

Exercise 4.13 Calculate ∇ Â u for the vector u expressed in cylindrical coordinates. For the cylindrical coordinates the result is h1 ¼ 1, h2 ¼ r, h3 ¼ 1 and (ur, uθ, uz). The determinant is given by the expression (4.4.13)

gr rgθ gz

1 ∂ ∂ ∂ ∇ Â u ¼ r ∂ r ∂ θ ∂ z x x x ur ruθ uz

which development provides 1 ∂uz ∂uθ ∂ur ∂uz 1 ∂ruθ 1 ∂ur ∇ Â u ¼ À g þ À gθ þ À g r ∂θ ∂z r ∂z ∂r r ∂r r ∂θ z

Exercise 4.14 Calculate ∇ Â u for the vector u expressed in spherical coordinates. For the spherical coordinates the result is h1 ¼ 1, h2 ¼ r, h3 ¼ r sin ϕ and (ur, uϕ, uθ). The determinant given by expression (4.4.13) 206 4 Differential Operators

ϕ gr rgκ r sin gθ

1 ∂ ∂ ∂ ∇ Â u ¼ 2 ϕ ϕ θ r sin ∂xr ∂x ∂x

ur ruϕ r sin ϕuθ which development provides ∂ ϕ ∂ ∂ ϕ ∂ ∇ 1 uθ sin uϕ 1 uθr sin 1 ur Â u ¼ ϕ ∂ϕ À ∂θ gr þ ϕ ∂ À ∂θ r sin r sin r r 1 ∂ruϕ ∂ur gϕ þ À gθ r ∂r ∂ϕ

Exercise 4.15 Let r be the position vector of the point in space E3, show that: (a) ∇ Â r ¼ 0; (b) ϕ(r)r is irrotational. (a) With the deﬁnition of curl ∂ ∂r ∇ Â r ¼ g Â r ¼ g Â i ∂xi i ∂xi

but

∂r ¼ g ∂xi i

then

∇ : : : Á r ¼ gi Â gi ¼ 0QE D

(b) A condition that a vectorial function must fulﬁll so that the ﬁeld that it represents is irrotational is

∇ Â ½¼ϕðÞr r 0

and putting

ϕðÞ¼r ψ

it follows that hi 0 ∇ Â ½¼ϕðÞr r gradϕ Â r þ ϕ∇ Â r ¼ ϕ ðÞr gradr Â r þ ϕðÞÁr ∇ Â r 4.5 Successive Applications of the Nabla Operator 207

but

∇ Â r ¼ 0

then 0 1 ∇ Â ½¼ϕðÞr r ϕ ðÞr r Â r r

r Â r ¼ 0

thus

∇ Â ½¼ϕðÞr r 0Q:E:D:

4.5 Successive Applications of the Nabla Operator

The operator ∇ can be applied successively to a ﬁeld. The number of combinations of two out of the three differential operators, the gradient, the divergence, and the curl, are 32 ¼ 9 types of double operators. The combinations ∇ Á ðÞ∇ Á u and ∇ Â ðÞ∇ Á u have no mathematical meaning.

4.5.1 Basic Relations

(1) ∇ Á ðÞ∇ Â u

The curl of a vectorial function is given by

∇ εk‘m∂ k : : Â u ¼ ‘umgk ¼ w ¼ w gk ð4 5 1Þ

k k‘m w ¼ ε ∂‘um ð4:5:2Þ 208 4 Differential Operators

then ÀÁ ∂ wkg ∂wk ∂g ∇ Á ðÞ¼∇ Â u gi Á k ¼ gi Á g þ wk k ∂xi ∂xi k ∂xi

and with expression

∂g k ¼ Γ n g ∂xi ki n

it follows that ∂wk ∇ Á ðÞ¼∇ Â u gi Á g þ wkΓ n g ∂xi k ki n

The permutation of indexes n $ k in the second member in parenthesis provides ∂wk ∇ Á ðÞ¼∇ Â u þ wnΓ k gi Á g ∂xi ni k

and with

i δ i g Á gk ¼ k

the result is ∂wk ∇ Á ðÞ¼∇ Â u þ wnΓ k ¼ ∂ wk ∂xk nk k

Substituting expression (4.5.2)

ÀÁ k‘m k‘m k‘m e ∇ Á ðÞ¼∇ Â u ∂k ε ∂‘um ¼ ε ∂kðÞ¼∂‘um pffiffiffi ∂kðÞ∂‘um g

and interchanging the indexes i, j, k ¼ 1, 2, 3 cyclically

1 ∇ Á ðÞ¼∇ Â u pffiffiffi ðÞð∂k∂‘um À ∂‘∂kum 4:5:3Þ g

whereby

∇ Á ðÞ¼∇ Â u 0 ð4:5:4Þ 4.5 Successive Applications of the Nabla Operator 209

Vector ∇ Â u represents a vectorial field associated to the vectorial function u. Expression (4.5.4) defines the condition of existence for this function. The property of the field defined by the curl of the vectorial function u shows that the divergence of this field is null, i.e., the field is solenoidal. In a reciprocal way for a solenoidal field ∇ Á ðÞ¼∇ Â u 0 a solenoidal vector v can be determined, such that v ¼ ðÞ∇ Â u . In this case the vector v derives from the potential function u, being linked to this function. (2) ∇ Â ðÞ∇ϕ

The gradient of a scalar function ϕ(xi) is given by

∂ϕ ðÞ¼∇ϕ u ¼ gk ¼ u gk ð4:5:5Þ ∂xk k

then

∇ ∇ϕ ∇ εijk∂ Â ðÞ¼u ¼ jukgi

it follows that ! ! ∂ϕ ∂2ϕ ijk ∂2ϕ ijk ijk e ∇ Â ðÞ¼∇ϕ ε ∂j g ¼ ε g ¼ pffiffiffi g ∂xk i ∂xj∂xk i g ∂xj∂xk i

Interchanging the indexes i, j, k ¼ 1, 2, 3 cyclically ! 1 ∂2ϕ ∂2ϕ ∇ Â ðÞ¼∇ϕ pffiffiffi À g g ∂xj∂xk ∂xk∂xj i

∂2ϕ ∂2ϕ ¼ ∂xj∂xk ∂xk∂xj

it results in

∇ Â ðÞ¼∇ϕ 0 ð4:5:6Þ

The field that fulfills the condition given by expression (4.5.6) is called a conservative field, i.e., every vectorial field with potential is an irrotational field. Let ∇ϕ ¼ u the result is ∇ Â ðÞ¼∇ϕ ∇ Â u ¼ 0, then 210 4 Differential Operators

∂u ∂u i ¼ j ∂xj ∂xi

i i As uidx is an exact differential it follows that for a scalar function ϕ(x ): ÀÁ ∂ϕ ∂ϕ ϕ xi ¼ dxi ) u À dxi ¼ 0 ∂xi i ∂xi

whereby

∂ϕ u ¼ i ∂xi

This analysis shows that the vector u can be considered as the gradient of a scalar function ϕ(xi) as long as it fulfills the condition ∇ Â u ¼ 0. Expression (4.5.6) can be demonstrated changing only the order of the operations, for ðÞ∇ Â ∇ ϕ ¼ 0, where the term in parenthesis indicates the cross product of a vector by itself, which results in the null vector. The condition ∇ Â u ¼ 0 being u ¼ ∇ϕðÞxi , where the scalar field defined by the function ϕ(xi) is divided into families of level surfaces ϕðÞ¼xi C, which do not intersect, so they form level surface “layers,” leads to the denomination of lamellar field. (3) ∇ Â ðÞ∇ Â u

For ∇ Â ðÞ∇ Â u using the Grassmann identity

u Â ðÞ¼v Â w ðÞu Á w v À ðÞu Á v w

whereby

∇ Â ðÞ¼∇ Â u ∇∇ðÞÀÁ u ∇ Á ðÞ∇u ð4:5:7Þ

In terms of the vector coordinates it follows that ÀÁ k ‘j t t ∇ Â u ¼ εtjk ∂‘u g g ¼ w ¼ wtg ÀÁ k ‘j wt ¼ εtjk ∂‘u g ∂ðÞw gt ∂w ∂gt ∇ Â w ¼ gs Â t ¼ gs Â t gt þ w ∂xs ∂xs t ∂xs ∂gt ¼ÀΓ t gn ∂xs sn 4.5 Successive Applications of the Nabla Operator 211 ∂w ∂w ∇ Â w ¼ gs Â t gt À w Γ t gn ¼ t À w Γ n gs Â gt ∂xs t sn ∂xs n st gs Â gt ¼ εrstg r ∂w ∇ Â w ¼ εrst t À w Γ n g ∂xs n st r ∂ εrst ¼ 0 s ÀÁ ∇ εrst ∂ εrstε ∂ ∂ k ‘j Â w ¼ ðÞswt gr ¼ ijk s ‘u g gr ‘ ∂ g j ¼ 0 s ÀÁ ∇ εrstε ∂ ∂ k ‘j Â w ¼ tjk s ‘u g gr

and with

εrstε δrs δrs δ rδ s δ sδ r tjk ¼ jk jk ¼ j k À j k

it follows that hiÀÁ ÀÁ ∇ ∇ δ rδ s ∂ ∂ k ‘j δ sδ r ∂ ∂ k ‘j Â ðÞ¼Â u j k s ‘u g À j k s ‘u g gr

whereby ÂÃÀÁÀÁ ∇ ∇ ∂ ∂ k ‘r ∂ ∂ r ‘j : : Â ðÞ¼Â u k ‘u g À j ‘u g gr ð4 5 8Þ

hiÀÁ ∇ ∇ ∂ ∂r k ‘r ∂ ∂j r : : Â ðÞ¼Â u k u g À j u gr ð4 5 9Þ

For the Cartesian coordinates the result is ! ∂2 ∂2 ∇ ∇ uk ur : : Â ðÞ¼Â u À gr ð4 5 10Þ ∂xk∂xr ∂2xj

(4) ∇∇ðÞÁ u

For the gradient of a vector ∂ui ∇ Á u ¼ þ ujΓ i ∂xi ji 212 4 Differential Operators

then ∂ ∂ui ∇∇ðÞ¼Á u gm þ ujΓ i ∂xm ∂xi ji

The development provides ! ∂2ui ∂uj ∂Γ i ∇∇ðÞ¼Á u gm þ Γ i þ uj ji ∂xm∂xi ∂xm ji ∂xm

or ÀÁ i m ∇∇ðÞ¼Á u ∂m ∂iu g ð4:5:11Þ

and with

0 u ¼ ϕ ðÞr

du 0 ¼ ϕ ðÞr dr

it follows that ÀÁ ∇∇ mk∂ ∂ i ðÞ¼Á u g m iu gk

whereby ÀÁ ∇∇ ∂k ∂ i : : ðÞ¼Á u iu gk ð4 5 12Þ

Exercise 4.16 Let ϕ(xi) and ψ(xi) be scalar functions, show that: (a) ∇ Â ðÞ¼ψ∇ϕ þ ϕ∇ψ 0; (b) ∇ Á ðÞ¼∇ϕ Â ∇ψ 0; (c) trðÞ¼∇ u ∇ Á u. (a) Putting

∇ϕ ¼ u ∇ψ ¼ v

then

∇ Â ðÞ¼ψ∇ϕ þ ϕ∇ψ ∇ Â ðÞ¼ψu þ ϕv ∇ Â ψu þ ∇ Â ϕv

and with the expression shown in Exercise 4.12 it follows that

∇ Â ψu ¼ ∇ψ Â u þ ψ Â ∇u ¼ ∇ψ Â ∇ϕ þ ψ∇ Â ∇ϕ ∇ Â ϕv ¼ ∇ϕ Â v þ ϕ Â ∇v ¼ ∇ϕ Â ∇ψ þ ϕ∇ Â ∇ψ 4.5 Successive Applications of the Nabla Operator 213

and with expression (4.5.6)

∇ Â ∇ϕ ¼ ∇ Â ∇ψ ¼ 0

and

∇ϕ Â ∇ψ ¼À∇ψ Â ∇ϕ

then

∇ Â ðÞ¼ψ∇ϕ þ ϕ∇ψ 0Q:E:D:

(b) Putting

∇ϕ ¼ u ∇ψ ¼ v ∇ Â ðÞ¼∇ϕ Â ∇ψ 0

then

∇ Á ðÞ¼∇ϕ Â ∇ψ ∇ Á ðÞu Â v

With expression deducted in Exercise 4.7b it follows that

∇ Á ðÞ¼u Â v v Á ∇ Â u À u Á ∇ Â v

∇ Â ðÞ¼∇ϕ Â ∇ψ 0

and with expression (4.5.6)

∇ Â ∇ϕ ¼ ∇ Â ∇ψ ¼ 0

then

∇ Á ðÞ¼∇ϕ Â ∇ψ 0Q:E:D:

i k ∇ u ¼ ðÞ∂iuk g g

the result for i ¼ k is

trðÞ¼∇ u ∂iui1

and comparing this result with expression (4.3.6) 214 4 Differential Operators

∇ Á u ¼ ∂iui

it is veriﬁed that

tr ðÞ¼∇ u ∇ Á u Q:E:D:

4.5.2 Laplace Operator

The combination of the divergence and the gradient, in this order, deﬁnes the Laplace operator or Laplacian

2 k k ∇ ¼ ∇ Á ∇ ¼ Δ ¼ Dk Á D ¼ ∂k∂ ¼ divgrad ¼ lap ð4:5:13Þ

A few authors denominate this operator of differential parameter of the second order of Beltrami, and use the spelling Δ 2 to represent it. With the expression the contravariant derivative

k kj ∂ ¼ g ∂j it follows that for the Laplacian of an arbitrary tensor ÀÁ ÀÁ ∂ ∂k ÁÁÁ ∂ ∂k ÁÁÁ ∂ kj∂ ÁÁÁ kj∂ ∂ ÁÁÁ ∂j∂ ÁÁÁ k TÁÁÁ ¼ k TÁÁÁ ¼ k g jTÁÁÁ ¼ g k jTÁÁÁ ¼ jTÁÁÁ that shows that the Laplacian operator is commutative. For Cartesian coordinates the covariant and contravariant derivatives are equal

∂ÁÁÁ ∂ÁÁÁ ∂ ¼ ∂k ¼ k ∂xk ∂xk k ∂k ¼ ∂ resulting for the Laplacian ð ð m dϕðÞ¼r 1 dr r2

4.5.2.1 Laplacian of a Scalar Function

The Laplacian of the scalar function ϕ(xi) expresses in a curvilinear coordinate system, with covariant derivative given by 4.5 Successive Applications of the Nabla Operator 215

m ϕðÞ¼r 1 þ m r 2 thus

HðÞ¼ÁÁÁ gi∇ gj∇ðÞÁÁÁ

The development of the covariant derivative of the term in parenthesis provides ÀÁ ϕ xi

The contracted Christoffel symbol ÀÁpffiffiffi 1 ∂ g Γ k ¼ pffiffiffi mk g ∂xm provides ÀÁpffiffiffi ∂ ∂ϕ ∂ϕ 1 ∂ g ∇2ϕ ¼ gkj þ gmj pffiffiffi ∂xk ∂xj ∂xj g ∂xm whereby ∂ ∂ ∂ ∂ ∂ ∂ HðÞ¼ϕ gi gj ϕ ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 2 3 ∂2ϕ ∂2ϕ ∂2ϕ 6 7 6 ∂ 1∂ 1 ∂ 1∂ 2 ∂ 1∂ 3 7 6 x x x x x x 7 6 7 6 ∂2ϕ ∂2ϕ ∂2ϕ 7 ¼ 6 7gi gj 6 ∂ 2∂ 1 ∂ 2∂ 2 ∂ 2∂ 3 7 6 x x x x x x 7 4 ∂2ϕ ∂2ϕ ∂2ϕ 5 ∂x3∂x1 ∂x3∂x2 ∂x3∂x3 or ÀÁ 2 ik ∇ ϕ ¼ g ∂jkϕ ð4:5:14Þ it follows that ! ∂2ϕ ∂ϕ ∇2ϕ ¼ gik À Γ m ð4:5:15Þ ∂xj∂xk ∂xm jk 216 4 Differential Operators

In vectorial notation

∇ Á ðÞ¼∇ϕ ∇2ϕ ð4:5:16Þ or

divðÞ¼ gradϕ ∇2ϕ ð4:5:17Þ

δ In space E3 and in orthogonal Cartesian coordinates the result is gij ¼ ij, then the Laplacian of a scalar function is the sum of its derivatives of the second order

∂2ϕ ∇2ϕ ¼ ð4:5:18Þ ∂xj∂xj

4.5.3 Properties

The Laplacian of the sum of two scalar functions ϕ(xi) and ψ(xi) provides

∇2ðÞ¼ϕ þ ψ ∇ Á ∇ðÞ¼ϕ þ ψ ∇ Á ðÞ¼∇ϕ þ ∇ψ ∇ Á ∇ϕ þ ∇ Á ∇ψ whereby

∇2ðÞ¼ϕ þ ψ ∇2ϕ þ ∇2ψ

For the function mϕ(xi), where m is a scalar, this operator provides

∇2ðÞ¼mϕ ∇ Á ∇ðÞ¼mϕ ∇ Á m∇ðÞ¼ϕ m∇ Á ∇ðÞϕ whereby

∇2ðÞ¼mϕ m∇ Á ∇ðÞϕ

These two demonstrations prove that the Laplacian is a linear operator. The gradient of the product of two scalar functions is given by

∇ðÞ¼ϕψ ψ∇ðÞþϕ ϕ∇ðÞψ then

∇ Á ∇ðÞ¼ϕψ ∇ Á ½ψ∇ðÞþϕ ϕ∇ðÞψ 4.5 Successive Applications of the Nabla Operator 217

Putting

∇ϕ ¼ u ∇ψ ¼ v thus

∇ Á ½¼ψ∇ðÞþϕ ϕ∇ðÞψ ∇ Á ðÞψu þ ϕv

Applying the distributive property of the divergence to this expression, and using the expression deducted in Exercise 4.7a it follows that

∇ Á ðÞ¼ψu þ ϕv ∇ Á ψu þ ∇ Á ϕv ∇ Á ψu ¼ ∇ψ Á u þ ψ∇u ∇ Á ϕv ¼ ∇ϕ Á v þ ϕ∇v then

∇2ðÞ¼ϕψ ðÞþ∇ψ Á u þ ψ∇u ðÞ∇ϕ Á v þ ϕ∇v ¼ ðÞþv Á u þ ψ∇u ðÞu Á v þ ϕ∇v ¼ ψ∇u þ ϕ∇v þ 2vðÞÁ u but

∇u ¼ ∇∇ϕ ¼ ∇2ϕ ∇v ¼ ∇∇ψ ¼ ∇2ψ whereby substituting

∇2ðÞ¼ϕψ ψ∇2ðÞþϕ ϕ∇2ðÞþψ 2∇ϕ∇ψ

An equation involving the Laplacian of a scalar function that appears in various problems of physics and engineering, the Laplace equation, is given by ÀÁ ∇2ϕ xi ¼ 0 ð4:5:19Þ

The function ϕðÞ¼xi x4z þ 3xy2 À zxy þ 1 that fulﬁlls this equation is said to be harmonic. In addition to satisfying the Laplace equation it must be regular in the domain D, with partial derivatives of the ﬁrst order continuous in the interior and in the boundary of D, and derivatives of the second order also continuous in D, which can be discontinuous in the boundary of this domain. The successive applying of the Laplacian to a scalar function (r; z sin θ;eθ cos z ) results in the bi-harmonic equation ÀÁ ϕ xi ¼ xy þ yz þ xz ð4:5:20Þ 218 4 Differential Operators

For

∂ÁÁÁ ð4:5:21Þ ∂t2 where ψ(xi) is a scalar function, this partial differential equation is called Poisson’s equation. As a consequence of the deﬁnition of the Laplacian the result is that ∇2m ¼ 0, where m is a scalar. The Laplacian of a scalar function ϕ(xi) is a scalar, then its physical components are equal to its ordinary components.

4.5.4 Orthogonal Coordinate Systems

With the gradient of the scalar function ϕ(xi):

ÀÁ ∂ϕ ∇ϕ xi ¼ gi ¼ u ∂xi and the orthogonal components of the vectorial function u given by ∂ ∇ 1 uih1h2h3 Á u ¼ i h1h2h3 ∂x hi results for the Laplacian of this function expressed in an orthogonal coordinate system ∂ ∂ϕ ∇2ϕ 1 h1h2h3 : : ¼ i i ð4 5 22Þ h1h2h3 ∂x hi ∂x where h1, h2, h3 are the components of the metric tensor.

4.5.5 Laplacian of a Vector

With expression (4.5.7)

∇2u ¼ ∇∇ðÞÀÁ u ∇ Â ðÞ∇ Â u and substituting expressions (4.5.12) and (4.5.9) this expression becomes 4.5 Successive Applications of the Nabla Operator 219

ÀÁhiÀÁ ∇2 ∂k ∂ i ∂ ∂r k ∂ ∂j r : : u ¼ gk iu À k u À j u gr ð4 5 23Þ

The change of the indexes k ! r in the ﬁrst term to the right and indexes k ! i in the second term to the right provides hiÀÁÀÁ ∇2 ∂r∂ i ∂ ∂r i ∂ ∂j r u ¼ iu À i u þ j u gr

r i r i ∂ ∂iu ¼ ∂i∂ u thus ∇2 ∂ ∂j r : : u ¼ j u gr ð4 5 24Þ

4.5.6 Curl of the Laplacian of a Vector

The curl of the Laplacian of a vector is ∇ Â ∇2u, and it can be developed by means of the Grassmann formula

∇2u ¼ divgradu ¼ ∇∇ Á u À ∇ Â ∇ Â u ð4:5:25Þ or

∇ Â ∇ Â u ¼ ∇∇ Á u À ∇2u

The curl of this expression is given by

∇ Â ∇ Â ∇ Â u ¼ ∇ Â ∇∇ Á u À ∇ Â ∇2u ð4:5:26Þ or

∇ Â ∇ Â ∇ Â u ¼ ∇∇ Á ðÞÀ∇ Â u ∇ Á ∇∇ðÞÂ u ð4:5:27Þ

Expressions (4.5.4) and (4.5.6) show, respectively, that

∇ Á ðÞ¼∇ Â u 0

∇ Â ∇ϕ ¼ 0 220 4 Differential Operators whereby the result for expression (4.5.26)is

∇ Â ∇ Â ∇ Â u ¼ ∇ Â ∇∇ Á u À ∇ Â ∇2u ¼ ∇ Â ∇ϕ À ∇ Â ∇2u ¼À∇ Â ∇2u and for expression (4.5.27)

∇ Â ∇ Â ∇ Â u ¼À∇ Á ∇∇ðÞÂ u

The result of these two expressions is

∇ Â ∇2u ¼ ∇2ðÞ∇ Â u ð4:5:28Þ or

rotlapu ¼ laprotu ð4:5:29Þ

It is concluded that the operators ∇2 and ∇Â are commutative when applied to vector u.

4.5.7 Laplacian of a Second-Order Tensor

The gradient of a second-order tensor is given by

∇ ∂ ij m T ¼ mT g gi gp and the divergence of the tensor deﬁned by the previous expression stays

∇ ∇ k ∂ ∂ ij m ∂ ∂ ij m k Á T ¼ g Á k mT g gi gp ¼ k mT g gi gp Á g ∂ ∂ ij m δ k ¼ k mT g gi p whereby

∇ ∇ ∂ ∂ ij m : : Á T ¼ p mT g gi ð4 5 30Þ is a second-order tensor. Exercise 4.17 Calculate ∇2ϕ for the scalar function ϕ(xi) expressed in cylindrical coordinates. The tensorial expression that defines the Laplacian is 4.5 Successive Applications of the Nabla Operator 221 1 ∂ pffiffiffi ∂ϕ ∇2ϕ ¼ pffiffiffi ggkj g ∂xk ∂xj and for the cylindrical coordinates

1 g11 ¼ 1 g22 ¼ g33 ¼ 1 r2 ∂ϕ 1 ∂ϕ ∂ϕ ∇ϕ ¼ g þ gθ þ g ∂r r r ∂θ ∂z z it follows that 1 ∂ ∂ϕ ∂ 1 ∂ϕ ∂ ∂ϕ ∇2ϕ ¼ r þ þ r r ∂r ∂r ∂θ r ∂ϕ ∂z ∂z then

1 ∂ϕ ∂2ϕ 1 ∂2ϕ ∂2ϕ ∇2ϕ ¼ þ þ þ r ∂r ∂r2 r2 ∂θ2 ∂z2

Exercise 4.18 Calculate ∇2ϕ for the scalar function ϕ(xi) expressed in spherical coordinates. The tensorial expression that defines the Laplacian is 1 ∂ pffiffiffi ∂ϕ ∇2ϕ ¼ pffiffiffi ggkj g ∂xk ∂xj and for the spherical coordinates

1 1 g11 ¼ 1 g22 ¼ g33 ¼ r2 r2 sin 2ϕ ∂ϕ 1 ∂ϕ 1 ∂ϕ ∇ϕ ¼ g þ gϕ þ gθ ∂r r r ∂ϕ r sin ϕ ∂xθ it follows that 1 ∂ ∂ϕ ∂ 1 ∂ϕ ∂ 1 ∂ϕ ∇2ϕ ¼ r2 sin ϕ þ r2 sin ϕ þ r2 sin ϕ r2 sin ϕ ∂r ∂r ∂ϕ r2 ∂ϕ ∂θ r2 sin2 ϕ ∂θ then 222 4 Differential Operators

2 ∂ϕ ∂2ϕ 1 ∂ ∂ϕ 1 ∂2ϕ ∇2ϕ ¼ þ þ sin ϕ þ r ∂r ∂r2 r2 sin ϕ ∂ϕ ∂ϕ r2 sin 2ϕ ∂θ2

ExerciseÀÁ 4.19 Let r be the position vector of the point in space E3, show that: (a) ∇2 x ∇2 n nÀ2 ∇2ϕ ϕ00 2 ϕ0 ∇2 r3 ¼ 0; (b) ðÞ¼r r nnðÞþ 3 r r; (c) ðÞ¼r ðÞþr r ðÞr ; (d) for ϕ ϕ m1 ðÞ¼r 0 the result is ðÞ¼r r þ m2, where m1, m2 are constant. (a) With the deﬁnition of Laplacian ! x ∂2 ∂2 ∂2 x ∇2 ¼ þ þ r3 ∂x2 ∂y2 ∂z2 r3

and for the derivative with respect to the variable x ∂2 x ∂ ∂ x ∂ 1 3x ∂r ¼ ¼ À ∂x2 r3 ∂x ∂x r3 ∂x r3 r4 ∂x

but ∂r 2r ¼ 2x ∂x

then ∂2 x ∂ 1 3x x 3 ∂r 6x 15x2 ∂r 9x 15x3 ¼ À ¼À À þ ¼À þ ∂x2 r3 ∂x r3 r4 r r4 ∂x r5 r6 ∂x r5 r2

In an analogous way for the other derivatives it follows that ∂2 y 3x 15xy ∂2 y 3x 15xz ¼À þ ¼À þ ∂y2 r3 r5 r7 ∂z2 r3 r5 r7

Then

3 ∇2 x 9x 15x 3x 15xy 3x 15xz 3 ¼À 5 þ 2 À 5 þ 7 À 5 þ 7 r r r r r r r x ∇2 ¼ 0Q:E:D: r3

(b) Putting

∇2ðÞ¼rnr ∇∇½Á ðÞrnr 4.5 Successive Applications of the Nabla Operator 223 it follows that ÂÃÀÁ ∇2ðÞ¼rnr ∇∇½¼ðÞÁrn r þ rn∇ Á r ∇ nrnÀ3r Á r þ 3rn ÂÃÀÁ ¼ ∇ nrnÀ3r2 þ 3rn ¼ ðÞn þ 3 ∇rn

then

∇2ðÞ¼rnr ðÞn þ 3 nrÀ2r Q:E:D:

∇2ϕðÞ¼r ∇ Á ½∇ϕðÞr

it follows that hi 0 1 0 1 0 1 0 ∇ Á ½¼∇ϕðÞr ∇ Á ϕ ðÞr ∇r ¼ ∇ Á ϕ ðÞr r ¼ ϕ ðÞr ∇ Á r þ r Á ∇ ϕ ðÞr r r r

but

∇ Á r ¼ 3

so 3 0 d 1 0 ∇ Á ½¼∇ϕðÞr ϕ ðÞr ∇ Á r þ r Á ϕ ðÞr ∇r r dr r 3ϕ0 1 ϕ0 1ϕ00 1 ¼ ðÞþr r ÁÀ2 ðÞþr ðÞr r r r r r 3 0 1 0 1 00 1 ¼ ϕ ðÞþr À ϕ ðÞþr ϕ ðÞr r Á r r r2 r r 3 0 1 0 1 00 1 ¼ ϕ ðÞþr À ϕ ðÞþr ϕ ðÞr r2 r r2 r r

then

2 0 00 ∇ Á ½¼∇ϕðÞr ϕ ðÞþr ϕ ðÞr r

(d) Let

00 00 2 0 00 2 0 ϕ ðÞr 2 ∇2ϕðÞ¼r ϕ ðÞþr ϕ ðÞ)r ϕ ðÞþr ϕ ðÞ¼r 0 ) ¼À r r ϕ0 ðÞr r 224 4 Differential Operators

Putting

0 du 0 u ¼ ϕ ðÞr ) ¼ ϕ ðÞr dr

then

du 2 ¼À dr u r

and integrating ð ð du 2 ¼À dr u r

it follows that ÀÁ m ‘nuðÞ¼À‘nr2 þ ‘nmðÞ¼‘n 1 1 r2

or 0 m 0 m ‘nϕ ðÞ¼r ‘n 1 ) ϕ ðÞ¼r 1 r2 r2

Integrating ð ð m dϕðÞ¼r 1dr r2

then m ϕðÞ¼r 1 þ m Q:E:D: r 2

4.6 Other Differential Operators

4.6.1 Hesse Operator

The operator defined on a scalar field, given by the tensorial product of two nabla operators applied to the scalar function that field represents 4.6 Other Differential Operators 225

HðÞ¼ÁÁÁ gi∇ gj∇ðÞÁÁÁ ð4:5:31Þ

i In matrix form the scalar function ϕ(x ) in Cartesian coordinates in the space E3 is ∂ ∂ ∂ ∂ ∂ ∂ HðÞ¼ϕ gi gj ϕ ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3 2 3 ∂2ϕ ∂2ϕ ∂2ϕ 6 7 6 ∂x1∂x1 ∂x1∂x2 ∂x1∂x3 7 6 7 6 ∂2ϕ ∂2ϕ ∂2ϕ 7 6 7 i j : : ¼ 6 7g g ð4 5 32Þ 6 ∂x2∂x1 ∂x2∂x2 ∂x2∂x3 7 4 5 ∂2ϕ ∂2ϕ ∂2ϕ ∂x3∂x1 ∂x3∂x2 ∂x3∂x3

This operator is a symmetric second-order tensor, is called Hessian or Hesse operator in homage to Ludwig Otto Hesse (1881–1874).

4.6.2 D’Alembert Operator

The differential operator deﬁned by the expression

1 ∂ÁÁÁ □ ¼ ∇2ÁÁÁþ ð4:5:33Þ c2 ∂t2

∂ÁÁÁ where c is a scalar and denotes the differentiation with respect to the time t,is ∂t2 called D’Alembert or D’Alembertian operator in homage to Jean Le Rond d’Alembert (1717–1783). The applying of this operator to a ﬁeld represented by the scalar function that depends on the position vector and the time provides as a result the scalar function

ÀÁ ÀÁ1 ∂ϕðÞxi; t □ ϕ xi; t ¼ ∇2ϕ xi; t þ ð4:5:34Þ c2 ∂t2

If the ﬁeld is represented by a vectorial function the result is the vector

ÀÁ ÀÁ1 ∂uðÞxi; t □ u xi; t ¼ ∇2u xi; t þ ð4:5:35Þ c2 ∂t2

The notation □... was initially applied by Cauchy to represent the Laplacian. The D’Alembertian is the four-dimensional equivalent to the Laplacian. 226 4 Differential Operators

Problems 4.1 Calculate the gradient of the scalar functions: 2 2 (a) ϕðÞ¼xi xy þ yz þ xz; (b) ϕðÞ¼xi xex þy . 2 2 2 2 Answer: (a) ðÞy þ z i þ ðÞx þ z j þ ðÞx þ y k; (b)ðÞ 1 þ 2x2 ex þy i þ 2xyex þy j 2 2 4.2 Calculate the directional derivative of the scalar function ϕ ¼ 2ðÞx1 þ 3ðÞx2 2 þðÞx3 at the point (2; 1; 3) in the direction of vector uðÞ1; 0; À2 . Answer: À1:789. 4.3 Calculate div ui with the vector u expressed in cylindrical coordinates by its covariant components (r; z sin θ;eθ cos z). Answer: divu ¼ 2 þ 1 z cos θ À eθ sin z i ðÞr 2 4.4 Show that ∇ Á ðÞ¼ϕψu ϕ∇ψ Á u þ ψ∇ϕ Á u þ ϕψ∇ Á u, where ϕ, ψ are scalar functions and u is a vectorial function. 4.5 Calculate the curl of the following vectorial ﬁelds: (a) y2i þ z2j þ x2k; (b) xyzðÞxi þ yj þ zk . Answer: (a) À2ðÞzi þ xj þ yk ; (b) ðÞxz2 À xy2 i þ ðÞx2y À yz2 j þ ðÞy2z À x2z k. 4.6 Calculate the Laplacian of the function ϕðÞ¼xi x4z þ 3xy2 À zxy þ 1. Answer: ∇2ϕ ¼ 12x2z À 6x. Chapter 5 Riemann Spaces

5.1 Preview

The space provided with metric is called Riemann space, for which the tensorial formalism is based on the study in its first fundamental form, being complemented by the definition of curvature and by the concept of geodesics, which allows expanding the basic conceptions of the Euclidian geometry for this type of space with N dimensions. In the Riemann spaces the covariant derivatives of tensors are equal to the partial derivatives when the coordinates are Cartesian, but the problem arises of researching how these derivatives behave when the coordinate system is curvilinear. The analysis of this derivative leads to the definition of curvature of the space, which is the fundamental parameter for the development of a consistent study of the Riemann spaces EN. The concepts and expressions of Tensor Calculus are essential for the formulation of the Theory of General Relativity, and it is for this theory just as the Integral and Differential Calculus is for the Classic Mechanics.

5.2 The Curvature Tensor

The Euclidian geometry is grounded on the basic concepts of point, straight line, and plane, and in various axioms. In this geometry a curved line is deﬁned in the Euclidian space E2 as the one that is not a straight line, and in the Euclidian space E3 a curved surface is deﬁned as the one that is not a plane. The curvature is an intrinsic characteristic of the space, so it is not a property measurable by comparison between distinct spaces.

The conception of a Riemann geometry for the space EN is grounded on the basic concepts of the Euclidian geometry in space E3, which generalization is carried out by means of deﬁning the metric for the space EN, given by

2 ε i j ds ¼ gijdx dx where ε ¼Æ1 is a functional indicator. The space in which metric can be writed as an Euclidian metric, positive and definite, is called a flat space, otherwise it is called space with curvature. The concept of curvature of the space EN was firstly conceived by Riemann as a generalization of the study of a surface’s curvature developed by Gauß. Riemann presented his results in a paper in 1861, published only in 1876. Christoffel in 1869 and R. Lipschitz in four papers published in 1869, 1870 (two articles), and 1877 obtained the same results as Riemann when studying the transformationX ÀÁ of the i j 2 i 2 quadratic differential formula gijdx dx to the Euclidian metric ds ¼ dx . i The curvature analysis of the Riemann space EN was carried out by Ricci-Curbastro and Levi-Civita who deducted the expression of the curvature tensor in a very formal and concise approach, which was also obtained by Christoffel, whose deduction has an extensive algebrism. In 1917 Tulio Levi-Civita, and after Jan Arnoldus Schouten (1918) and Karl Hessenberg found independently an interpretation for the curvature tensor associating it to the concept of parallel transport of vectors.

5.2.1 Formulation

The covariant derivative of a tensor is a tensor, just as when repeating this differentiation will provide a new tensor. However, the differentiation order with respect to the variables must be considered in this analysis. For a function ϕ(xi) of class C2 that represents a scalar ﬁeld exists the derivative ∂ϕðÞxi ∂xk that represents a covariant vector. Differentiating again with respect to the variable xj results by means of the partial differentiation rule of Differential Calculus

∂2ϕðÞxi ∂2ϕðÞxi ¼ ∂xk∂xj ∂xj∂xk

In this case, the covariant derivative is commutative. However, for tensors which components are functions of class C2 represented in curvilinear coordinate systems this independence of the differentiation order in general is not veriﬁed. It is concluded that only the condition of the functions being class C2 is not enough to ensure this independence. 5.2 The Curvature Tensor 229

For the case of a covariant vector ui the result for its covariant derivative is the tensor with variance (0, 2):

∂ui ‘ ∂ u ¼ À u‘Γ ð5:2:1Þ j i ∂xj ij and with

∂jui ¼ Tij it follows that for the covariant derivative of this tensor with respect to the variable xk

∂Tij ‘ ‘ ∂ T ¼ À T‘ Γ À T ‘Γ ð5:2:2Þ k ij ∂xk j ik i jk

The substitution of expression (5.2.1) provides ÀÁ ÀÁ ∂∂jui ‘ ‘ ∂ T ¼ À ∂ u‘ Γ À ðÞ∂‘u Γ k ij ∂ k j ik i jk x ∂ ∂ ∂ ∂ ui ‘ u‘ m ‘ ui m ‘ ¼ À u‘Γ À À u Γ Γ À À u Γ Γ ∂xk ∂xj ij ∂xj m ‘j ik ∂x‘ m i‘ jk it follows that

2 ∂Γ ‘ ∂ ui ∂u‘ ‘ ij ∂u‘ ‘ ∂ T ¼ ∂ ∂ u ¼ À Γ À u‘ À Γ k ij j k i ∂ k∂ j ∂ k ij ∂ j ∂ j ik x x x x x ð5:2:3Þ ‘ ∂u ‘ ‘ þ u Γ mΓ À i Γ þ u Γ mΓ m ‘j ik ∂x‘ jk m i‘ jk that represents a tensor with variance (0, 3). The inversion of the differentiation order provides

∂2 ∂ ∂Γ ‘ ∂ ∂ ui u‘ ‘ ik u‘ ‘ m ‘ ui ‘ ∂ ∂ u ¼ À Γ À u‘ À Γ þ u Γ Γ À Γ k j i ∂xj∂xk ∂xj ik ∂xj ∂xk ij m ‘k ij ∂x‘ kj Γ mΓ ‘ : : þ um i‘ kj ð5 2 4Þ

In Differential Calculus the differentiation order does not change the result obtained then

∂2u ∂2u i ¼ i ∂xj∂xk ∂xk∂xj 230 5 Riemann Spaces

Subtracting expression (5.2.4) from expression (5.2.3) and considering the symmetry of the Christoffel symbols ! ∂Γ ‘ ∂Γ ‘ ik ij m ‘ m ‘ ∂ ∂ u À ∂ ∂ u ¼ u‘ À þ u Γ Γ À Γ Γ j k i k j i ∂xj ∂xk m ‘j ik ‘k ij and with the permutation of the dummy indexes m $ ‘ in the second term to the right "# ! ∂Γ ‘ ∂Γ ‘ ik ij ‘ m ‘ m ∂ ∂ u À ∂ ∂ u ¼ u‘ À þ Γ Γ À Γ Γ j k i k j i ∂xj ∂xk mj ik mk ij

Putting

‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ R ¼ ik À ij þ Γ Γ m À Γ Γ m ð5:2:5Þ ijk ∂xj ∂xk mj ik mk ij results in

∂ ∂ ∂ ‘ j kui À kjui ¼ u‘Rijk

‘ The quotient law is used for verifying if the variety Rijk is a tensor, carrying out ‘ the inner product of vector u‘ by Rijk:

‘ ‘ Rijku‘ ¼ Rijk‘ ¼ Rijk

The transformation law of tensors to the variety Rijk is given by

∂xi ∂xj ∂xk R ¼ R pqr ∂xp ∂xq ∂xr ijk for the vector u‘ the result of the transformation law is

∂x‘ u ¼ u‘ m ∂xm that substituted in previous expression provides

i j k ‘ ∂x ∂x ∂x ∂x ‘ R ¼ R u‘ pqr ∂xp ∂xq ∂xr ∂xm ijk

i In the coordinate system X the variety Rpqr is given by 5.2 The Curvature Tensor 231

‘ Rpqr ¼ Rpqru‘ whereby i j k ‘ ‘ ∂x ∂x ∂x ∂x ‘ R À R u‘ ¼ 0 pqr ∂xp ∂xq ∂xr ∂xm ijk

As u‘ is an arbitrary vector it results in

i j k ‘ ‘ ∂x ∂x ∂x ∂x ‘ R ¼ R pqr ∂xp ∂xq ∂xr ∂xm ijk

‘ that represents the transformation law of tensor with variances (1, 3), as Rijk is a tensor. The tensor defined by expression (5.2.5) is called Riemann–Christoffel curvature tensor, Riemann–Christoffel mixed tensor, or Riemann–Christoffel tensor of the second kind, or simply curvature tensor. This tensor defines a tensorial field that depends only on the metric tensor and its derivatives up to the second ‘ order, and classifies the space, for thus Rijk 6¼ 0 the result is a space with curvature.

5.2.2 Differentiation Commutativity

The formulation of an analogous analysis for a contravariant vector ui, which generates a mixed tensor with variance (1, 1), is carried out by calculating ﬁrstly the covariant derivative of this vector with respect to the coordinate xj:

i ∂u ‘ ∂ ui ¼ þ u Γ i ¼ T i ð5:2:6Þ j ∂xj j‘ j

The covariant derivative of the second order of this vector with respect to the coordinate xk is given by

ÀÁ ∂ i T ‘ ‘ ∂ ∂ ui ¼ ∂ T i ¼ j þ T Γ i À T iΓ k j k j ∂xk j ‘k ‘ jk

Substituting expression (5.2.6) in this expression ÀÁ i ‘ i ∂ ∂u ‘ ∂u ‘ ∂u ‘ ∂ ∂ ui ¼ þ u Γ i þ þ umΓ Γ i À þ umΓ i Γ k j ∂xk ∂xj j‘ ∂xj jm ‘k ∂x‘ ‘m jk whereby 232 5 Riemann Spaces

ÀÁ 2 i ‘ ∂Γ i ‘ i ∂ u ∂u ‘ ‘ ∂u ‘ ∂u ‘ ∂ ∂ ui ¼ þ Γ i þ u j þ Γ i þ umΓ Γ i À Γ k j ∂xk∂xj ∂xk j‘ ∂xk ∂xj ‘k jm ‘k ∂x‘ jk mΓ i Γ ‘ : : À u ‘m jk ð5 2 7Þ

The inversion of the differentiation is obtained interchanging indexes j $ k,so

ÀÁ 2 i ‘ i ‘ i ∂ u ∂u ‘ ∂Γ ∂u ‘ ∂u ‘ ∂ ∂ ui ¼ þ Γ i þ u k‘ þ Γ i þ umΓ Γ i À Γ j k ∂xj∂xk ∂xj k‘ ∂xj ∂xk ‘j km ‘j ∂x‘ kj mΓ i Γ ‘ : : À u ‘m kj ð5 2 8Þ

As in the partial derivative the order of differentiation does not change the result

∂2ui ∂2ui ¼ ∂xk∂xj ∂xj∂xk and subtracting expression (5.2.7) from expression (5.2.8)

ÀÁ ÀÁ 2 i ‘ ∂Γ i ‘ ∂ u ∂u ‘ ‘ ∂u ∂ ∂ ui À ∂ ∂ ui ¼ þ Γ i þ u j þ Γ i k j j k ∂xk∂xj ∂xk j‘ ∂xk ∂xj ‘k i ‘ ∂u ‘ ‘ þ umΓ Γ i À Γ À umΓ i Γ jm ‘k ∂x‘ jk ‘m jk

2 i ‘ i ‘ ∂ u ∂u ‘ ∂Γ ∂u À þ Γ i þ u k‘ þ Γ i ∂xj∂xk ∂xj k‘ ∂xj ∂xk ‘j i ‘ ∂u ‘ ‘ þ umΓ Γ i À Γ À umΓ i Γ km ‘j ∂x‘ kj ‘m kj and with the symmetry of the Christoffel symbols ! ÀÁ ÀÁ ∂Γ i i ‘ ‘ ∂Γ ‘ ‘ ∂ ∂ ui À ∂ ∂ ui ¼ u j À k‘ þ um Γ Γ i À Γ Γ i k j j k ∂xk ∂xj jm ‘k km ‘j

The permutation of indexes ‘ $ m in the last two terms provides ! ÀÁ ÀÁ ∂Γ i i ‘ ∂Γ ‘ ‘ ∂ ∂ ui À ∂ ∂ ui ¼ j À k þ m Γ i À Γ mΓ i u k j j k ∂xk ∂xj j‘ mk k‘ mj 5.2 The Curvature Tensor 233 and putting

i i ∂Γ ‘ ∂Γ R i ¼ j À k‘ þ Γ mΓ i À Γ mΓ i ð5:2:9Þ ‘kj ∂xk ∂xj j‘ mk k‘ mj it results in ÀÁ ÀÁ ∂ ∂ i ∂ ∂ i i ‘ : : k ju À j ku ¼ R‘kju ð5 2 10Þ

The permutation of indexes j $ k provides ÀÁ ÀÁ ∂ ∂ i ∂ ∂ i i ‘ j ku À k ju ¼ R‘jku where

i i ∂Γ ∂Γ ‘ R i ¼ k‘ À j þ Γ mΓ i À Γ mΓ i ð5:2:11Þ ‘jk ∂xj ∂xk k‘ mj j‘ mk

i ∂ ∂ k ∂ ∂ k This analysis shows that R‘jk ¼ 0 ) j ku ¼ k ju , i.e., the space is ﬂat. The necessary and sufﬁcient condition so that the differentiation commutativity be valid i is that the tensor R‘jk be null.

i 5.2.3 Antisymmetry of Tensor R‘jk

The comparison of expressions (5.2.9) and (5.2.11) shows that the Riemann– Christoffel curvature tensor is antisymmetric with respect to the last two indexes

i i R‘kj ¼ÀR‘jk

i 5.2.4 Notations for Tensor R‘jk

Putting the indexes in the sequence i, j, k, ‘ the result in tensorial notation is

‘ k j i : : R ¼ Rijkg‘ g g g ð5 2 12Þ and rewriting the Riemann–Christoffel curvature tensor as 234 5 Riemann Spaces

‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ R ¼ ik À ij þ Γ mΓ À Γ mΓ ijk ∂xj ∂xk ik mj ij mk the result in symbolic form by means of determinants is

∂ ∂ Γ m Γ m ‘ ik ij R ¼ ∂xj ∂xk þ ð5:2:13Þ ijk Γ ‘ Γ ‘ Γ ‘ Γ ‘ mk mj ij ik

‘ 5.2.5 Uniqueness of Tensor Rijk

ij The metric tensor gij and its conjugated tensor g are unique in a Riemann space, then their partial derivatives of the first and second order the Christoffel symbols of i this space are unique at pointx 2EN. Thus it is verified that expression (5.2.11) does i not ensure that tensor R‘jk is the only tensor that can be expressed by the derivatives of the first and second order of the metric tensor. However, the covariant derivatives of a contravariant vector with respect to the i coordinates of a referential system are unique at point x 2EN, and having the Riemann–Christoffel curvature tensor with variance (1, 3) obtained by means of these derivatives, it is concluded that it is unique in the point being considered. Expressions (5.2.5) and (5.2.11) obtained in distinct manners indicate this tensor’s uniqueness. i For the points x 2EN in which the Christoffel symbols are null, it is verified that i R‘jk is expressed by means of a linear combination of the derivatives of the second order of the metric tensor.

5.2.6 First Bianchi Identity

The Riemann–Christoffel curvature tensor

‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ R ¼ ik À ij þ Γ m Γ À Γ mΓ ijk ∂xj ∂xk ikΓ mj ij mk and the cyclic permutations of indexes i, j, k generate the expressions

∂Γ ‘ ∂Γ ‘ ‘ ‘ ‘ R ¼ ji À jk þ Γ mΓ À Γ mΓ jki ∂xk ∂xi ji mk jk mi 5.2 The Curvature Tensor 235

∂Γ ‘ ‘ ‘ ∂Γ ‘ ‘ R ¼ kj À ki þ Γ mΓ À Γ mΓ kij ∂xi ∂xj kj mi ki mj

The sum of these three expressions provides the ﬁrst Bianchi identity for the Riemann–Christoffel curvature tensor

‘ ‘ ‘ : : Rijk þ Rjki þ Rkij ¼ 0 ð5 2 14Þ

5.2.7 Second Bianchi Identity

The covariant derivative of a tensor with variance (1, 3) is given by

j ∂T ‘ ∂ T j ¼ p m À T j Γ q À T j Γ q À T j Γ q þ T q Γ j k p‘m ∂xk q‘m pk pqm ‘k p‘q mk p‘m kq whereby for the Riemann–Christoffel curvature tensor yields it follows that

‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ R ¼ ik À ij þ Γ mΓ À Γ mΓ ijk ∂xj ∂xk ik mj ij mk

The covariant derivative with respect to the coordinate xp is given by

2 ‘ ∂2Γ ‘ m ∂Γ ‘ ∂Γ m ‘ ∂ Γ ∂Γ ‘ ‘ ∂Γ ∂ R ‘ ¼ ik À ij þ ik Γ þ Γ m mj À ij Γ À Γ m mk p ijk ∂xp∂xj ∂xp∂xk ∂xp mj ik ∂xp ∂xp mk ij ∂xp m Γ ‘ ‘ Γ m ‘ Γ m ‘ Γ m þRijk mp À Rmjk ip À Rimk jp À Rijm kp and with the cyclic permutation of indexes j, k, p it follows that

∂2Γ ‘ 2 ‘ ∂Γ m ‘ m ∂Γ ‘ ∂ Γ ‘ ∂Γ ∂Γ ‘ ∂ R ‘ ¼ ip À ik þ ip Γ þ Γ m mk À ik Γ À Γ m mp j ikp ∂xj∂xk ∂xj∂xp ∂xj mk ip ∂xj ∂xj mp ik ∂xj m Γ ‘ ‘ Γ m ‘ Γ m ‘ Γ m þ Rikp mj À RmkpΓ ij À Rimp kj À Rikm pj ∂2Γ ‘ ∂2Γ ‘ ∂Γ m ∂Γ ‘ ∂Γ m ∂Γ ‘ ‘ ‘ ∂ R ‘ ¼ ij À ip þ ij Γ þ Γ m mp À ip Γ À Γ m mj k ipj ∂xk∂xp ∂xk∂xj ∂xk mp ij ∂xk ∂xk mj ip ∂xk m Γ ‘ ‘ Γ m ‘ Γ m ‘ Γ m þ Ripj mk À Rmpj ik À Rimj pk À Ripm jk

The sum of these three expressions provides 236 5 Riemann Spaces

2 ‘ ∂2Γ ‘ m ∂Γ ‘ ∂Γ m ∂ Γ ∂Γ ‘ ‘ ∂ R ‘ þ ∂ R ‘ þ ∂ R ‘ ¼ ik À ij þ ik Γ þ Γ m mj À ij Γ p ikj j ikp k ipj ∂xp∂xj ∂xp∂xk ∂xp mj ik ∂xp ∂xp mk ‘ ∂Γ ‘ ‘ ‘ ‘ ÀΓ m mk þ R m Γ À R Γ m À R Γ m À R Γ m ij ∂xp ijk mp mjk ip imk jp ijm kp ∂2Γ ‘ 2 ‘ ∂Γ m ‘ m ∂ Γ ‘ ∂Γ ∂Γ ‘ þ ip À ik þ ip Γ þ Γ m mk À ik Γ ∂xj∂xk ∂xj∂xp ∂xj mk ip ∂xj ∂xj mp ∂Γ ‘ ‘ ‘ ‘ ‘ ÀΓ m mp þ R m Γ À R Γ m À R Γ m À R Γ m ik ∂xj ikp mj mkp ij imp kj ikm pj ∂2Γ ‘ ∂2Γ ‘ ∂Γ m ∂Γ ‘ ∂Γ m ‘ ‘ þ ij À ip þ ij Γ þ Γ m mp À ip Γ ∂xk∂xp ∂xk∂xj ∂xk mp ij ∂xk ∂xk mj ∂Γ ‘ ‘ ‘ ‘ ‘ ÀΓ m mj þ R m Γ À R Γ m À R Γ m À R Γ m ip ∂xk ipj mk mpj ik imj pk ipm jk and with the equalities

∂2Γ ‘ ∂2Γ ‘ ∂2Γ ‘ ∂2Γ ‘ ∂2Γ ‘ ∂2Γ ‘ ik ¼ ik ij ¼ ij ip ¼ ip ∂xp∂xj ∂xj∂xp ∂xp∂xk ∂xk∂xp ∂xj∂xk ∂xk∂xj the previous expression stays

m ∂Γ ‘ ∂Γ m ‘ ∂Γ ‘ ‘ ∂Γ ∂ R ‘ þ ∂ R ‘ þ ∂ R ‘ ¼ ik Γ þ Γ m mj À ij Γ À Γ m mk p ikj j ikp k ipj ∂xp mj ik ∂xp ∂xp mk ij ∂xp m Γ ‘ ‘ Γ m ‘ Γ m ‘ Γ m þRijk mp À Rmjk ip À Rimk jp À Rijm kp ∂Γ m ‘ m ∂Γ ‘ ‘ ∂Γ ∂Γ ‘ þ ip Γ þ Γ m mk À ik Γ À Γ m mp ∂xj mk ip ∂xj ∂xj mp ik ∂xj m Γ ‘ ‘ Γ m ‘ Γ m ‘ Γ m þRikp mj À Rmkp ij À Rimp kj À Rikm pj ∂Γ m ∂Γ ‘ ∂Γ m ∂Γ ‘ ‘ ‘ þ ij Γ þ Γ m mp À ip Γ À Γ m mj ∂xk mp ij ∂xk ∂xk mj ip ∂xk m Γ ‘ ‘ Γ m ‘ Γ m ‘ Γ m þRipj mk À Rmpj ik À Rimj pk À Ripm jk

Putting the Christoffel symbols in evidence and considering the antisymmetry of ‘ ‘ ‘ ‘ the Riemann–Christoffel curvature tensor, i.e., Rimk ¼ÀRikm, Rijm ¼ÀRimj, ‘ ‘ Γ m Γ m Γ m Rimp ¼ÀRipm, and the symmetry of the Christoffel symbols, i.e., jp ¼ pj , kp Γ m Γ m Γ m ¼ pk, kj ¼ jk it follows that 5.2 The Curvature Tensor 237 ∂Γ m ∂Γ m ∂Γ m ∂Γ m ∂ R ‘ þ ∂ R ‘ þ ∂ R ‘ ¼ Γ ‘ R m À ik þ ij þ Γ ‘ R m þ ik À ip p ikj j ikp k ipj mp ijk ∂xj ∂xk mj ikp ∂xp ∂xk ! ∂Γ m ∂Γ m ∂Γ ‘ ∂Γ ‘ þΓ ‘ R m À ij þ ip À Γ m R ‘ À mk þ mj mk ipj ∂xp ∂xj ip mjk ∂xj ∂xk ! ! ∂Γ ‘ ∂Γ ‘ ∂Γ ‘ ∂Γ ‘ ÀΓ m R ‘ þ mk À mp À Γ m R ‘ À mj þ mp ij mkp ∂xp ∂xk ik mpj ∂xp ∂xj

The expressions of the tensors are given by

∂Γ m ∂Γ m R m ¼ ik À ij þ Γ q Γ m À Γ qΓ m ijk ∂xj ∂xk ik qj ij qk ∂Γ m ∂Γ m R m ¼ ip À ik þ Γ q Γ m À Γ q Γ m ikp ∂xk ∂xp ip qk ik qp ∂Γ m ∂Γ m R m ¼ ij À ip þ Γ qΓ m À Γ q Γ m ipj ∂xp ∂xj ij qp ip qj ‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ R ¼ mk À mj þ Γ q Γ À Γ q Γ mjk ∂xj ∂xk mk qj mj qk ∂Γ ‘ ‘ ‘ ∂Γ ‘ ‘ R ¼ mp À mk þ Γ q Γ À Γ q Γ mkp ∂xk ∂xp mp qk mk qp ∂Γ ‘ ∂Γ ‘ ‘ ‘ ‘ R ¼ mj À mp þ Γ q Γ À Γ q Γ mpj ∂xp ∂xj mj qp mp qj that substituted in previous expression provide ∂Γ m ∂Γ m ∂Γ m ∂Γ m ∂ R ‘ þ∂ R ‘ þ∂ R ‘ ¼ Γ ‘ ik À ij þΓ q Γ m ÀΓ qΓ m À ik þ ij p ikj j ikp k ipj mp ∂xj ∂xk ik qj ij qk ∂xj ∂xk ∂Γ m ∂Γ m ∂Γ m ∂Γ m þΓ ‘ ip À ik þΓ q Γ m ÀΓ q Γ m þ ik À ip mj ∂xk ∂xp ip qk ik qp ∂xp ∂xk ∂Γ m ∂Γ m ∂Γ m ∂Γ m þΓ ‘ ij À ip þΓ qΓ m ÀΓ q Γ m À ij þ ip mk ∂xp ∂xj ij qp ip qj ∂xp ∂xj ! ‘ ∂Γ ‘ ‘ ∂Γ ‘ ∂Γ ‘ ‘ ∂Γ ÀΓ m mk À mj þΓ q Γ ÀΓ q Γ À mk þ mj ip ∂xj ∂xk mk qj mj qk ∂xj ∂xk ! ∂Γ ‘ ‘ ‘ ∂Γ ‘ ∂Γ ‘ ‘ ∂Γ ÀΓ m mp À mk þΓ q Γ ÀΓ q Γ þ mk À mp ij ∂xk ∂xp mp qk mk qp ∂xp ∂xk ! ∂Γ ‘ ∂Γ ‘ ∂Γ ‘ ∂Γ ‘ ‘ ‘ ÀΓ m mj À mp þΓ q Γ ÀΓ q Γ À mj þ mp ik ∂xp ∂xj mj qp mp qj ∂xp ∂xj 238 5 Riemann Spaces

Simplifying ∂ ‘ ∂ ‘ ∂ ‘ Γ ‘ Γ q Γ m Γ qΓ m pRikj þ jRikp þ kRipj ¼ mp ik qj À ij qk Γ ‘ Γ q Γ m Γ q Γ m þ mj ip qk À ik qp Γ ‘ Γ qΓ m Γ q Γ m þ mk ij qp À ip qj Γ m Γ q Γ ‘ Γ q Γ ‘ À ip mk qj À mj qk Γ m Γ q Γ ‘ Γ q Γ ‘ À ij mp qk À mk qp Γ m Γ q Γ ‘ Γ q Γ ‘ À ik mj qp À mp qj and with the permutation of the dummy indexes m $ q in the ﬁrst six terms it follows that

∂ ‘ ∂ ‘ ∂ ‘ Γ ‘ Γ mΓ q Γ ‘ Γ mΓ q pRikj þ jRikp þ kRipj ¼ qp ik mj À qp ij mk Γ ‘ Γ mΓ q Γ ‘ Γ mΓ q þ qj ip mk À qj ik mp Γ ‘ Γ mΓ q Γ ‘ Γ mΓ q þ qk ij mp À qk ip mj Γ mΓ q Γ ‘ Γ mΓ q Γ ‘ À ip mk qj þ ip mj qk Γ mΓ q Γ ‘ Γ mΓ q Γ ‘ À ij mp qk þ ij mk qp Γ mΓ q Γ ‘ Γ mΓ q Γ ‘ À ik mj qp þ ik mp qj whereby

∂ ‘ ∂ ‘ ∂ ‘ : : pRikj þ jRikp þ kRipj ¼ 0 ð5 2 15Þ that is called second Bianchi identity.

5.2.8 Curvature Tensor of Variance (0, 4)

The Riemann–Christoffel curvature tensor generates a curvature tensor expressed ‘ in covariant components. With the multiplying of tensor Rijk by the metric tensor gp‘ it follows that 5.2 The Curvature Tensor 239 ! ∂Γ ‘ ∂Γ ‘ ‘ ik ij m ‘ m ‘ g ‘R ¼ g ‘ À þ Γ Γ À Γ Γ p ijk p ∂xj ∂xk ik mj ij mk or ∂ Γ ‘ ∂ Γ ‘ gp‘ ∂ gp‘ ∂ ‘ ik gp‘ ‘ ij gp‘ ‘ m ‘ m ‘ g ‘R ¼ À Γ À þ Γ þ g ‘Γ ΓΓ À g ‘Γ Γ p ijk ∂xj ∂xj ik ∂xk ∂xk ij p ik mj p ij mk

Ricci’s identity allows writing

∂ ∂ gp‘ gp‘ ¼ Γ ‘ þ Γ‘ ¼ Γ ‘ þ Γ‘ ∂xj pj, j, p ∂xk pk, k, p then ‘ ‘ ∂ g ‘Γ ∂ g ‘Γ ÀÁÀÁ ‘ p ik p ij ‘ ‘ g ‘R ¼ À þ Γ Γ ‘ þ Γ‘ þ Γ Γ ‘ þ Γ‘ p ijk ∂xj ∂xk ik pj, j, p ij pk, k, p Γ Γ m Γ Γ m À mk, p ij þ mj, p ik ÀÁÀÁ ∂Γik, p ∂Γij, p ‘ ‘ ¼ À À Γ Γ ‘ þ Γ‘ þ Γ Γ ‘ þ Γ‘ ∂xj ∂xk ik pj, j, p ij pk, k, p Γ Γ m Γ Γ m À mk, p ij þ mj, p ik and replacing indexes m ! ‘ in the last two terms ÀÁÀÁ ‘ ∂Γik,p ∂Γij,p ‘ ‘ ‘ ‘ g ‘R ¼ À À Γ Γ ‘ þ Γ‘ þ Γ Γ ‘ þ Γ‘ À Γ‘ Γ þ Γ‘ Γ p ijk ∂xj ∂xk ik pj, j,p ij pk, k,p k,p ij j,p ik ∂Γik,p ∂Γij,p ‘ ‘ ¼ À þ Γ Γ ‘ À Γ Γ ‘ ∂xj ∂xk ij pk, ik pj, whereby the result for the Riemann–Christoffel curvature tensor with variance (0, 4) or Riemann–Christoffel of ﬁrst tensor type is

∂Γik, p ∂Γij, p ‘ ‘ R ¼ À þ Γ Γ ‘ À Γ Γ ‘ ð5:2:16Þ pijk ∂xj ∂xk ij pk, ik pj, which in tensorial notation is written as

p i j k R ¼ Rpijkg g g g ð5:2:17Þ and in symbolic form by means of determinants stays 240 5 Riemann Spaces

∂ ∂ Γ ‘ Γ ‘ ∂ j ∂ k ij ik : : Rpijk ¼ x x þ ð5 2 18Þ Γpj, ‘ Γpk, ‘ Γij, p Γik, p

In tensorial notation the Riemann–Christoffel tensors, mixed and covariant, are represented by R.

5.2.9 Properties of Tensor Rpijk

For the Riemann–Christoffel covariant tensor the ﬁrst Bianchi identity provides ‘ ‘ ‘ g‘p Rikj þ Rjki þ Rkij ¼ 0 whereby the following cyclic property results

Rpikj þ Rpjki þ Rpkij ¼ 0 ð5:2:19Þ

Considering the antisymmetry of the Riemann–Christoffel tensor with variance (1, 3) the result is

‘ ‘ gp‘Rijk ¼Àgp‘Rikj ) Rpijk ¼ÀRpikj then the Riemann–Christoffel tensor with variance (0, 4) is antisymmetric in the last two indexes. Rewriting expression (5.2.16)

∂Γik, p ∂Γij, p ‘ ‘ R ¼ À þ Γ Γ ‘ À Γ Γ ‘ pijk ∂xj ∂xk ij pk, ik pj, and with expressions 1 ∂g ∂g ∂g 1 ∂g ∂g ∂g Γ ¼ pk þ ip À ik Γ ¼ jp þ ip À ij ik, p 2 ∂xi ∂xk ∂xp ij, p 2 ∂xi ∂xj ∂xp Γ Γ q Γ Γ q pk, ‘ ¼ gq‘ pk pj, ‘ ¼ gq‘ pj it follows that ∂ 1 ∂g ∂g ∂g ∂ 1 ∂g ∂g ∂g R ¼ pk þ ip À ik À jp þ ip À ij pijk ∂xj 2 ∂xi ∂xk ∂xp ∂xk 2 ∂xi ∂xj ∂xp Γ q Γ ‘ Γ q Γ ‘ þ gq‘ pk ij À gq‘ pj ik 5.2 The Curvature Tensor 241 ! ∂2 ∂2 ∂2 ∂2 1 gik gpk gji gpj q ‘ q ‘ R ¼ þ À À þ g ‘ Γ Γ À Γ Γ pijk 2 ∂xj∂xp ∂xj∂xi ∂xk∂xp ∂xk∂xi q pk ij pj ik ð5:2:20Þ

The expression (5.2.20) allows calculating the components of the tensor Rpijk directly in terms of the metric tensor. With the permutation of indexes i $ p in expression (5.2.20) ! ∂2 2 ∂2 ∂2 1 g ∂ g g g ‘ ‘ R ¼ pk þ ik À jp À ij þ g Γ q Γ À Γ q Γ ipjk 2 ∂xj∂xi ∂xj∂xp ∂xk∂xi ∂xk∂xp q‘ pk ij pj ik and with the permutation of the dummy indexes q $ ‘ this expression becomes ! ∂2 ∂2 ∂2 ∂2 1 gpk gik gjp gij ‘ q ‘ q R ¼ þ À À þ g‘ Γ Γ À Γ Γ ipjk 2 ∂xj∂xi ∂xj∂xp ∂xk∂xi ∂xk∂xp q ik pj ij pk

Considering the symmetry of the metric tensor it is verified that the term to the right represents the components ÀRpijk, then Ripjk ¼ÀRpijk, i.e., the tensor is antisymmetric in the first two indexes. These analyses show that the tensor Rpijk is antisymmetric in the first two and the last two indexes. The permutation of indexes p $ j, i $ k in expression (5.2.20) leads to ! ∂2 ∂2 ∂2 ∂2 1 gji gki gjp gpk q ‘ q ‘ R ¼ À À þ þ g ‘ Γ Γ À Γ Γ pijk 2 ∂xp∂xk ∂xp∂xj ∂xi∂xk ∂xi∂xj q ji kp jp ki

The symmetry of the metric tensor gives Rpijk ¼ Rjkpi. It is concluded that the tensor Rpijk is symmetric for the permutation of the pair of initial indexes for the pair of ﬁnal indexes.

5.2.10 Distinct Algebraic Components of Tensor Rpijk

The number of components of tensor Rpijk in the Riemann space EN cannot be obtained counting the equations Rpikj þ Rpjki þ Rpkij ¼ 0 and considering the components antisymmetric Rpijk ¼ÀRipjk, Rpijk ¼ÀRpikj and the symmetric components Rpijk ¼ Rjkpi, because these two equations overlap. The methodology used to carry out this counting is given by means of classifying the tensor components into four groups, as a function of the number of repeated indexes:

(a) The four indexes are equal Riiii (b) The initial pair of indexes is equal to the second pair Ripip 242 5 Riemann Spaces

Case (a) must fulfill the antisymmetry of tensor Rpijk that provides Riiii ¼ÀRiiii then Riiii ¼ 0. The components are null when the four indexes are equal. For case (b) only two indexes are different: Ripip having that these components differ from the componentsÀÁRippi solely in the sign, and by the antisymmetry the result is Rpipi ¼ÀRippi ¼À ÀRipip ¼ Ripip. There is a number of components for Ripip as many as the different pair of indexes, i.e., i 6¼ p. For index i there are N distinct combinations, and for index p there are ðÞN À 1 distinct combinations, N and considering the antisymmetry of the tensor for these last indexes 2 ðÞN À 1 different combinations result. This number of combinations corresponds to the N number of the 2 ðÞN À 1 distinct combinations. There is no reduction of components due to the symmetry Rpijk ¼ Rjkpi. The first Bianchi identity is satisfied, for

Rpipi þ Rppii þ Rpiip ¼ Rpipi þ 0 À Rpipi ¼ 0

N does not reduce the number of components. Therefore, in this case only 2 ðÞN À 1 independent components are non-null. Case (c) has components of the kind Rppik. In this case there are N combinations for the index p, ðÞN À 1 combinations for index i, and ðÞN À 2 combinations for index k. The number of combinations for the indexes provides the number of tensor components. The antisymmetry does not reduce the number of components, for Rppik ¼ 0 and Rpipk ¼ 0, and the first Bianchi identity is satisfied. Considering the symmetryRpipk ¼ Rpkpi the number of components is reduced by half, whereby there N are 2 ðÞN À 1 ðÞN À 2 independent and non-null components. Admitting the four indexes different there are, for example, the components R1234, R2314, R3124. With methodology analogous to the previous case, it is verified that the indexes p, i, j, k can be selected in NNðÞÀ 1 ðÞN À 2 ðÞN À 3 modes. Considering the antisymmetries Rpijk ¼ÀRipjk and Rpijk ¼ÀRpikj, the combination of indexes is N reduced to 4 ðÞN À 1 ðÞN À 2 ðÞN À 3 modes. The symmetry Rpijk ¼ Rjkpi reduces to N half these combinations, then having 8 ðÞN À 1 ðÞN À 2 ðÞN À 3 modes. The first Bianchi identity is given by ÀÁ Rpikj þ Rpjki þ Rpkij ¼ 0 ) Rpikj ¼À Rpjki þ Rpkij that shows that the different combinations of the indexes are related among themselves, for a component can be expressed in terms of the other two. Therefore, 2 the total number of combinations of indexes is reduced in 3, and the total number of 2 N non-null independent components for this case is 3 8 ðÞN À 1 ðÞN À 2 ðÞN À 3 . The consideration of all the cases that were analyzed leads to 5.2 The Curvature Tensor 243

N N N 0 þ ðÞþN À 1 ðÞN À 1 ðÞþN À 2 ðÞN À 1 ðÞN À 2 ðÞN À 3 2 2 12 ÀÁ N2 2 whereby there are 12 N À 1 independent and non-null components for the tensor Rpijk. The expressions that provide the Christoffel symbols for the orthogonal coordinate systems are

1 ∂g Γ k ¼ 0 Γ k ¼À ii ij ii ∂ k 2gkk x ÀÁpffiffiffiffiffi ÀÁpffiffiffiffiffi ∂ ‘n g ∂ ‘n g Γ i ¼ ii Γ i ¼ ii ij ∂xj ii ∂xi and with expression (5.2.20) that defines the Riemann–Christoffel curvature tensor with variance (0, 4) it results for the components of this tensor, where the indexes p, i, j, k indicate no summation: – Four different indexes

Rpijk ¼ 0 ð5:2:21Þ

– i ¼ j and the other three indexes different 0 ffiffiffiffiffiffi ÀÁ1 2pffiffiffiffiffi pffiffiffiffiffi ∂ ‘ p pffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi ∂ g ∂ g n gpp ∂ g ∂ ‘n g R ¼ g @ ii À ii À ii kk A ð5:2:22Þ piik ii ∂xp∂xk ∂xp ∂xk ∂xk ∂xp

– p ¼ k, i ¼ j, p 6¼ i (two different indexes) pffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffipffiffiffiffiffiffi ∂ 1 ∂ g ∂ 1 ∂ g 1 ∂ g ∂ g R ¼ g g ffiffiffiffiffiffi ii þ ffiffiffiffiffi kk þ ii kk kiik ii kk ∂ k p ∂ k ∂ i p ∂ i ∂ m ∂ m x gkk x x gii x gmm x x ð5:2:23Þ withm 6¼ pandm ¼ ito fulfill the condition of having two different pairs of indexes, with the summation carried out only for the index m.

Table 5.1 shows four Riemann spaces EN and the independent and non-null components of tensor Rpijk. For the Riemann space E1 the only component of tensor RpijkÀÁis R1111, which by N2 2 means of its antisymmetry will always be null. Expression 12 N À 1 proves this nullity. It is concluded that this tensor express only the internal properties of the space and not the way how this space is embedded in the Riemann spaces EN, N > 1, for this characteristic veriﬁes that in E1 a curved line has null curvature, seen that R1111 ¼ 0. 244 5 Riemann Spaces

Table 5.1 Independent and non-null components of tensor Rpijk

Dimension of space EN 23 4 5 Number of components 16 81 256 625 Independent and non-null components 1 6 20 50 of Rpijk

Kinds of components R1212 Riþ1 iþ2 jþ1 jþ2 Rpipi, Rppik, Rpipi, Rppik, Rpijk Rpijk

For the Riemann space E2 the tensor Rpijk has null components when three or more indexes are equal. Only one component cannot be null: R1212. By means of the symmetry and the antisymmetry it is veriﬁed that R1212 ¼ÀR2112 ¼ÀR1221 ¼ R2121. This component is given by ! ∂2 ∂2 ∂2 ÀÁ 1 g12 g11 g22 q ‘ q ‘ R ¼ 2 À À þ g ‘ Γ Γ À Γ Γ ð5:2:24Þ 1212 2 ∂x1∂x2 ∂x2∂x2 ∂x1∂x1 q 12 12 11 22

For the Riemann space E3 the six components of tensor Rpijk are: – Three components with two repeated indexes

R1212 R1313

R2323

– Three components with only one index repeated (three indexes are different)

R1213 R1223(¼R2123) R1323(¼R3132)

For the Riemann space E4 there are 21 non-null components of tensor R‘ijk which are: – Six components with two repeated indexes

R1212 R1313 R1414

R2323 R2424 R3434

– Twelve components with only one index repeated (three indexes are different)

R1213 R1214 R1223 R1224 R1314 R1323 R1334 R1424 R1434

R2324 R2334 R2434 5.2 The Curvature Tensor 245

– Three components with only one index repeated (three indexes are different)

R1234 R1324 R1423

having that R1234 þ R1423 À R1324 ¼ 0, then there are 20 independent non-null components. The non-null components of tensor R‘ijk for the Riemann space E5 are: – Ten components with two repeated indexes

R1212 R1313 R1414 R1515

R2323 R2424 R2525

R3434 R3535

R4545

– Thirty components with only one index repeated (three indexes are different)

R1213 R1214 R1215 R1314 R1315 R1415

R2123 R2124 R2125 R2324 R2325 R2425

R3132 R3134 R3135 R3234 R3235 R3435

R4142 R4143 R4145 R4243 R4245 R4345

R5152 R5153 R5154 R5253 R5254 R5354

– Ten components in which all the indexes are different

R1234 R1235 R1245 R1345 R2345

R1324 R1325 R1425 R1435 R2435

5.2.11 Classiﬁcation of Spaces

As a function of the values assumed by the Riemann–Christoffel tensors the spaces ‘ ‘ ; are classified as: (a) flat: Rijk ¼ Rijkm ¼ 0; (b) curved space Rijk 6¼ 0 Rijkm 6¼ 0. i The condition R‘jk ¼ Rijkm ¼ 0 indicates that the space is flat with the compo- 2 i j nents of its metric tensor gij being constant. If the metric ds ¼ gijdx dx is definite

> positive, i.e., gij 0, this space is Euclidian, then it is possible to carry out a linear transformation of the coordinates xi to the coordinates xi for which the result is δ i gij ¼ j , so the metric is

2 δ i i j 1 1 2 2 m m : : ds ¼ j dx dx ¼ dx dx þ dx dx þÁÁÁþdx dx ð5 2 25Þ 246 5 Riemann Spaces

i The vectors of base ei of this new coordinate system X form a set of orthogonal δ i directions, thus j ¼ ei Á ei, and deﬁne an Euclidian space EM. Consider the Rie- i mann space EN with the coordinates x , i ¼ 1, 2, ...N, EN EM, with M > N, which coordinates are xk, k ¼ 1, 2, ..., M. Let the functions M be independent in terms of the coordinates xk, so as to have the metric ÀÁ 2 i j k 2 i j k k ds ¼ gijdx dx ¼ dx ) gijdx dx ¼ dx dx

By means of the transformation law for coordinates it follows that

∂xk ∂xk dxk ¼ dxi dxk ¼ dxj ∂ i ∂ j x x ∂xk ∂xk ∂xk ∂xk g dxidxj ¼ dxi dxj ) g À dxidxj ¼ 0 ij ∂xi ∂xj ij ∂xi ∂xj

As dxi and dxj are arbitrary, provides

∂xk ∂xk g ¼ ij ∂xi ∂xj

N that deﬁnes 2 ðÞN þ 1 independent differential equations as a function of k < N M unknowns x . In this case M 2 ðÞN þ 1 is the condition in order to have N EN EM. For N ¼ 1 the result is M 2.

5.3 Riemann Curvature

5.3.1 Deﬁnition

The study of the Riemann space EN is carried by means of the definition of the Riemann K curvature, which is more effective for the formulations of analyses than the Riemann–Christoffel curvature tensor Rpijk, for it considers the directions of the space. For establishing a general formulation, valid for the Riemann spaces EN with undefined metric, with the unit vectors ui and vi, linearly independents, defined in a i point x 2EN, and the expression

wi ¼ aui þ bvi ð5:3:1Þ that deﬁnes a coplanar vector with these two unit vectors, where a, b, are scalars that assume arbitrary values. The elementary displacements in the directions deﬁned by i i the vectors w determine a plane π that contains the point x 2EN. 5.3 Riemann Curvature 247

It is admitted that ui and vi deﬁne coplanar vectors

i i i w ¼ a1u þ b1v ð5:3:2Þ

i i i r ¼ a2u þ b2v ð5:3:3Þ where a1, b1, a2, b2 are scalars, and putting εðÞ¼Æu 1 and εðÞ¼Æv 1 as functional indicators of these unit vectors, and having wi and ri vectors mutually orthogonal it follows that

ε k ‘ 2 k ‘ 2 k ‘ 2ε 2ε ðÞ¼w gk‘w w ¼ a1u u þ b1v v ¼ a1 ðÞþu b1 ðÞv ε k ‘ 2ε 2ε ðÞ¼r gk‘r r ¼ a2 ðÞþu b2 ðÞv and with the condition of orthogonality

k ‘ ε ε gk‘w r ¼ ðÞu a1a2 þ ðÞv b1b2 ¼ 0 whereby ÂÃÂÃ ε ε 2ε 2ε 2ε 2ε ε ε 2 ðÞw ðÞ¼r a1 ðÞþu b1 ðÞv a2 ðÞþu b2 ðÞv À ½ðÞu a1a2 þ ðÞv b1b2 2 ¼ εðÞu εðÞv ðÞa1b2 À a2b1 ð5:3:4Þ

As the functional indicators assume the values Æ1:

a1b2 À a2b1 ¼Æ1 ð5:3:5Þ whereby

εðÞu εðÞ¼v εðÞw εðÞr ð5:3:6Þ

Consider two orthogonal unit vectors u and v that determine the plane π that i contains the point x 2EN, thus the Riemann curvature is deﬁned by

k ‘ m n K ¼ εðÞu εðÞv Rk‘mnu v u v ð5:3:7Þ

5.3.2 Invariance

For the other pair of orthogonal vectors w and r coplanar with u and v, there is in an analogous way for the Riemann curvature

e k ‘ m n K ¼ εðÞw εðÞr Rk‘mnw r w r 248 5 Riemann Spaces and with expressions (5.3.4)–(5.3.6) it follows that

k ‘ m n k ‘ m εðÞu εðÞv ðÞa1b2 À a2b1 Rk‘mnw r w r ¼ εðÞu εðÞv Rk‘mnu v u v Ke ¼ K thus the Riemann curvature does not depend on the pair of unit vectors used to deﬁne it, then K is an invariant.

5.3.3 Normalized Form

The obtaining of an expression for the Riemann curvature can be carried out admitting that the Riemann space EN is isotropic, in which the isotropic tensor is deﬁned by

Tij‘m ¼ Agijg‘m þ Bgi‘gjm þ Cgimgj‘

i where A, B, C are scalars that depend on the point x 2EN. Assuming that tensor Tij‘m is the curvature tensor Rij‘m the result is : : Rij‘m ¼ Agijg‘m þ Bgi‘gjm þ Cgimgj‘ ð5 3 8Þ and the antisymmetry of tensor Rij‘m allows writing Riiii ¼ 0, Riijj ¼ 0, Rii‘m ¼ Rij‘‘ ¼ 0, and with expression (5.3.8) it follows that

2 Riiii ¼ Agiigii þ Bgiigii þ Cgiigii ¼ giiðÞ¼A þ B þ C 0 A þ B þ C ¼ 0 ) B þ C ¼ÀA

Riijj ¼ Ag g þ Bg g þ Cg g ¼ Ag g þ ðÞB þ C g g ii jj ij ij ij ij ii jj ij ij 2 : : ¼ Agiigjj À gij ¼ 0 ð5 3 9Þ : : Rij‘‘ ¼ Agijg‘‘ þ Bgi‘gj‘ þ Cgi‘gj‘ ¼ Agijg‘‘ À gi‘gj‘ ¼ 0 ð5 3 10Þ

The minors of det gi‘ cannot all be simultaneously null, then in expressions (5.3.9) and (5.3.10) the result is A ¼ 0 and B ¼ÀC, whereby : : Rij‘k ¼ Bgi‘gjm À gimgj‘ ð5 3 11Þ 5.3 Riemann Curvature 249

Let

k ‘ m n K ¼ εðÞu εðÞv Rk‘mnu v u v or

k ‘ m n K ¼ Rk‘mnu v u v ð5:3:12Þ and substituting expression (5.3.11) it is concluded that B ¼ K is the Riemann i curvature in x 2EN,so : : Rij‘m ¼ Kgi‘gjm À gimgj‘ ð5 3 13Þ

The expression of the Riemann curvature for the isotropic space EN, with N > 2, in terms of the generalized Kronecker delta and the Ricci pseudotensor

p p ...p p ...p p p ...p ε ...... ε 1 2 m mþ1 n N 2 !δ 1 2 n i1i2 imimþ1 in ¼ ðÞÀ i1i2...in takes the form

p p ...p p ...p ε ...... ε 1 2 m mþ1 n ... i1i2 imimþ1 in p1p2 pn Rij‘k ¼ K ¼ Kδ ... ð5:3:14Þ ðÞN À 2 ! i1i2 in

The normalized Riemann curvature is established admitting that the vectors u i and v form an angle α and deﬁne a tangent plane π in point x 2EN. The norm of the vector perpendicular to this plane is given by

kku Â v 2 ¼ kku 2kkv 2 sin 2α and with the square of the dot product of these two vectors it follows that

ðÞu Á v 2 ¼ kku 2kkv 2 cos 2α ¼ cos 2α ÀÁ kku Â v 2 ¼ kku 2kkv 2 1 À cos 2α ¼ kku 2kkv 2 À ðÞu Á v 2

In terms of the components of these vectors

2 k m 2 ‘ n kku ¼ gkmu u kkv ¼ g‘nv v then

2 k m ‘ n k n m ‘ k ‘ m n kku Â v ¼ gkmu u g‘nv v À gknu v gm‘u v ¼ u v u v ðÞgkmg‘n À gkngm‘

Expression (5.3.12) in its normalized form is 250 5 Riemann Spaces

ÀÁ k ‘ m n i; ; Rk‘mnu v u v : : Kx u v ¼ k ‘ m n ð5 3 15Þ ðÞgkmg‘n À gkngm‘ u v u v or

ÀÁ k‘ mn i; ; Rk‘mnA A : : Kx u v ¼ k‘ mn ð5 3 16Þ ðÞgkmg‘n À gkngm‘ A A where Ak‘ ¼ ukv‘, Amn ¼ umvn represent the plane π deﬁned by the vectors u, v. This expression highlights that the Riemann curvature K(xi; u, v) of the Riemann space EN relative to the plane π deﬁned by the vectors u and v depends on the point i x 2π EN. k ‘ m n In the numerator of expression (5.3.15) the product Rk‘mnu v u v is an invariant. Putting

Gk‘mn ¼ gkmg‘n À gkngm‘ it is veriﬁed that Gk‘mn is a tensor, for it is obtained by means of algebraic operations with the metric tensor. The permutation of the tensor indexes Gk‘mn shows that this tensor has the same properties of symmetry and antisymmetry as tensor Rk‘mn. For an orthogonal coordinate system exists gij ¼ 0 for i 6¼ j, and the non-null components of this tensor are given by Gijij ¼ giigjj, where the indexes do not indicate summation. k ‘ m n The inner product Gk‘mnu v u v generates a scalar, then expression (5.3.15) represents an invariant, highlighting the demonstration that Ke ¼ K.

5.4 Ricci Tensor and Scalar Curvature

The Riemann–Christoffel curvature tensor Rpijk allows obtaining tensors of lower order by means of theirs various contractions. To obtain a non-null tensor ﬁrst an index of a pair of indexes are contracted with an index of another pair of indexes, being possible the contractions: 1–3; 1–4; 2–3; 2–4. The contraction of this tensor mp generates the Ricci tensor, thus the multiplying of tensor Rpijk by g provides

mp mp ‘ δ m ‘ m g Rpijk ¼ g gp‘Rijk ¼ ‘ Rijk ¼ Rijk and with the contraction m ¼ k the result is

k Rijk ¼ Rij 5.4 Ricci Tensor and Scalar Curvature 251

Then the Riemann–Christoffel curvature tensor with variance (1, 3) provides two Ricci tensors, one of variance (0, 2) and another of variance (1, 1). The second contraction gives a scalar with important properties, called scalar curvature. The Ricci tensor is essentially the only contraction of the Riemann–Christoffel tensor.

5.4.1 Ricci Tensor with Variance (0, 2)

The contraction of the curvature tensor

‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ R ¼ ik À ij þ Γ mΓ À Γ mΓ ijk ∂xj ∂xk ik mj ij mk in indexes ‘ ¼ k provides

‘ ∂Γ ‘ ∂Γ ‘ ‘ R ¼ i‘ À ij þ Γ mΓ À Γ mΓ ð5:4:1Þ ij ∂xj ∂x‘ i‘ mj ij m‘

In determinants form the result is

∂ ∂ Γ m Γ m i‘ ij R ¼ ∂xj ∂x‘ þ ð5:4:2Þ ij Γ ‘ Γ ‘ Γ ‘ Γ ‘ m‘ mj ij i‘ and with the expressions ÀÁpffiffiffi ÀÁpffiffiffi ‘ ∂ ‘n g ‘ ∂ ‘n g Γ ¼ Γ ¼ i‘ ∂xi m‘ ∂xm it follows that ÀÁffiffiffi ÀÁffiffiffi p ∂Γ ‘ p ∂ ∂ ‘n g ‘ ∂ ‘n g R ¼ À ij þ Γ mΓ À Γ m ij ∂xj ∂xi ∂x‘ i‘ mj ij ∂xm whereby for the Ricci tensor with variance (0, 2) the result is ÀÁffiffiffi ÀÁffiffiffi 2 p ∂Γ ‘ p ∂ ‘n g ‘ ∂ ‘n g R ¼ À ij þ Γ mΓ À Γ m ij ∂xj∂xi ∂x‘ i‘ mj ij ∂xm or 252 5 Riemann Spaces

2 ∂Γ ‘ 1 ∂ ðÞ‘ng ‘ 1 ∂ðÞ‘ng R ¼ À ij þ Γ mΓ À Γ m ð5:4:3Þ ij 2 ∂xj∂xi ∂x‘ i‘ mj 2 ij ∂xm

If g < 0 it is enough to change g for Àg in the expression (5.4.3). The permutation of indexes j $ i leads to

2 ∂Γ ‘ 1 ∂ ðÞ‘ng ‘ 1 ∂ðÞ‘ng R ¼ À ji þ Γ mΓ À Γ m ji 2 ∂xi∂xj ∂x‘ j‘ mi 2 ji ∂xm

As the Christoffel symbols are symmetric and the order of differentiation in the ﬁrst term of the previous expression is independent of the sequence in which it is N carried out, it is concluded that the Ricci tensor Rij is symmetric, so it has 2 ðÞN þ 1 distinct components. ‘ ‘ ‘ ‘ The contractions that can be carried out in tensor Rijk are: R‘jk, Ri‘k, Rij‘. Con- ‘ ‘ ‘ sidering the antisymmetry of curvature tensor Rijk ¼ÀRikj and with k ¼ the result ‘ ‘ ‘ ‘ is Rij‘ ¼ÀRi‘j, whereby Rij ¼ÀRi‘j. The contraction Ri‘k generates the Ricci tensor ‘ Rij with sign changed, then it is enough to consider only the contraction Rij‘ to ‘ obtain tensor Rij, which contains components independent of Rijk in the more adequate form of a symmetric tensor. ‘ ‘ The contraction of tensor Rijk in the indexes i ¼ is given by

‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ ‘ R ¼ ‘k À j þ Γ mΓ À Γ mΓ ‘jk ∂xj ∂xk ‘k mj ‘j mk and with the expressions ÀÁpffiffiffi ÀÁpffiffiffi ‘ ∂ ‘n g ‘ ∂ ‘n g Γ ¼ Γ ¼ ‘k ∂xk ‘j ∂xj the result is ÀÁpffiffiffi ÀÁpffiffiffi ‘ ∂ ∂ ‘n g ∂ ∂ ‘n g ‘ ‘ R ¼ À þ Γ mΓ À Γ mΓ ‘jk ∂xj ∂xk ∂xk ∂xj ‘k mj ‘j mk

The permutation of indexes ‘ $ m in the last term and the symmetry of the Christoffel symbols allow writing ÀÁpffiffiffi ÀÁpffiffiffi ‘ ∂ ∂ ‘n g ∂ ∂ ‘n g ‘ ‘ R ¼ À þ Γ mΓ À Γ Γ m ‘jk ∂xj ∂xk ∂xk ∂xj ‘k mj mj ‘k and as 5.4 Ricci Tensor and Scalar Curvature 253

ÀÁpffiffiffi ÀÁpffiffiffi ∂ ∂ ‘n g ∂ ∂ ‘n g ¼ ∂xj ∂xk ∂xk ∂xj then

‘ R‘jk ¼ Rjk ¼ 0

It is concluded that the contraction of the Riemann–Christoffel curvature tensor ‘ ‘ Rijk in the indexes ¼ i generates the null tensor.

5.4.2 Divergence of the Ricci Tensor with Variance Ricci (0, 2)

‘ The calculation of the divergence of tensor Rijk is carried out considering the second Bianchi identity

∂ ‘ ∂ ‘ ∂ ‘ ‘Rijk þ jRik‘ þ kRi‘j ¼ 0 in which the contraction of the indexes ‘ ¼ k provides

∂ k ∂ k ∂ k ∂ ∂ k ‘Rijk þ jRik‘ þ kRi‘j ¼ 0 ) ‘Rij þ jRi‘ þ divRi‘j ¼ 0 whereby ÀÁ k ∂ ∂ divRi‘j ¼À ‘Rij þ jRi‘ and with the ordination of the indexes ÀÁ ‘ ∂ ∂ : : divRijk ¼À jRik þ kRij ð5 4 4Þ

5.4.3 Bianchi Identity for the Ricci Tensor with Variance (0, 2)

An identity analogous to the second Bianchi identity can be obtained for the Ricci tensor. Rewriting expression (5.2.15)

∂ ‘ ∂ ‘ ∂ ‘ pRikj þ jRikp þ kRipj ¼ 0 254 5 Riemann Spaces and with the relations

‘ ‘ ‘ ‘ ‘ ‘ gi Rkj ¼ Rikj gi Rkp ¼ Rikp gi Rpj ¼ Ripj it follows that

‘∂ ∂ ‘ ‘∂ ∂ ‘ ‘∂ ∂ ‘ gi pRkj ¼ pRikj gi iRkp ¼ iRikp gi kRpj ¼ kRipj

The sum of these three expressions provides ÀÁ ‘ ∂ ∂ ∂ ∂ ‘ ∂ ‘ ∂ ‘ gi pRkj þ iRkp þ kRpj ¼ pRikj þ iRikp þ kRipj

As the term to the right is the second Ricci identity it results in

∂pRkj þ ∂iRkp þ ∂kRpj ¼ 0

The changes of the indexes j ! i, k ! j, p ! k allow the ordination of the same, then

∂kRij þ ∂iRjk þ ∂jRki ¼ 0 ð5:4:5Þ that is called Bianchi identity for the Ricci tensor of covariant components.

5.4.4 Scalar Curvature

ij The multiplying of the Ricci tensor Rij by the conjugate metric tensor g provides

ij R ¼ g Rij ð5:4:6Þ that deﬁnes the scalar curvature, which is the trace of the Ricci tensor, also called Ricci curvature or invariant curvature of the Riemann space EN.

5.4.5 Geometric Interpretation of the Ricci Tensor with Variance (0, 2)

Let the Riemann curvature

ÀÁ k ‘ m n i; ; Rk‘mnu v u v Kx u v ¼ k ‘ m n ðÞgkmg‘n À gkngm‘ u v u v 5.4 Ricci Tensor and Scalar Curvature 255 where u, v are orthogonal unit vectors, the result thereof is ÀÁÀÁ k ‘ m n k m ‘ n g g‘ u v u v ¼ g u u g‘ v v km n ÀÁkm ÀÁn k ‘ m n k n m ‘ gkngm‘u v u v ¼ gknu v gm‘u v but

k m ‘ n k n m ‘ gkmu u ¼ g‘nv v ¼ 1 gknu v ¼ gm‘u v ¼ 0 then

ÀÁ k ‘ m n i Rk‘mnu v u v k ‘ m n K ¼ Kx; u; v ¼ ¼ R ‘ u v u v uv 1 Â 1 À 0 k mn where the notation Kuv is adopted by convenience of graphic representation. If the unit vectors u, v are linearly dependent, the result is K ¼ 0. The summation of all the N components of vector u is given by

XN XN XN k ‘ m n k m ‘ n Kuv ¼ Rk‘mnu v u v ¼ u u Rk‘mnv v vj¼1 vj¼1 vj¼1 but

XN ‘ ‘ v vn ¼ g n vj¼1

‘ whereby the contraction Ri‘j generates the Ricci tensor Rij with the sign changed, then

XN k m ‘n k m n k m Kuv ¼Àu u g Rk‘mn ¼Àu u Rkmn ¼Àu u Rkm vj¼1

Putting

XN k m Ku ¼ Kuv ¼Àu u Rkm ð5:4:7Þ vj¼1 where Ku is the sum of the Riemann curvature for the space EN determined by the components of vector u and each ðÞN À 1 directions which are mutually orthogonal to them. This expression is independent of these directions and deﬁnes the mean curvature of EN in the direction of this vector. 256 5 Riemann Spaces

In expression (5.4.7) when carrying out the summation on the N directions mutually orthogonal, it follows that

XN XN k m Ku ¼À u u Rkm ui¼1 ui¼1 XN ukum ¼ gkm ui¼1 XN km Ku ¼Àg Rkm ¼ÀR ð5:4:8Þ ui¼1

Expression (5.4.8) shows that the sum of the mean curvatures in the Riemann space EN for mutually orthogonal directions are independent of the directions deﬁned by the vectors u, v, being equal to the scalar curvature.

5.4.6 Eigenvectors of the Ricci Tensor with Variance (0, 2)

The Ricci tensor Rij is symmetric and has in each point of the Riemann space EN a system of linearly independent equations that deﬁne principal directions (eigenvectors). Let the Riemann curvature

ÀÁ k ‘ m n i; ; Rk‘mnu v u v Kx u v ¼ k ‘ m n ðÞgkmg‘n À gkngm‘ u v u v where the vectors are orthogonal and only v is a unit vector, so ÀÁÀÁ k ‘ m n k m ‘ n gkmg‘nu v u v ¼ gkmu u g‘nv v ÀÁÀÁ k ‘ m n k n m ‘ gkngm‘u v u v ¼ gknu v gm‘u v ¼ 0 then

ÀÁ k ‘ m n i; ; Rk‘mnu v u v Kx u v ¼ k m ‘ n ðÞgkmu u ðÞg‘nv v but as v is a unit vector the result is

‘ n ‘ n ‘n g‘nv v ¼ 1 ) v v ¼ g 5.4 Ricci Tensor and Scalar Curvature 257 whereby

ÀÁ ‘n k m i; ; Rk‘mng u u Kx u v ¼À k m gkmu u thereof

ÀÁ k m i; ; Rkmu u : : Ku ¼ Kx u v ¼À k m ð5 4 9Þ gkmu u is the normalized mean curvature, where the index indicates that u is not unit vector. The calculation of the eigenvalues is carried out by means of the equations system

k m ðÞRkm þ Kugkm u u ¼ 0 with extreme values given by the condition

∂ ÂÃ ðÞR þ K g ukum ¼ 0 ∂uk km u km which developed stays

∂R ∂K 2ðÞR þ K g um þ km ukum þ u g ukum ¼ 0 km u km ∂uk ∂uk km

k and as the Ricci tensor Rij does not depend on vector u the result is

∂K 2ðÞR þ K g um þ u g ukum ¼ 0 km u km ∂uk km

∂Ku For the extreme values of Ku the result is ∂xk ¼ 0, whereby the equations system

m ðÞRkm þ Kugkm u ¼ 0 allows determining the principal directions (eigenvectors) of the Ricci tensor Rij.

5.4.7 Ricci Tensor with Variance (1, 1)

The Ricci tensor in terms of its mixed components is given by

i im : : Rj ¼ g Rmj ð5 4 10Þ 258 5 Riemann Spaces

An important expression that relates the Ricci tensor with variance (1, 1) with the derivative of the scalar curvature can be obtained by means of the second Bianchi identity

∂ ‘ ∂ ‘ ∂ ‘ pRijk þ jRikp þ kRipj ¼ 0

‘ ‘ where with the antisymmetry Rikp ¼ÀRipk the result is

∂ ‘ ∂ ‘ ∂ ‘ pRijk À jRipk þ kRipj ¼ 0

The contraction of these tensors in indexes ‘ ¼ k provides

∂ ∂ ∂ k pRij À jRip þ kRipj ¼ 0

Multiplying by gip it follows that

ip∂ ip∂ ip∂ k g pRij À g jRip þ g kRipj ¼ 0 ∂ ip ∂ ip ∂ ip k pg Rij À jg Rip þ kg Ripj ¼ 0 ∂R ∂R ∂ R p À þ ∂ R k ¼ 0 ) ¼ ∂ R p þ ∂ R k p j ∂xj k j ∂xj p j k j

The change of the dummy indexes p ! k provides

∂R ¼ 2∂ R k ∂xj k j whereby

1 ∂R ∂ R k ¼ ð5:4:11Þ k j 2 ∂xj

ij For the Riemann space EN, with N > 2, multiplying expression (5.4.5)byg the result is

ij ij ij ij ij ij g ∂kRij þ g ∂iRjk þ g ∂jRki ¼ 0 ) ∂kg Rij þ ∂ig Rjk þ ∂jg Rki ¼ 0 and having curvature R a scalar function at its partial derivative is equal to its covariant derivative, then

∂R þ ∂ R i þ ∂ R j ¼ 0 ∂xk i k j k 5.4 Ricci Tensor and Scalar Curvature 259 and with the change of indexes j ! i the result is

∂R þ 2∂ R i ¼ 0 ∂xk i k and with

1 ∂R ∂ R i ¼ i k 2 ∂xk it follows that

∂R 1 ∂R ∂R þ 2 Á ¼ 0 ) ¼ 0 ∂xk 2 ∂xk ∂xk then the scalar curvature is constant for this kind of space. The purpose of the supposition N > 2 will be clariﬁed by expression (5.6.10), obtained when analyzing the scalar curvature in the Riemann space E2. i i δ i α β α β Exercise 5.1 For the tensorial expression Tj ¼ Rj þ j ðÞR þ , where , are α ∂ i scalars, calculate the value of so that the covariant derivative iTj is null. ∂ i The null covariant derivative iTj is given by hi ∂ i ∂ i ∂ δ i α β iTj ¼ iRj þ i j ðÞR þ

∂ δ i having i j ¼ 0 it follows that

∂ i ∂ i α∂ iTj ¼ iRj þ iR ¼ 0

With the expression (5.4.11) 1 ∂R 1 ∂R ∂ R i ¼ ) ∂ T i ¼ þ α ¼ 0 i j 2 ∂xj i j 2 ∂xj

∂ ∂R α 1 for iR ¼ ∂xj, and as this derivative assumes any values the result is ¼À2.

5.4.8 Notations

In Table 5.2, in which the Tulio Levi-Civita notation was inserted, there is a compilation of the evolution of the notation for the Riemann–Christoffel curvature tensors and for the Ricci tensor. The notations that make use of (,) or (;) seek to 260 5 Riemann Spaces

Table 5.2 Notations for the Riemann–Christoffel curvature tensors and Ricci tensor Riemann–Christoffel curvature tensor Mixed variance Covariant components (1, 3) components (0, 4) Author Ricci tensorX Brillouin i R ‘ m Rj; k‘ ij, k Rj‘ ¼ Rj,m‘ Xm Appe-Thiry Ri R ‘ m jk‘ ijk Rjk ¼ Rjkm Xm Weyl i F ‘ m Fjk‘ ijk Rj‘ ¼ Fjm‘ Xm Eddington- i B ‘ m Bjk‘ jk i Gjk ¼ Bjmk Becquerel Xm Galbrun i R ‘ m Rj‘k ij k Rjk ¼ Rjmk Xm Juvet R i R ‘ m j ‘k ji k Rjk ¼ Rjmk Xm Cartan i R ‘ m Rj;‘k ji, k Rj‘ ¼ Rj‘m m Christoffel and ( ji; k‘)(ji, k‘)– Bianchi

Levi-Civita {ji, k‘}(ji, k‘) αj‘

indicate the properties of symmetry and antisymmetry of the Riemann–Christoffel tensors. In the case of using (.) it indicates the index, or the position and the index that will be lowered or raised. The only difference between the two notations of Christoffel and Bianchi is the change of the point and comma (;) for the comma (,). Currently these two forms of spelling were abandoned. It is stressed that several authors have opted for different positioning of the indexes. The Weyl notation, with the change of the letter F for R (Riemann), was the one that became consecrated in the current literature. Γ i δ i ∂ϕ δ i ∂ψ ϕ ψ Exercise 5.2 In a coordinates system let jk ¼ j ∂xk þ k ∂xj, where , are i ψ ‘ i functions of position. Calculate: (a) Rjk‘; (b) Rjk for ¼À naðÞix . (a) Substituting the expression

∂ϕ ∂ψ Γ i ¼ δ i þ δ i jk j ∂xk k ∂xj

in the expression of the Riemann–Christoffel curvature tensor

i i ∂Γ ‘ ∂Γ R i ¼ j À jk þ Γ i Γ r À Γ i Γ r jk‘ ∂xk ∂x‘ rk j‘ r‘ jk 5.4 Ricci Tensor and Scalar Curvature 261

it follows that ∂ ∂ϕ ∂ψ ∂ ∂ϕ ∂ψ R i ¼ δ i þ δ i À δ i þ δ i jk‘ ∂xk j ∂x‘ ‘ ∂xj ∂x‘ j ∂xk k ∂xj ∂ϕ ∂ψ ∂ϕ ∂ψ þ δ i þ δ i δ r þ δ r j ∂xk k ∂xr j ∂x‘ ‘ ∂xj ∂ϕ ∂ψ ∂ϕ ∂ψ À δ i þ δ i δ r þ δ r r ∂x‘ ‘ ∂xr j ∂xk k ∂xj ∂2ϕ ∂2ψ ∂2ϕ ∂2ψ ∂ϕ ∂ϕ ∂ϕ ∂ψ ¼ δ i þ δ i À δ i À δ i þ δ iδ r þ δ iδ r j ∂xk∂x‘ ‘ ∂xk∂xj j ∂x‘∂xk k ∂x‘∂xj j j ∂xk ∂x‘ j ‘ ∂x‘ ∂xj ∂ψ ∂ϕ ∂ψ ∂ψ ∂ϕ ∂ϕ ∂ϕ ∂ψ ∂ψ ∂ϕ þδ iδ r þ δ iδ r À δ iδ r À δ iδ r À δ iδ r k j ∂xr ∂x‘ k ‘ ∂xr ∂xj r j ∂x‘ ∂xk r k ∂x‘ ∂xj ‘ j ∂xr ∂xk ∂ψ ∂ψ Àδ iδ r ‘ k ∂xr ∂xj ∂ϕ ∂ϕ ∂ψ ∂ψ ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ϕ ∂ϕ R i ¼ δ i þ δ i þ δ i þ δ i À δ i jk‘ j ∂xk ∂x‘ k ∂xj ∂x‘ k ∂x‘ ∂xj ‘ ∂xk ∂xj j ∂x‘ ∂xk ∂ψ ∂ψ ∂ϕ ∂ψ ∂ϕ ∂ψ ∂2ϕ ∂2ψ Àδ i À δ i À δ i þ δ i þ δ i ‘ ∂xj ∂xk ‘ ∂xk ∂xj k ∂x‘ ∂xj j ∂x‘∂xk ‘ ∂xj∂xk ∂2ϕ ∂2ψ Àδ i À δ i j ∂ k∂ ‘ k ∂ j∂ ‘ x x x x! ! ∂ψ ∂ψ ∂2ψ ∂ψ ∂ψ ∂2ψ R i ¼ δ i À À δ i À jk‘ k ∂xj ∂x‘ ∂xj∂x‘ ‘ ∂xj ∂xk ∂xj∂xk

i ψ then Rjk‘ only depends on the function . i (b) For ψ ¼À‘naðÞix the partial derivatives result

2 ∂ψ a ∂ ψ a a‘ ¼À j ) ¼ j ∂ j i ∂ j∂ ‘ i 2 x aix x x ðÞaix ∂ψ a‘ ∂ψ ∂ψ a a‘ ¼À ) ¼ j ∂ ‘ i ∂ j ∂ ‘ i 2 x aix x x ðÞaix

and substituting this derivatives in the expression obtained in item (a) it follows that 262 5 Riemann Spaces ! ! ∂ψ ∂ψ ∂2ψ ∂ψ ∂ψ ∂2ψ R i ¼ δ i À À δ i À jk‘ k ∂xj ∂x‘ ∂xj∂x‘ ‘ ∂xj ∂xk ∂xj∂xk "#"#

i i aja‘ aja‘ i ajak ajak R ‘ ¼ δ À À δ‘ À ¼ 0 jk k i 2 i 2 i 2 i 2 ðÞaix ðÞaix ðÞaix ðÞaix

whereby

i : : : Rjki ¼ Rjk ¼ 0 Q E D

5.5 Einstein Tensor

The tensor Rijk‘, the second Bianchi identity, the Ricci tensor Rij and the scalar curvature R allow obtaining a second-order tensor with peculiar characteristics. Let the second Bianchi identity

∂mRijk‘ þ ∂kRij‘m þ ∂‘Rijmk ¼ 0 and with the antisymmetry of the Riemann–Christoffel curvature tensor Rijk‘

∂mRijk‘ À ∂kRijm‘ À ∂‘Rjimk ¼ 0 and multiplying by gi‘ and gjk it follows that

i‘ jk i‘ jk i‘ jk g g ∂mRijk‘ À g g ∂kRijm‘ À g g ∂‘Rjimk ¼ 0 jk∂ ‘ jk∂ ‘ i‘∂ k g mRjk‘ À g kRjm‘ À g ‘Rimk ¼ 0 whereby in terms of the Ricci tensor

jk jk i‘ g ∂mRjk À g ∂kRjm À g ∂‘Rim ¼ 0

The change of the dummy index ‘ ! k in the last term provides

jk∂ jk∂ ik∂ ∂ jk ∂ jk ∂ ik g mRjk À g kRjm À g kRim ¼ 0 ) mRjk À kRjm À kRim ¼ 0

The contractions of the curvature tensors provide

∂ ∂ k ∂ k ∂ ∂ k mR À kRm À kRm ¼ 0 ) mR ¼ 2 kRm 5.5 Einstein Tensor 263 whereby

1 ∂ R k ¼ ∂ R ð5:5:1Þ k m 2 m is the divergence of a tensor, which can be written under the form 1 ∂ R k À δ k R ¼ 0 ð5:5:2Þ k m 2 m where the terms in parenthesis deﬁne the Einstein tensor with variance (1, 1)

1 G k ¼ R k À δ k R ð5:5:3Þ m m 2 m

The Einstein tensor can be written as a function of its covariant components, so 1 G ¼ g G k ¼ g R k À δ kR ð5:5:4Þ ij ik j ik j 2 j thus

1 G ¼ R À g R ð5:5:5Þ ij ij 2 ij

By means of this expression it is veriﬁed that the Einstein tensor is generated only by the metric tensor and the Ricci tensor. As Rij and gij are two symmetric tensors then Einstein tensor is symmetric. For the contravariant components of this tensor the result is

1 Gij ¼ Rij À gijR ð5:5:6Þ 2

The divergence of the Einstein tensor is given by

1 1 ∂ G j ¼ ∂ R i À δ j ∂ R ¼ ∂ R i À ∂ R i i i j i 2 i i j 2 j but

1 ∂ R i ¼ ∂ R i j 2 j then

∂ i : : iGj ¼ 0 ð5 5 7Þ 264 5 Riemann Spaces

Thus for any Riemann space the divergence of the Einstein tensor is null, and with the contraction of this tensor it follows that

1 1 G i ¼ R i À δ iR ¼ R À NR i i 2 i 2 1 G ¼À ðÞN À 2 R ð5:5:8Þ 2

For the Riemann space E2 it is veriﬁed that G ¼ 0. i i δ i Exercise 5.3 Show that the tensor of the kind Tj ¼ Rj þ j m, being m a scalar function, has the characteristics of an Einstein tensor. ∂ i The divergence of this tensor given by jTj ¼ 0 stays ∂ i ∂ i δ i∂ ∂ i jTj ¼ jRj þ j jm ¼ j Rj þ m ¼ 0 and with expression (5.4.11)

1 ∂ R i ¼ ∂ R j j 2 j substituted in this expression 1 1 1 ∂ T i ¼ ∂ R þ m ¼ 0 ) R þ m ¼ k ) m ¼ k À R j j j 2 2 1 1 2 where k1 is a constant. The substitution of this expression in the expression of i tensor Tj provides 1 T i ¼ R i À δ i R þ k j j j 2 2 where k2 ¼Àk1. Thus this tensor has the same characteristics of the Einstein tensor deﬁned by expression (5.5.3).

5.6 Particular Cases of Riemann Spaces

Some kinds of Riemann spaces will be analyzed in this item with speciﬁc characteristics that make them important: the Riemann space E2, the Riemann space with constant curvature, the Minkowski space, and the conformal space. 5.6 Particular Cases of Riemann Spaces 265

5.6.1 Riemann Space E2

In the Riemann space E2 the Ricci tensor Rij is deﬁned by its components "# R11 R12 Rij ¼ R21 R22 as R12 ¼ R21 and the metric tensor in matrix form is given by "# g11 g12 gij ¼ g21 g22 where g12 ¼ g21. The Ricci tensor written in terms of the Riemann–Christoffel curvature tensor with variance (0, 4), and considering the symmetry and the metric tensor is given by

kp pk Rij ¼ g Rpijk ¼ g Ripkj and the development provides

11 12 21 22 Rij ¼ g Ri11j þ g Ri12j þ g Ri21j þ g Ri22j whereby the result for component R11 is

11 12 21 22 R11 ¼ g R1111 þ g R1121 þ g R1211 þ g R1221

As the tensor Rpijk is antisymmetric in the ﬁrst two and the last two indexes, i.e., Rpijk ¼ÀRipjk and Rpijk ¼ÀRpikj it follows that

22 R11 ¼ 0 þ 0 þ 0 þ g R1221

22 22 Let g ¼ detgij and G the cofactor of g :

G22 g g22 ¼ ¼ 11 g g whereby

g11 R11 R1212 R11 ¼ ðÞ)ÀR1212 ¼À g g11 g

Proceeding in an analogous way for component R22: 266 5 Riemann Spaces

11 12 21 22 R22 ¼ g R2112 þ g R2122 þ g R2212 þ g R2222 11 R22 ¼ g R2112 þ 0 þ 0 þ 0 11 R22 ¼Àg R1212 G11 g g11 ¼ ¼ 22 g g g R ¼À 22 R 22 g 1212 whereby

R R 22 ¼À 1212 g22 g

For component R12, it follows that

11 12 21 22 21 R12 ¼ g R1112 þ g R1122 þ g R1212 þ g R1222 ¼ 0 þ 0 þ g R1212 þ 0 21 R12 ¼ g R1212 G21 g g21 ¼ ¼ 12 g g g R ¼À 12 R 12 g 1212 thus

R R 12 ¼À 1212 g12 g and with the symmetries Rij ¼ Rji and gij ¼ gji the result for component R21 is

R R 21 ¼À 1212 g21 g

The analysis developed shows that

R R R R R K ¼ 11 ¼ 22 ¼ 12 ¼ 21 ¼À 1212 g11 g22 g12 g21 g

These equalities indicate that in the Riemann space E2 the components of the Ricci tensor Rij are proportional to the components of the metric tensor gij and to its derivatives, and are independent of the directions considered. It is veriﬁed that the Riemann curvature does not vary with the orientation considered, then all the points 5.6 Particular Cases of Riemann Spaces 267

of the space E2 are isotropic. This, in general, is not valid for spaces with dimension N > 2. The scalar K in Riemann space E2 is called Gauß curvature. This analysis allows writing the components of the Ricci tensor as a function of the component R1212 and of the metric tensor, thus

R R ¼À 1212 g ð5:6:1Þ ij g ij

5.6.2 Gauß Curvature

Expression (5.6.1) is valid only for the Riemann space E2. The knowledge of the properties of the surfaces in the Euclidian space E3 is not useful for understanding the properties of the Riemann spaces EN, with N > 3. For N ¼ 2 several simpliﬁ- cations are admitted in the formulation of the expression of Rij, so the conclusions obtained for the Riemann space E2 cannot be generalized for the spaces of dimensions N > 3. The scalar curvature allows expressing the Riemann–Christoffel tensor Rpijk as a function of the components of the metric tensor. With the non-null components R1212, ¼ÀR2121, ¼ÀR1221 ¼ R2112, and the expression of the scalar curvature it follows that

R R 2 R R ¼ gijR ¼Àgijg 1212 ¼Àδ i 1212 ¼À R ) R ¼À g ij ij g i g g 1212 1212 2 and the development provides

R g11 g12 R R1212 ¼À ¼À ðÞg11g22 À g12g21 2 g21 g22 2

The other non-null components are obtained by means of the indexes in this expression, and considering the symmetry of tensor Rpijk it follows that

R R ¼À ðÞg g À g g 2121 2 22 11 21 12 R R ¼À ðÞg g À g g 1221 2 12 21 11 22 R R ¼À ðÞg g À g g 2121 2 21 12 22 11 268 5 Riemann Spaces then R R ‘ ¼À g g À g g ð5:6:2Þ ijk 2 ik j‘ i‘ jk or : : Rijk‘ ¼ÀKgikgj‘ À gi‘gjk ð5 6 3Þ

The Gauß curvature, that in general depends on the coordinates of the point considered, is determined by

1 K ¼ R ð5:6:4Þ 2 that can be obtained as a function of the Riemann–Christoffel curvature tensor with variance (0, 4), and with the Ricci pseudotensor for the Riemann space E2

ffiffiffi ij p ij e εij ¼ geij ε ¼ pffiffiffi g and with the expression

R K ¼ 1212 g then

Rijk‘ ¼ Kεijεk‘ ð5:6:5Þ

The multiplication of both members of this expression by εijεk‘ provides

ij k‘ ij k‘ ε ε Rijk‘ ¼ Kεijεk‘ε ε and as

ε εij δ i ij ¼ i ¼ 2 thus

1 ij k‘ K ¼ R ‘ε ε ð5:6:6Þ 4 ijk this expression shows that the Gauß curvature is an invariant. 5.6 Particular Cases of Riemann Spaces 269

5.6.3 Component R1212 in Orthogonal Coordinate Systems

For the orthogonal coordinate systems in the Riemann space EN expression (5.2.24) provides the component ! ∂2 ∂2 ∂2 ÀÁ 1 g12 g11 g22 q ‘ q ‘ R ¼ 2 À À þ g ‘ Γ Γ À Γ Γ 1212 2 ∂x1∂x2 ∂x2∂x2 ∂x1∂x1 q 12 12 11 22 or more explicitly ! 1 ∂2g ∂2g ÀÁÀÁ R ¼À 11 þ 22 þ g Γ1 Γ1 À Γ1 Γ1 þ g Γ2 Γ2 À Γ2 Γ2 1212 2 ∂x2∂x2 ∂x1∂x1 11 12 12 11 22 22 12 12 11 22

The Christoffel symbols for these coordinates systems are given by 8 > 1 ∂g > Γ1 11 < 11 ¼ 1 ∂ 2g ∂x Γ k Γ i 1 gii 11 – i ¼ j ¼ k ) ij ¼ ii ¼ ∂ j ) 2gii x > 1 ∂g :> Γ2 ¼ 22 22 ∂ 2 8 2g22 x > 1 ∂g > Γ2 11 < 11 ¼À 2 ∂ 2g ∂x Γ k Γ k 1 gii 22 – i ¼ j 6¼ k ) ij ¼ ii ¼À ∂ k ) 2gkk x > 1 ∂g :> Γ1 ¼À 22 22 ∂ 1 8 2g11 x > 1 ∂g > Γ1 11 < 12 ¼ 2 ∂ 2g ∂x Γ k Γ i 1 gii 11 – i ¼ k 6¼ j ) ij ¼ ij ¼ ∂ j ) 2gii x > 1 ∂g :> Γ2 ¼ 22 12 ∂ 1 2g22 x – For i 6¼ j, j 6¼ k, i 6¼ k it results in Γij, k ¼ 0 so !"# 1 ∂2g ∂2g 1 ∂g 2 ∂g ∂g R ¼À 11 þ 22 þ 11 þ 11 22 1212 ∂ 2∂ 2 ∂ 1∂ 1 ∂ 2 ∂ 1 ∂ 1 2 x x x x 4g11 x x x "# 1 ∂g 2 ∂g ∂g þ 22 þ 11 22 ∂ 1 ∂ 2 ∂ 2 4g22 x x x 1 ∂ 1 ∂g ∂ 1 ∂g ¼À ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi 22 þ ffiffiffiffiffiffiffiffiffiffiffiffi 11 p ∂ 1 p ∂ 1 ∂ 2 p ∂ 2 2 g11g22 x g11g22 x x g11g22 x 270 5 Riemann Spaces or ∂ ∂ ∂ ∂ 1 1 g22 1 g11 R1212 ¼À pffiffiffi pffiffiffi þ pffiffiffi ð5:6:7Þ 2 g ∂x1 g ∂x1 ∂x2 g ∂x2

Exercise 5.4 Calculate the components of tensors Rijk‘, Rij, and the Gauß curvature 2 2 1 2 2 2 2 for the space E2 defined by the fundamental form ds ¼ c ðÞdx À f ðÞt ðÞdx where c2 is a constant. The metric tensor and conjugated metric tensor are given, respectively, by "#"# 2 À2 c 0 ij c 0 gij ¼ g ¼ 0 Àf 2ðÞt 0 Àf À2ðÞt then pffiffiffi g ¼ c2f 2i2 ) g ¼ cf i where i2 ¼À1 is the imaginary number and with expression (5.6.8) ∂ ∂ ∂ ∂ 1 1 g22 1 g11 R1212 ¼À pffiffiffi pffiffiffi þ pffiffiffi 2 g ∂x1 g ∂x1 ∂x2 g ∂x2 it follows that 1 ∂ 1 ∂g 1 ∂ 1 ÀÁ 1 ∂ 2f_ R ¼À 22 ¼À ÁÀ2f f_ ¼ 1212 2cf i ∂x1 cf i ∂x1 2cf i ∂x1 cf i 2cf i ∂x1 ci €f €f ¼ ¼À c2fi2 c2f

For the components of the Ricci tensor it follows that

pk Rij ¼ g Ripkj € € 22 1 f f R11 ¼ g R1212 ¼À À ¼ f 2 c2f c2f 3 1 €f €f R ¼ g11R ¼ À ¼À 22 1212 c2 c2f c4f 12 R12 ¼ R21 ¼ g R1212 ¼ 0 and for the Gauß curvature it results in

€f € R À 2 f K ¼ 1212 ¼ c f ¼ g c2f 2i2 c4f 3 5.6 Particular Cases of Riemann Spaces 271

5.6.4 Einstein Tensor

For the particular case in which the metric, the metric tensor, and its conjugated tensor are given, respectively, by ÀÁÀÁ ÀÁÀÁ 2 2 ds2 ¼ hx1; x2 dx1 þ hx1; x2 dx2 2 3 "# 1 6 0 7 h 0 ij 6 h 7 g ¼ g ¼ 4 5 ij 0 h 1 0 h

1; 2 > 2 where hxðÞx 0 is a function of the coordinates, g ¼ detgij ¼ h , and the Ricci tensor is expressed by

pk 11 12 21 22 Rij ¼ g Ripkj ¼ g Ri11j þ g Ri12j þ g Ri21j þ g Ri22j then

1 ÀÁ R ¼ R þ R ij h i11j i22j

Developing this expression and with the symmetry of tensor Ripkj it follows that

1 1 1 1 R ¼ ðÞR þ R ¼ R R ¼ ðÞR þ R ¼ R 11 h 1111 1221 h 1221 22 h 2112 2222 h 2112 1 1 R ¼ ðÞ¼R þ R 0 R ¼ ðÞ¼R þ R 0 12 h 1112 1222 21 h 2111 2221

Let the scalar curvature

ij 11 12 21 22 11 22 R ¼ g Rij ¼ g R11 þ g R12 þ g R21 þ g R22 ¼ g R11 þ 0 þ 0 þ g R22 11 22 ¼ g R11 þ g R22 and with the components of the Ricci tensor as a function of the components of tensor Ripkj it follows that

1 1 1 1 R ¼ R þ R h h 1221 h h 2112

As Ripkj ¼ Rpijk it results for the scalar curvature 272 5 Riemann Spaces

1 2 R ¼ ðÞ¼R1221 þ R1221 R1221 h2 h2 then

h2 R ¼ R 1221 2 and with the substitution of this expression in the expressions of the components of the Ricci tensor it follows that

1 h2 h R R ¼ R ¼ R ¼ g 11 h 2 2 2 11 1 h2 h R R R ¼ R ¼ g 22h 2 2 2 22

R12 ¼ R21 ¼ 0

These expressions allow relating the Ricci tensor with the scalar curvature and with the metric tensor, thus

R R ¼ g ð5:6:8Þ ij 2 ij and with the deﬁnition of the scalar curvature given by expression (5.4.6) and with the previous expression it follows

R R R R ¼ gijR ¼ gijg ¼ δ i ¼ N ij ij 2 i 2 2 or N R 1 À ¼ 0 ð5:6:9Þ 2 then for the Riemann space E2 it is veriﬁed that Rij ¼ R ¼ 0. Consider the Einstein tensor given by its covariant components

1 1 1 G ¼ R À g R ¼ÀKg À g R ¼ÀKg À g ðÞ¼À2K 0 ij ij 2 ij ij 2 ij ij 2 ij then the tensor Gij is null for the Riemann space E2. 5.6 Particular Cases of Riemann Spaces 273

5.6.5 Riemann Space with Constant Curvature

i The Riemann curvature in point x 2EN, in general, depends on this point in which it is deﬁned and the vectors u and v that establish the plane π with respect to which it is calculated. It is admitted that this dependency does not exist, i.e., the space is isotropic, then the relation of the isotropy of the space with the Riemann curvature is established by the following theorem.

Schur Theorem

If all the points of a neighborhood in the Riemann space EN, being N > 2, are isotropic, then the curvature K is constant in all this neighborhood.

To prove the validity of this theorem, let expression (5.3.13) be rewritten as

Rijk‘ ¼ Gijk‘K ð5:6:10Þ with

Gijk‘ ¼ gikgj‘ À gi‘gjk 6¼ 0

m valid in the neighborhood of point x of Riemann space EN. The covariant derivative of expression (5.6.11) with respect to variable xm is given by

∂mRijk‘ ¼ Gijk‘∂mK ð5:6:11Þ

∂ gij ∂ ‘ with mGijk ¼ 0, because, in general, ∂xm ¼ 0. With the permutation of indexes in the expression (5.6.12)

∂kRij‘m ¼ Gij‘m∂kK ð5:6:12Þ

∂‘Rijmk ¼ Gijmk∂‘K ð5:6:13Þ

The sum of expressions (5.6.12)–(5.6.14) provides

∂mRijk‘ þ ∂kRij‘m þ ∂‘Rijmk ¼ Gijk‘∂mK þ Gij‘m∂kK þ Gijmk∂‘K but the left side of expression is the second Bianchi identity thus

Gijk‘∂mK þ Gij‘m∂kK þ Gijmk∂‘K ¼ 0 and multiplying the terms of this expression by gikgj‘ it follows 274 5 Riemann Spaces ik j‘ ∂ ik j‘ δ kδ ‘ δ kδ ‘ 2 g g Gijk‘ mK ¼ g g gikgj‘ À gi‘gjk ¼ k ‘ À ‘ k ¼ N À N ik j‘ ∂ ik j‘ δ kδ ‘ δ k δ ‘ δ k δ k g g Gij‘m kK ¼ g g gi‘gjm À gimgj‘ ¼ ‘ m À m ‘ ¼ m À N m ik j‘ ∂ ik j‘ δ k δ ‘ δ kδ ‘ δ ‘ δ ‘ g g Gijmk ‘K ¼ g g gimgjk À gikgjm ¼ m k À k m ¼ m À N m

The sum of these three terms provides ÀÁÀÁÀÁ 2 ∂ δ k δ k ∂ δ ‘ δ ‘ ∂ N À N mK þ m À N m kK þ m À N m ‘K ¼ 0 it follows that ÀÁ 2 N À N ∂mK þ ðÞ1 À N ∂mK þ ðÞ1 À N ∂mK ¼ 0 whereby ÂÃÀÁ 2 N À N þ 21ðÞÀ N ∂mK ¼ 0 ð5:6:14Þ

m For N > 2 this expression is null only if ∂mK ¼ 0, and as x is an arbitrary coordinate it is concluded that K is constant in the neighborhood of this point in the Riemann space EN, which proves the Schur theorem. Expression (5.3.13), where K is a constant is the necessary and sufﬁcient condition so that the curvature of the Riemann space EN is independent of the orientation considered.

5.6.6 Isotropy

Another characteristic of this type of space is related with a scalar curvature. Let expression (5.3.13) be rewritten as

Rijk‘ ¼ Kgikgj‘ À gi‘gjk and multiplied by g‘i ‘i ‘i δ ‘ δ ‘ Rjk ¼ g Rijk‘ ¼ Kg gikgj‘ À gi‘gjk ¼ K k gj‘ À ‘ gjk ¼ Kgjk À Ngjk then

: : Rjk ¼ KðÞ1 À N gjk ð5 6 15Þ 5.6 Particular Cases of Riemann Spaces 275

For the scalar curvature it follows that

k kj kj δ k Rk ¼ g Rjk ¼ g KðÞ1 À N gjk ¼ KðÞ1 À N k whereby

R ¼ KðÞ1 À N N ð5:6:16Þ

This formulation shows that in the Riemann space E2 the tensor Rijk‘ leads to the Gauß curvature K, which is the reason for adopting the denomination curvature tensor by extension of this particular case for Riemann spaces of N dimensions. For the Riemann space EN, where N > 2, in which the Ricci tensor results from the substitution of expression (5.6.17) in expression (5.6.16), thus

R R ¼ g ð5:6:17Þ ij N ij

R where the ratio N deﬁnes a scalar. The space in which the Ricci tensor is proportional to the metric tensor is called the Einstein space. The scalar curvature of the Einstein space is given by

K gpiR ¼ gpig ij N ij following for the Ricci tensor with variance (1, 1)

K R p ¼ δ p j N i

The covariant derivative of this expression with respect to variable xp is given by

p K ∂δ ∂ R p ¼ j ¼ 0 p j N ∂xj and with expression (5.4.11)

1 ∂R ∂ R p ¼ ¼ 0 p j 2 ∂xj whereby

∂R ¼ 0 ð5:6:18Þ ∂xj then the Einstein space has constant curvature, i.e., is isotropic. The multiplying of expression (5.6.18) by vector uj allows researching the eigenvalues of the Ricci tensor, thus 276 5 Riemann Spaces R R R R uj ¼ g uj ¼ u ) R À δ uj ¼ 0 ij N ij N i ij N ij

R where the scalar curvature is constant then the eigenvalues are equal to N. In this case the eigenvectors of tensor Rij are undetermined.

Exercise 5.5 Calculate the components of the curvature tensor Rijk‘, of the Ricci tensor Rij, the scalar curvature and the Gauß curvature K for the bidimensional spherical space which metric is given by ÀÁ ds2 ¼ r2 dφ2 þ sin 2φdθ2

The metric tensor, the determinant g, and the conjugated tensor of gij are given, respectively, by 2 3 "# 1 2 6 2 0 7 r 0 4 2 ij 6 r 7 g ¼ g ¼ r sin φ g ¼ 4 5 ij 2 2φ 1 0 r sin 0 r2 sin 2φ

For the partial derivatives of the metric tensor the result is g11, 1 ¼ g22, 2 ¼ 0, following for the Christoffel symbols

Γ1 Γ2 Γ2 Γ2 Γ2 11 ¼ 22 ¼ 11 ¼ 12 ¼ 21 ¼ 0 g11g 1 ∂ðÞr2 sin 2φ cos φ Γ1 ¼ Γ1 ¼ 11, 2 ¼ ¼À 12 21 2 2r2 sin 2φ ∂φ sin φ g11g 1 ∂ðÞr2 sin 2φ Γ1 ¼À 22, 1 ¼ ¼Àsin φ Á cos φ 22 2 2r2 ∂φ thus

i i ∂Γ ‘ ∂Γ R i ¼ j À jk þ Γ mΓ i À Γ mΓ i jk‘ ∂xk ∂x‘ j‘ mk jk m‘ ∂Γ1 R ¼ g R m ¼ g R1 ¼ g 22 À Γ2 Γ1 1212 1m 212 11 212 11 ∂x1 21 22 ∂ cos φ R ¼ r2 ðÞÀÀ sin φ Á cos φ ÁÀðÞsin φ Á cos φ ¼ r2 sin 2φ 1212 ∂φ sin φ R 1 1 K ¼ 1212 ¼ r2 sin 2φ ¼ g r4 sin 2φ r2

For the Ricci tensor it follows that 5.6 Particular Cases of Riemann Spaces 277

1 1 g11 ¼ g22 ¼ r2 r2 sin 2φ pq Rij ¼ g Ripkj 1 R ¼ g22R ¼ r2 sin 2φ ¼ 1 11 1212 r2 sin 2φ 1 R ¼ g11R ¼ r2 sin 2φ ¼ sin 2φ 22 1212 r2 12 R12 ¼ R21 ¼ g R1212 ¼ 0

4 2 As R1212 ¼ Kg the space is curved, and with g ¼ r sin φ results in

1 K ¼ r2 then if the radius r is large K ! 0, i.e., the Gauß curvature is small. i i δ i Exercise 5.6 For the tensorial equation Aj ¼ Rj þ j ðÞaR þ b , where a and b are constants, and R is the scalar curvature, calculate the value of a for which the ∂ i condition iAj ¼ 0 exists. The derivative of the equation given with respect to the variable xj stays ∂R ∂ A i ¼ ∂ R i þ ∂ δ i ðÞþaR þ b δ i a þ 0 i j i j i j j ∂xi

∂ δ i where i j ¼ 0, then

∂R ∂ A i ¼ ∂ R i þ δ ia i j i j j ∂xi

∂ i and with the condition iAj ¼ 0 it follows that

∂R ∂R ∂ R i þ δ ia ¼ ∂ R i þ a ¼ 0 i j j ∂xi i j ∂xj

Having

1 ∂R ∂ R i ¼ i j 2 ∂xj that substituted in the previous expression provides 278 5 Riemann Spaces ∂R 1 þ a ¼ 0 ∂xj 2

∂R As ∂xj 6¼ 0 it results in 1 a ¼À 2

> Exercise 5.7 Analyze the curvature of the Riemann spaceEN, N 2, which ‘ ρδ‘ δ ‘ Riemann–Christoffel curvature tensor is given by Rijk ¼ j gik À k gij , where ρ is a constant. The contraction of the curvature tensor in the indexes ‘ ¼ k provides ‘ ρδ‘ δ ‘ Rij‘ ¼ j gi‘ À ‘ gij it follows that ‘ ρ ρ Rij‘ ¼ gij À Ngij ¼ ðÞ1 À N gij

With the constant

σ ¼ ρðÞ1 À N it results in

‘ σ Rij‘ ¼ gij

The multiplying of the members by gij it follows that

ij σ ij σδi ρ g Rij ¼ g gij ¼ i ¼ ðÞ1 À N N then for an Einstein space the scalar curvature is constant.

Exercise 5.8 Calculate the components of the Riemann–Christoffel tensor for E2, which metric is given by

ds2 ¼ dx2 þ GxðÞ; y dy2

The metric tensor and the conjugated metric tensor are given, respectively, by 5.6 Particular Cases of Riemann Spaces 279

"#2 3 10 10 g ¼ gij ¼ 4 1 5 ij 0 Gx; y 0 ðÞ GxðÞ; y

The derivatives of the metric tensor are g11, x ¼ g11, x ¼ g12, x ¼ g21, x ¼ g11, xx ¼ g12, xx ¼ g21, xx ¼ g11, yy ¼ g12, yy ¼ g21, yy ¼ 0 ; ; ; g22, x ¼ GxðÞy , x g22, y ¼ GxðÞy , y g22, xx ¼ GxðÞy , xx and the Christoffel symbols stay

Γ1 Γ1 Γ1 Γ2 11 ¼ 12 ¼ 21 ¼ 11 ¼ 0 1 1 1 Γ1 ¼ g1kΓ ¼ g1k Àg ¼ g11 À GxðÞ; y ¼À GxðÞ; y 22 22, k 2 22, k 2 , x 2 , x 2 2 2k 2K 1 221 1 Γ ¼ Γ ¼ g Γ12, k ¼ g g ¼ g GxðÞ; y ¼À GxðÞ; y 12 21 2 2k, 1 2 , x 2GxðÞ; y , x 2 2k 2K 1 221 1 Γ ¼ g Γ22, k ¼ g g ¼ g GxðÞ; y ¼À GxðÞ; y 22 2 2k, 2 2 , y 2GxðÞ; y , y

The Riemann–Christoffel curvature tensor with variance (0, 4) is given by ! ∂2 ∂2 ∂2 ∂2 1 gik gpk gji gpj q ‘ q ‘ R ¼ þ À À þ g ‘Γ Γ À g ‘Γ Γ pijk 2 ∂xj∂xp ∂xj∂xi ∂xk∂xp ∂xk∂xi q pk ij q pj ik

In space E2 this tensor has a single independent non-null component, then ! ∂2 ∂2 ∂2 ∂2 1 g12 g21 g22 g11 q ‘ q ‘ R ¼ þ À À þ g ‘Γ Γ À g ‘Γ Γ 1212 2 ∂x1∂x2 ∂x2∂x1 ∂x1∂x1 ∂x2∂x2 q 21 12 q 22 11 ! 1 ∂2g R ¼ À 22 þ g Γ2 Γ2 1212 2 ∂x1∂x1 22 21 12 hihi 1 1 2 R1212 ¼ ÀGxðÞ; y þ GxðÞ; y GxðÞ; y 2 , xx 4G2ðÞx; y , x hi 1 1 2 R ¼À GxðÞ; y þ GxðÞ; y 1212 2 , xx 4GxðÞ; y , x 280 5 Riemann Spaces

5.6.7 Minkowski Space

The Riemann space that links three coordinates deﬁned by lengths and a fourth coordinate related to time is called the Minkowski space. The metric of this Riemann space E4 is deﬁned by ÀÁ ÀÁ ÀÁ ÀÁ 2 2 2 2 ds2 ¼ dxidxi ¼ dx1 þ dx2 þ dx3 þ dx4 where the fourth coordinate is x4 ¼ ict, where i2 ¼À1 is the imaginary number, c is a constant, and t is the time variable, so for the fundamental form the result is ÀÁ ÀÁ ÀÁ 2 2 2 ds2 ¼ dx1 þ dx2 þ dx3 À c2ðÞdt 2 ð5:6:19Þ and the metric tensor is given by 2 3 100 0 6 7 6 7 6 010 07 g ¼ 6 7 ð5:6:20Þ ij 4 001 05 000Àc2

This tensor is not positive definite, so the Minkowski space is not Euclidian. It is ∂ gij verified promptly that ∂xk ¼ 0, 8i, j, k ¼ 1, 2, 3, 4, then all the Christoffel symbols ‘ are null, whereby Rijk ¼ 0, which shows that this space is flat. It is stressed that every Euclidian space is flat, but not every flat space is Euclidian, as the case of the Minkowski space. The fundamental form and the metric tensor of the Minkowski space in spherical coordinates are given, respectively, by

ds2 ¼ dr2 þ r2dφ2 þ r2 sin 2φdθ2 À c2dt2 ð5:6:21Þ 2 3 10 0 0 6 7 6 2 7 6 0 r 007 g ¼ 6 7 ð5:6:22Þ ij 4 00r2 sin 2φ 0 5 00 0 Àc2

Exercise 5.9 Calculate the components of the Riemann–Christoffel tensor of the 2 2 2 space deﬁned by metric ds2 ¼ ðÞdx1 þ ðÞdx2 þ ðÞdx3 À eÀtðÞdt 2. The metric tensor and its conjugated tensor are given, respectively, by 5.6 Particular Cases of Riemann Spaces 281 2 3 2 3 100 0 1000 6 7 6 7 6 7 6 7 6 010 07 ij 6 01007 g ¼ 6 7 g ¼ 6 7 ij 4 001 05 4 00105 000eÀt 000et

The unique non-null Christoffel symbol of second kind is

1 1 1 Γ ¼ g ¼À eÀt ) Γ4 ¼ g44Γ ¼À 44, 4 2 44,4 2 44 44,4 2

As the Riemann–Christoffel curvature tensor is deﬁned by expression

‘ ∂Γ ‘ ‘ ∂Γ ‘ ‘ R ¼ ik À ij þ Γ Γ m À Γ Γ m ijk ∂xj ∂xk mj ik mk ij

Γ4 ‘ and 44 is a constant value thus Rijk ¼ 0, then this is a ﬂat space.

5.6.8 Conformal Spaces

5.6.8.1 Initial Concept

A functional relation is called conformal when the domain D of a set of complex variables in a plane generates a contradomain of values of complex variables in another plane, preserving the angle and the direction between the curves that intersect. This concept is generalized for the case of the variables in the Riemann e space EN and in the conformal spaceEN. Consider these two spaces and a coordinate i e system X , with the relation between its metric tensors gij, gij given by

e 2ϕ : : gij ¼ e gij ð5 6 23Þ

i i 3 where x 2D EN, ϕðÞx > 0 is a scalar function of class C . The angles between two vectors u, v tangent to two curves in these two Riemann spaces are given by 282 5 Riemann Spaces

g uivj g uivj g uivj cos α ¼ ij ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiij p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiij p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε k m ε k m ε k m k m kku Á kkv Á gkmu u Á gkmv v gkmu u gkmv v g uivj ge uivj e2ϕg uivj cos αe ¼ ij ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiij p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiijp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e e ε e k m ε e k m ε 2ϕ k m ε 2ϕ k m kku Á kkv Á gkmu u Á gkmv v Á e gkmu u Á e gkmv v g uivj ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiij p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε k m k m gkmu u gkmv v ð5:6:24Þ where ε ¼Æ1 is a functional operator. This expression shows that α ¼ αe, then the expression (5.6.24) represents a conformal transformation. e In spaces EN and EN the conjugated metric tensors are related by

ϕ geij ¼ eÀ2 gij ð5:6:25Þ and with the following expressions being valid in these spaces – Basis vectors

ϕ ei ¼ e2 ei ð5:6:26Þ À2ϕ ei ¼ e ei ð5:6:27Þ

– Norm of a vector qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e k m 2ϕ k m : : kku ¼ gkmu u ¼ e gkmu u ð5 6 28Þ

– Dot product of vectors

e k m 2ϕ k m : : u Á v ¼ gkmu v ¼ e gkmu v ð5 6 29Þ

5.6.8.2 Christoffel Symbols

e e Let the Christoffel symbol of ﬁrst kind be Γjk, m for the conformal space EN, which relates with the Riemann space EN by means of the expressions (5.6.24) and (5.6.26), then 5.6 Particular Cases of Riemann Spaces 283 ÀÁ Γe 1 e e e 1 2ϕ 2ϕ 2ϕ jk, m ¼ gjm, k þ gkm, j À gjk, m ¼ e gjm þ e gkm À e gjk 2 2 , k , j , m h 1 2ϕϕ 2ϕ 2ϕϕ 2ϕ ¼ 2e , kgjm þ e gjm, k þ 2e , jgkm þ e gkm, j 2 i 2ϕϕ 2ϕ À 2e , mgjk þ e gjk, m 1 ¼ e2ϕ g þ g À g þ ϕ g þ ϕ g À ϕ g 2 jm, k km, j jk, m , k jm , j km , m jk

e whereby the expression for this afﬁne connection in the conformal space EN stays hi Γe 2ϕ Γ ϕ ϕ ϕ : : jk, m ¼ e jk, m þ , kgjm þ , jgkm À , mgjk ð5 6 30Þ

For the Christoffel symbol of second kind the result is

Γe i eimΓe À2ϕ imΓe jk ¼ g jk, m ¼ e g jk, m it follows that hi Γe i À2ϕ im 2ϕ Γ ϕ ϕ ϕ jk ¼ e g e jk, m þ , kgjm þ , jgkm À , mgjk imΓ im ϕ ϕ ϕ ¼ g jk, m þ g , kgjm þ , jgkm À , mgjk

e whereby the expression for this afﬁne connection in the conformal space EN stays Γe i Γ i ϕ δ i ϕ δ i ϕ im : : jk ¼ jk þ , k j þ , j k À , mg gjk ð5 6 31Þ

Expressions (5.6.31) and (5.6.32) show that the Christoffel symbols are not invariant for the conformal transformation given by expression (5.6.24).

5.6.8.3 Riemann–Christoffel tensor

e The deﬁnition of the Riemann–Christoffel tensor in the conformal space E N is given by e 1 e m e n e m e n R ‘ ¼ eg þ ge À ge À ge þ ge Γ Γ À Γ Γ ð5:6:32Þ ijk 2 i‘, kj jk, ‘i j‘, ki ik, ‘j mn jk i‘ j‘ ik 284 5 Riemann Spaces

For the derivatives of the metric tensor it follows that

e 2ϕ e ϕ 2ϕ 2ϕ gi‘ ¼ e gi‘ ) gi‘, k ¼ 2 , ke gi‘ þ e gi‘, k e ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ 2ϕ gi‘, kj ¼ 4 , k , je gi‘ þ 2 , kje gi‘ þ 2 , ke gi‘, j þ 2 , je gi‘, k þ e gi‘, kj and in an analogous way

e ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ 2ϕ gjk, ‘i ¼ 4 , ‘ , ie gjk þ 2 , ‘ie gjk þ 2 , ‘e gjk, i þ 2 , ie gjk, ‘ þ e gjk, ‘i e ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ 2ϕ gj‘, ki ¼ 4 , k , ie gj‘ þ 2 , kie gj‘ þ 2 , ke gj‘, i þ 2 , ie gj‘, k þ e gj‘, ki e ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ ϕ 2ϕ 2ϕ gik, ‘j ¼ 4 , ‘ , je gik þ 2 , ‘je gik þ 2 , ‘e gik, j þ 2 , je gik, ‘ þ e gik, ‘j

For the Christoffel symbols by means of expression (5.6.32) it follows that Γe m Γ m ϕ δ m ϕ δ m ϕ mi jk ¼ jk þ , k j þ , j k À , ig gjk ÀÁ e n n n n nm Γ ¼ Γ þ ϕ ‘δ þ ϕ δ À ϕ g g i‘ i‘ , i , i ‘ , m i‘ Γe m Γ m ϕ δ m ϕ δ m ϕ mi j‘ ¼ j‘ þ , ‘ j þ , j ‘ À , ig gj‘ ÀÁ Γe n Γ n ϕ δ n ϕ δ n ϕ mn ik ¼ ik þ , k i þ , i k À , ng gik

The substitution of these expressions in expression (5.6.33) leads to 8 h 9 > ÀÁÀÁ> > R ‘ þ g ‘ ϕ À ϕ ϕ þ g ϕ ‘ À ϕ ϕ ‘ > > ijk i , jk , j , k jk , i , i , > < ÀÁÀÁi = e 2ϕ ϕ ϕ ϕ ϕ ϕ ϕ Rijk‘ ¼ e > Àgik , j‘ À , j , ‘ À gj‘ , ik À , i , k > > hi> > ÀÁÀÁ> : mnϕ ϕ mnϕ ϕ ; þ gi‘gjk g , m , n À gikgj‘ g , m , n

Putting

ϕ ϕ ϕ ϕ ϕ jk ¼ kj ¼ , jk À , j , k ϕ ϕ ϕ ϕ ϕ i‘ ¼ ‘i ¼ , i‘ À , i , ‘ ϕ ϕ ϕ ϕ ϕ j‘ ¼ ‘j ¼ , j‘ À , j , ‘ ϕ ϕ ϕ ϕ ϕ ik ¼ ki ¼ , ik À , i , k results in hi e 2ϕ ϕ ϕ ϕ ϕ mnϕ ϕ Rijk‘ ¼ e Rijk‘ þ gi‘ jk þ gjk i‘ À gik j‘ À gj‘ ik þ g , m , n gi‘gjk À gikgj‘ ð5:6:33Þ then the Riemann–Christoffel tensor is not invariant for the transformation as deﬁned by expression (5.6.24). 5.6 Particular Cases of Riemann Spaces 285

5.6.8.4 Ricci Tensor

e The deﬁnition of the Ricci tensor in the conformal space EN is given by

e ik e Rjk ¼ ge Rijk‘ ð5:6:34Þ and with the substitution of expression (5.6.26) in expression (5.6.35) it follows that hi e À2ϕ ik ϕ ϕ ϕ ϕ mnϕ ϕ Rj‘ ¼ e g Rijk‘ þ gi‘ jk þ gjk j‘ À gik j‘ À gj‘ ik þ g ,m ,n gi‘gjk À gikgj‘ e δ kϕ δ iϕ δ iϕ ik ϕ mnϕ ϕ δ k δ i Rj‘ ¼ Rj‘ þ ‘ jk þ j j‘ À i j‘ À g gj‘ ik þ g ,m ,n ‘ gjk À i gj‘

Putting

ϕ ikϕ ¼ g ik then ÀÁ ϕ mnϕ mn ϕ ϕ ϕ ¼ g mn ¼ g , mn À , m , n ð5:6:35Þ mnϕ ϕ mnϕ mnϕ g , m , n ¼ g , mn À g mn thus e ϕ ϕ ϕ ϕ mnϕ ϕ Rj‘ ¼ Rj‘ þ j‘ þ j‘ À N j‘ À gj‘ þ g , m , n gj‘ À Ngj‘ whereby

e ϕ mnϕ mnϕ ϕ : : Rj‘ ¼ Rj‘ À ðÞN À 2 j‘ À gj‘g , mn À ðÞN À 2 gj‘g , m , n ð5 6 36Þ

e is the expression for the Ricci tensor in the conformal space EN, which is not invariant for the transformation as deﬁned by expression (5.6.24).

5.6.8.5 Scalar Curvature

The deﬁnition of the scalar curvature is given by

e j‘ e R ¼ ge Rj‘ and with the substitution of expression (5.6.26) it follows that 286 5 Riemann Spaces hi e ej‘ ϕ mnϕ mnϕ ϕ R ¼ g Rj‘ À ðÞN À 2 j‘ À gj‘g , mn À ðÞN À 2 gj‘g , m , n hi À2ϕ j‘ ϕ mnϕ mnϕ ϕ ¼ e g Rj‘ À ðÞN À 2 j‘ À gj‘g , mn À ðÞN À 2 gj‘g , m , n whereby ÈÉÂÃ e À2ϕ ϕ ϕ ϕ mn : : R ¼ e R þÀ2ðÞN À 1 , mn À ðÞN À 1 ðÞN À 2 , m , n g ð5 6 37Þ

e is the expression for the scalar curvature in space EN, which is not invariant for the conformal transformation deﬁned by expression (5.6.24).

5.6.8.6 Weyl Tensor

Formulation

The research of a variety that remains invariant when passing from the space EN for e the conformal space EN led Hermann Weyl to conceive a tensor that has the same properties of the Riemann–Christoffel tensor, and were invariant when a conformal transformation deﬁned by expression (5.6.24) takes place. e Let the Riemann–Christoffel tensor in space EN be deﬁned by hi e 2ϕ ϕ ϕ ϕ ϕ mnϕ ϕ Rijk‘ ¼ e Rijk‘ þ gi‘ jk þ gjk j‘ À gik j‘ À gj‘ ik þ gi‘gjk À gikgj‘ g , m , n and with the expressions

eip e e p eip À2ϕ ip g Rijk‘ ¼ Rjk‘ g ¼ e g the result is e p p δ pϕ δ pϕ ip ϕ ϕ Rjk‘ ¼ Rjk‘ þ ‘ jk À k j‘ þ g gjk i‘ À gj‘ ik δ p δ pϕ mnϕ ϕ : : þ ‘ gjk À k j‘ g , m , n ð5 6 38Þ

mn The term g ϕ,mϕ,n can be eliminated, and with expression (5.6.36) it is possible to obtain the parameters ϕj‘, ϕjk, ϕi‘, ϕik in terms of the Ricci tensor, the scalar curvature, and the metric tensor, thus 1 e 1 mn mn ϕ ‘ ¼À R ‘ þ R ‘ À g ‘g ϕ À g ‘g ϕ ϕ ð5:6:39Þ j ðÞN À 2 j j ðÞN À 2 j , mn j , m , n 5.6 Particular Cases of Riemann Spaces 287 whereby mn 1 e 2ϕ ðÞN À 2 g ‘g ϕ ¼À Re þ R À g ‘ϕ ϕ j , mn 2ðÞN À 1 2 j , m , n and with e ϕ g ‘ e2 ¼ j gj‘ the result is mn 1 e ðÞN À 2 g ‘g ϕ ¼À Rge ‘ þ Rg ‘ À g ‘ϕ ϕ ð5:6:40Þ j , mn 2ðÞN À 1 j j 2 j , m , n

The substitution of expression (5.6.39) in expression (5.6.38) provides 1 e 1 e 1 mn ϕ ‘ ¼À R ‘ À R ‘ À Rge ‘ À Rg ‘ À g ‘g ϕ ϕ j ðÞN À 2 j j 2ðÞN À 1 ðÞN À 2 j j 2 j , m , n ð5:6:41Þ

The other parameters analogous to this parameter stay 1 1 1 ϕ ¼À Re À R À Rege À Rg À g gmnϕ ϕ jk ðÞN À 2 jk jk 2ðÞN À 1 ðÞN À 2 jk jk 2 jk , m , n 1 e 1 e 1 mn ϕ ‘ ¼À R ‘ À R ‘ À Rge ‘ À Rg ‘ À g ‘g ϕ ϕ i ðÞN À 2 i i 2ðÞN À 1 ðÞN À 2 i i 2 i , m , n 1 1 1 ϕ ¼À Re À R À Rege À Rg À g gmnϕ ϕ ik ðÞN À 2 ik ik 2ðÞN À 1 ðÞN À 2 ik ik 2 ik , m , n ð5:6:42Þ and with the substitution of expressions (5.6.42) and (5.6.43) in expression (5.6.39), and with expressions (5.6.24) and (5.6.26) results in e e p 1 p e p e e p e p R pe pe R À δ R ‘ À δ R þ ge ‘R À eg R þ δ g ‘ À δ‘ g jk‘ ðÞN À 2 k j ‘ jk j k jk ‘ ðÞN À 1 ðÞN À 2 k j jk p 1 p p p p ¼ R À δ R ‘ À δ R þ g ‘R À g R jk‘ ðÞN À 2 k j ‘ jk j k jk ‘ R p p þ δ g ‘ À δ g ðÞN À 1 ðÞN À 2 k j ‘ jk ð5:6:43Þ 288 5 Riemann Spaces

Putting p p 1 p p p p W ¼ R À δ R ‘ À δ R þ g ‘R À g R jk‘ jk‘ ðÞN À 2 k j ‘ jk j k jk ‘ ð5:6:44Þ R p p þ δ g ‘ À δ g ðÞN À 1 ðÞN À 2 k j ‘ jk veriﬁes that expression (5.6.44) represents an equality between tensors

e p p W jk‘ ¼ Wjk‘

p and shows that tensor Wjk‘ is preserved when a conformal transformation, i.e., this e tensor is invariant for the space EN. p Lowering the index of tensor Wjk‘,

p gpiWjk‘ ¼ Wijk‘ whereby 1 p p W ‘ ¼ R ‘ À g R ‘ À g ‘R þ g ‘R À g R ijk ijk ðÞN À 2 ik j i jk j k jk ‘ ð5:6:45Þ R þ g g ‘ À g ‘g ðÞN À 1 ðÞN À 2 ik j i jk deﬁnes the Weyl curvature tensor, and shows that the tensor Wijk‘ is obtained by means of decomposing the Riemann–Christoffel tensor Rijk‘ in their parts com- prised by the Ricci tensor and by the scalar curvature, then the Riemann–Christoffel tensor can be decomposed into irreducible components.

Properties of the Weyl Tensor

Expression (5.6.45) indicates that the tensor Wijk‘ has the same number of indepen- 1 dent components as the tensor Rijk‘, i.e., 12 NNðÞþ 1 ðÞN þ 2 ðÞN À 3 components. The tensorial sum given by expression (5.6.45) shows that the Weyl tensor has the same properties of symmetry and antisymmetry as tensor Rijk‘, then

Wijk‘ ¼ Wk‘ij

Wijk‘ ¼ÀWjik‘ ¼ÀWij‘k

These properties indicate that the ﬁrst Bianchi identity is valid for the Weyl tensor 5.6 Particular Cases of Riemann Spaces 289

‘ ‘ ‘ Wijk þ Wjki þ Wkij ¼ 0

Wijk‘ þ Wikj‘ þ Wik‘j ¼ 0

For the Riemann space E1 the result is Rijk‘ ¼ 0. For the bidimensional space E2 there is only the component R1212, and the curvature is deﬁned by the scalar curvature. For the tridimensional space E3 the six components of the curvature tensor are deﬁned by the Ricci tensor, having Wijk‘ ¼ 0. For the space EN, N > 3 the components of Rijk‘ are determined by the Ricci tensor and by the Weyl tensor.

Uniqueness of the Weyl tensor

Let expression (5.3.13) that determines the Riemann curvature K in point xi of the isotropic space EN, N > 3, and with expression (5.6.16) the result is

Rijk‘ ¼ Kgikgj‘ À gi‘gjk

Rjk ¼ KðÞ1 À N gjk thus

R gjk K ¼ jk ðÞ1 À N jk hi Rjkg 1 jk jk R ‘ ¼ g g ‘ À g ‘g ¼ R g g ‘ g À R g g g ‘ ijk ðÞ1 À N il j i jk ðÞ1 À N jk j ik jk jk i jk δ k jk Rjkg gj‘ ¼ Rjk ‘ ¼ Rj‘ Rjkg gjk ¼ Rjk 1 ÀÁ R ‘ ¼ R ‘g À R g ‘ ð5:6:46Þ ijk ðÞ1 À N j ik jk i

The Weyl tensor is deﬁned by the expression

1 ÀÁ W ‘ ¼ R ‘ À R ‘g À R g ‘ ð5:6:47Þ ijk ijk ðÞ1 À N j ik jk i

If Wijk‘ ¼ 0 in the isotropic Riemann space EN, where N > 3 the expression (5.6.48) is null, then the expression (5.6.47) is valid for this space. This is the necessary condition so that this space has constant Riemann curvature. To demonstrate that the Riemann curvature must be constant for the condition j‘ Wijk‘ ¼ 0 the multiplying of expression (5.6.47)byg is carried out, thus 290 5 Riemann Spaces

ÀÁ j‘ j‘ 1 j‘ j‘ 1 j g R ‘ ¼Àg R ‘ ¼ÀR ¼ g R ‘g À g R g ‘ ¼ Rg À R δ ijk jik ik ðÞ1 À N j ik jk i ðÞ1 À N ik jk i 1 ¼ ðÞRg À R ðÞ1 À N ik ik results in

R ÀR ðÞ¼1 À N Rg À R ) R ¼ g ik ik ik ik N ik

This last expression is identical to expression (5.6.17) that deﬁnes an Einstein space (isotropic space, whereby it has constant curvature), which proves that this condition is sufﬁcient for the Weyl tensor to be null.

Contraction of the Weyl Tensor

The contraction of index k of the Weyl tensor Wijk‘ stays m‘ m‘ 2 m‘ 2 m‘ g W ‘ ¼ g R ‘ À g R‘ g À R‘ g þ g Rg g‘ ijk ijk ðÞN À 2 j ik i jk ðÞN À 1 ðÞN À 2 ik j m m 2 m‘ m‘ 2 m‘ W ¼ R À g R‘ g À g R‘ g þ g Rg g‘ ijk ijk ðÞN À 2 j ik i jk ðÞN À 1 ðÞN À 2 ik j 2 2 W m ¼ R m À R mg À R kg þ Rg mg ijk ijk ðÞN À 2 j ik i jk ðÞN À 1 ðÞN À 2 j ik and for m ¼ k 2 2 W k ¼ R k À R kg À R kg þ Rg kg ijk ijk ðÞN À 2 j ik i jk ðÞN À 1 ðÞN À 2 j ik 2 W ¼ R þ Rg kg ij ij ðÞN À 1 ðÞN À 2 j ik

j The contraction Wijk ¼ 0 shows that the Weyl tensor is the portion of the Riemann–Christoffel curvature tensor for which all the contractions are null, i.e., trW ¼ 0.

Weyl Tensor in the Riemann Space E4

For the Riemann space E4 the Weyl tensor deﬁned by expression (5.6.46) stays 5.7 Dimensional Analysis 291

1 1 W ‘ ¼ R ‘ À R‘ g þ R g À R g À R‘ g þ Rgg À g g ð5:6:48Þ ijk ijk 2 j ik ki j‘ kj i‘ i jk 6 ik ‘j i‘ kj

The total of components of this tensor is 256, but only 10 are algebraically independent, which are a part of the 20 components of tensor Rijk‘, having that the other 10 are due to tensor Rij. The curvature of the Riemann space E4 is determined by tensor Wijk‘, for when Rij ¼ 0 the result is Rijk‘ ¼ Wijk‘, which indicates that if the Ricci tensor is null the space is not necessarily ﬂat. The Weyl tensor is the tensor with null trace that comprises the Ricci tensor with an extra condition of having Rij ¼ 0. It is, therefore, the tensor Rijk‘ with all the contractions removed.

Exercise 5.10 Show that the Riemann space E4, which Riemann–Christoffel α α tensor is Rijk‘ ¼ gikgj‘ À gi‘gjk , where is a constant, is ﬂat. The Riemann–Christoffel tensor is given by α Rijk‘ ¼ gikgj‘ À gi‘gjk and its contraction stays ik α ik g Rijk‘ ¼ g gikgj‘ À gi‘gjk αδi k α α Rj‘ ¼ i gj‘ À g‘ gjk ¼ 4gj‘ À g‘j ¼ 3 gj‘

For the scalar curvature the result is

j‘ α j‘ α R ¼ g Rj‘ ¼ 3 g gj‘ ¼ 12

Thus, with the substitution of these values in the expression for the Weyl tensor 1 1 W ‘ ¼ R ‘ À R‘ g þ R g À R g À R‘ g þ Rgg À g g ijk ijk 2 j ik ki j‘ kj i‘ i jk 6 ik ‘j i‘ kj whereby it is veriﬁed that Wijk‘ ¼ 0. The nullity of this Weyl tensor shows that this space is ﬂat.

5.7 Dimensional Analysis

The dimensions of the various parameters of the Riemann space EN are determined 2 ε i j as a function of the formula that expresses the metric ds ¼ gijdx dx for ds being a distance, its dimension will be a length [L]. With this expression the result for the 292 5 Riemann Spaces

Table 5.3 Dimensions of the aim parameters of the Riemann space EN Parameter Definition formula Dimensions qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Metric ε i j [L] ds ¼ gijdx dx Metric tensor ds2 2 À2 gij ¼ dxidxj ½L ½U Conjugated metric tensor gij ¼ 1 ½L À2½U 2 gij Christoffel symbols First kind Γ 1 ½L 2½U À1 ij,k ¼ 2 gik,j þ gjk,i À gij,k Γ m kmΓ À1 Second ij ¼ g ij,k ½U kind ∂Γ i ∂Γ i À2 Riemann–Christoffel cur- Variance j‘ ‘ R i ¼ À k þ Γ mΓ i À Γ mΓ i ½U vature tensor (1, 3) ‘kj ∂xk ∂xj j‘ mk k‘ mj ‘ 2 À4 Variance Rijkm ¼ g‘mRijk ½L ½U (0, 4) km À2 Ricci tensor Rij ¼ g Rijkm ½U k ‘ m n À2 Riemann curvature Rk‘mnu v u v K ¼ k ‘ m n ½L ðÞgkmg‘nÀgkngm‘ u v u v Scalar curvature R ¼ 2K À2 ÀÁ½L 1 2 À4 Weyl tensor (isotropic space) ‘ ‘ ‘ Wijk ¼ Rijk þ ðÞNÀ1 Rj gik À Rjkgi‘ ½L ½U

hi 2 À2 metric tensor is gij ¼ ½L ½U , where [U] represents a dimension for the coordinates dxi, dxj measured in a curvilinear or Cartesian coordinate system, whereby it can be an angle or a length. The conjugated metric tensor has dimensions ½¼gij ½L À2½U 2. For the other parameters the dimensions are shown in Table 5.3.

Problems

5.1. Calculate the components R1212 of the Riemann–Christoffel tensor for the 2 2 2 spaces deﬁned by the metrics: (a) ds2 ¼ÀðÞx2 ðÞdx1 þ ðÞdx2 ; (b) ds2 ¼ dr2 2 2 2 þr dθ À dt . Answer:aðÞR1212 ¼ 0; ðÞb R1212 ¼ 0. 5.2. Calculate the Ricci tensor and the scalar curvature of the space deﬁned by the metric ÀÁ ÀÁ 1 2 1 2 ds2 ¼ dx1 þ dx2 ðÞx2 2 ðÞx2 2

Answer: Rij ¼Àgij, R ¼À2. 5.3. Show that the metric hiÀÁ ÀÁÀÁ hiÀÁ ÀÁÀÁ 2 2 2 2 2 2 ds2 ¼ x1 þ x2 dx1 þ x1 þ x2 dx2

deﬁnes an Euclidian space. 5.7 Dimensional Analysis 293

5.4. Calculate the Gauß curvatures of the spaces E3 deﬁned by the metrics: hihi 2 2 2 2 2 2 2 (a) ds2 ¼ ðÞx1 þ ðÞx2 ðÞdx1 þ ðÞx1 þ ðÞx2 ðÞdx2 þ ðÞdx3 ÀÁ ÀÁ 2 2 2 2 r φ2 2φ θ2 (b) ds ¼ dr þ a sin ÀÁa Á ÀÁd þ sin Á d 2 2 2 2 r φ2 2φ θ2 (c) ds ¼ dr þ a sinh a Á d þ sin Á d 1 1 Answer: (a) K ¼ 0; (b) K ¼ a2; (c) K ¼Àa2. 5.5. Calculate the Riemann–Christoffel curvature tensor of the space deﬁned by the metric ÀÁ ÀÁ ÀÁ ÀÁ 2 2 2 4 2 ds2 ¼À dx1 À dx2 À dx3 þ eÀx dx4

‘ ‘ Answer: Rijk ¼ 0, 8i, j, k, ¼ 1, 2, 3, 4. 5.6. Calculate the Riemann–Christoffel curvature tensor and the Gauß curvature of the spaces E4 deﬁned by the metric ÀÁ ÀÁÀÁ ÀÁÀÁ ÀÁÀÁ 2 2 2 2 2 2 2 ds2 ¼ dx1 þ 4 x2 dx2 þ 4 x3 dx3 À 4 x4 dx4

Answer: Rpijk ¼ 0, K ¼ 0 Chapter 6 Geodesics and Parallelism of Vectors

6.1 Introduction

The shortest distance between two points located on a surface of the Riemann space EN is related to a curve of stationary value, which equation is obtained by means of the variational calculus. This curve is called geodesic. The checking of the existence of this type of curve is carried out from the basic concepts of the elementary geometry. In the Euclidian space E3, the shortest distance between two points is a straight line, and in this case the geodesic is unique. In the case of a sphere, the shortest between two points located on its surface is an arc of the circle, which radius is the radius of the sphere. The geodesic is not necessarily unique, for instance, (a) for two points diametrically opposite in the surface of a sphere, it has several geodesics, and (b) for a circular cylinder, the geodesics depend on the positions of the points on the surface and if the points are in a generatrix, the geodesic is a straight line; otherwise, the geodesic is a spiral or an arc of circle.

6.2 Geodesics

The idea of stationary length leads to the definition of geodesic as the curve which length is minimum, keeping the initial and final point fixed. The stationary length ðB between two points A and B is calculated by the variational condition δ ds ¼ 0. A In parametric form a curve in the Riemann space EN is defined by the continuous i i 2 function x ¼ x ðÞt of class C , where t0 t t1, and the distance between two points is determined by

© Springer International Publishing Switzerland 2016 295 E. de Souza Sa´nchez Filho, Tensor Calculus for Engineers and Physicists, DOI 10.1007/978-3-319-31520-1_6 296 6 Geodesics and Parallelism of Vectors

Fig. 6.1 Geodesic in the Riemann space EN x t + εx* t A B t = t0 x t t = t1

ðb rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx i dxj s ¼ g dt ð6:2:1Þ ij dt dt a

Figure 6.1 shows a surface of the Riemann space EN containing two curves represented by parametric equations, in which the parameter t assumes the values t0 and t1 in the extreme points A and B. Admitting continuous and derivable functions which cancel each other in these extreme points, neighboring curves x(t) and xtðÞ þεx*ðÞt exist, x*(t) being a continuous class C1 parametric function that represents the change of the tracing of this curve with respect to the curve x(t), and ε is a very small value. In the case of the curve with a minimum length, i.e., the geodesic, this coefficient cancels itself. The determination of the equation for the geodesic is carried out calculating the extreme value of the functional rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxi dxj F ¼ g ð6:2:2Þ ij dt dt

The variational calculus addresses this problem by means of the Euler–Lagrange formula d ∂F ∂F À ¼ 0 ð6:2:3Þ dt ∂x_ p ∂xp and with expressions (6.2.1) and (6.2.2) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds dxi dxj F ¼ ¼ g dt ij dt dt 6.2 Geodesics 297

With the notation x_ i, x_ j for the derivatives of the coordinates with respect to the parameter t, it follows that ∂F g 1 1 ¼ ij δ i x_ j þ x_ iδ j ¼ g δ i x_ j þ g x_ iδ j ¼ g x_ j þ g x_ i ∂x_ p 2F p p 2F ij p ij p 2F pj ip and the change of the indexes i ! j and the symmetry of the metric tensor gpj ¼ gjp provide ∂F 1 g x_ j ¼ g x_ j þ g x_ j ¼ jp ∂x_ p 2F pj jp F

For the other derivatives it follows that ÀÁ d ∂F d 1 1 ∂g 1 ¼ F g x_ j þ jp x_ mx_ j þ g €x j dt ∂x_ p dt jp F ∂x m F jp d ∂F 1 dðÞ‘nF ∂g ¼ À g x_ j þ jp x_ mx_ j þ g €x j ð6:2:4Þ dt ∂x_ p F dt jp ∂x m jp ∂F 1 ∂g ¼ ij x_ ix_ j ð6:2:5Þ ∂xp 2F ∂xp

Thus with the substitution of expressions (6.2.4) and (6.2.5) in expression (6.2.3) 1 dðÞ‘nF ∂g 1 ∂g À g x_ j þ jp x_ mx_ j þ g €x j À ij x_ ix_ j ¼ 0 F dt jp ∂x m jp 2F ∂xp it follows that 1 dðÞ‘nF ∂g ∂g 2 g x_ j À 2 jp x_ mx_ j À 2g €x j þ ij x_ ix_ j ¼ 0 2F dt jp ∂x m jp ∂xp and the change of the indexes m ! i provides

dðÞ‘nF ∂g ∂g 2 g x_ j À 2 jp x_ ix_ j À 2g €xj þ ij x_ ix_ j ¼ 0 dt jp ∂x i jp ∂xp whereby dðÞ‘nF ∂g ∂g 2 g x_ j À 2 jp À ij x_ ix_ j À 2g €x j ¼ 0 ð6:2:6Þ dt jp ∂x i ∂xp jp but with expression (2.4.30) 298 6 Geodesics and Parallelism of Vectors

∂g jp ¼ g Γ r þ g Γ r ð6:2:7Þ ∂x i rp ji jr pi ∂g ij ¼ g Γ r þ g Γ r ð6:2:8Þ ∂xp rj ip ir jp

The substitution of expressions (6.2.7) and (6.2.8) in expression (6.2.6) provides

dðÞ‘nF 2 g x_ j À 2g Γ rx_ ix_ j À 2g Γ r x_ ix_ j À g Γ r x_ ix_ j þ g Γ r x_ ix_ j À 2g €x j dt jp rp ji rj ip jr pi ir jp jp ¼ 0 then dðÞ‘nF 2 g x_ j À 2g Γ r À g Γ r þ g Γ r x_ ix_ j À 2g €x j ¼ 0 dt jp rp ji rj ip ir jp jp

Γ r _ i _ j In term gir jpx x the indexes i, j are dummies and then can be permutated:

Γ r _ i _ j Γ r _ j _ i gir jpx x ¼ gjr ipx x thus

dðÞ‘nF 2 g x_ j À 2g Γ rx_ ix_ j À 2g €xj ¼ 0 dt jp rp ji jp

1 pm and multiplying by 2 g

dðÞ‘nF δ mx_ j À δ mΓ rx_ ix_ j À δ m €x j ¼ 0 dt j r ji j

Γ m Γ m but with ji ¼ ij , it results in

d2x m dxi dx j dðÞ‘nF dxm þ Γ m ¼ ð6:2:9Þ dt2 ij dt dt dt dt

The N equations given by expression (6.2.9) are ordinary differential equations of the second order of the functions xm(s), and their solutions have 2N constants of integration. The solution xm(s) of this differential equation provides the expression of the geodesic in a surface of the Riemann space EN, which is determined if the m dxm initial values of x and ds are known, i.e., the coordinates of the point and the direction of the tangent vector to the geodesics in this point. Then the geodesic is unique. 6.2 Geodesics 299

6.2.1 Representation by Means of Curves in the Surfaces

The calculation of the geodesics can be carried out in an alternative way in a bidimensional space if the surface is represented by coordinates ξ1, ξ2, then with expression (6.2.9)

d2ξ1 dξ i dξj dðÞ‘nF dξ1 þ Γ1 ¼ ð6:2:10Þ dt2 ij dt dt dt dt

d2ξ2 dξ i dξj dðÞ‘nF dξ2 þ Γ2 ¼ ð6:2:11Þ dt2 ij dt dt dt dt ÀÁ ÀÁ ξ2 ξ1 ξ1 ξ2 Consider the functions ¼ f or ¼ÀÁg related with a curve in the surface, for example, with the function ξ2 ¼ f ξ1 , it follows that

dξ1 dξ2 dξ2 dξ1 ξ1 ¼ t ) ¼ 1 ) ¼ dt dt dξ1 dt d2ξ1 d2ξ2 d2ξ2 ξ2 ξ2 ÀÁ ¼ ðÞ)t ¼ 0 ) ¼ 2 dt2 dt2 dξ1

The development of expressions (6.2.10) and (6.2.11) and the substitution of expressions of the respective derivatives lead to the following differential equations:

2 dξ2 dξ2 dðÞ‘nF Γ1 þ 2Γ1 þ Γ1 ¼ ð6:2:12Þ 11 12 dξ1 22 dξ1 dξ1

2 3 d2ξ2 dξ1 ÀÁdξ1 dξ1 ÀÁ Γ1 Γ1 Γ2 Γ1 Γ1 Γ2 2 þ 22 þ 2 21 À 22 2 þ 11 À 2 21 2 À 11 2 ¼ 0 dξ1 dξ dξ dξ ð6:2:13Þ

The solutions of these differential equations provide the expression of the curve that represents the geodesic.

6.2.2 Constant Direction

In expression (6.2.9) the parameter t is arbitrary, being plausible to admit t ¼ s, then

_ i _ j dðÞ‘nF gijx x ¼ 1, i.e., F ¼ 1, thus dt ¼ 0, and the differential equation of the geodesics is simpliﬁed as 300 6 Geodesics and Parallelism of Vectors

d2x m dx i dxj þ Γ m ¼ 0 ð6:2:14Þ ds2 ij ds ds

dx i that can be expressed in another way. Let ds a unit tangent vector in each point of the geodesic, and then

dxi dxj g ¼ 1 ij ds ds

dxr will be a solution of the differential equation (6.2.14), which multiplied by 2gmr ds and when carrying out the sum with respect to the index m takes the form

dxr d2x m dxr dxi dxj 2g þ 2g Γ m ¼ 0 ð6:2:15Þ mr ds ds2 mr ds ij ds ds

As d dxm dx r dg dx m dxr d2x m dx r dxm d2x r g ¼ mr þ g þ g ds mr ds ds ds ds ds mr ds2 ds mr ds ds2 and with the permutation of the indexes m $ r in the last term to the right, it follows that d2x m dxr d dxm dxr dg dxm dxr 2g ¼ g À mr ð6:2:16Þ mr ds2 ds ds mr ds ds ds ds ds

Putting

dxr dxi dxj dx r dxi dx j 2g Γ m ¼ 2Γ mr ij ds ds ds ij, r ds ds ds and with the cyclic permutation of the indexes of the Christoffel symbol of ﬁrst kind Γij, r ¼ Γri, j, thus

dxr dxi dxj dx r dxi dx j 2g Γ m ¼ 2Γ mr ij ds ds ds ij, r ds ds ds

∂g ir ¼ Γ þ Γ ∂x j ij, r ri, j dx r dxi dx j ∂g dx r dxi dx j 2g Γ m ¼ 2 ir mr ij ds ds ds ∂x j ds ds ds whereby 6.2 Geodesics 301

dxr dx i dxj dg dxi dxj 2g Γ m ¼ ir ð6:2:17Þ mr ij ds ds ds ds ds ds

The substitution of expressions (6.2.16) and (6.2.17) in expression (6.2.15) provides d dxm dx r dg dxm dx r dg dxi dx j g À mr þ ir ¼ 0 ds mr ds ds ds ds ds ds ds ds and with the change of the indexes i ! m in the last term d dx m dxr g ¼ 0 ð6:2:18Þ ds mr ds ds that is another way of writing the differential equation of the geodesics, then its ﬁrst integral is given by

dxm dx r g ¼ constant ð6:2:19Þ mr ds ds

dxm dx r with constant ¼ 1. The terms ds and ds represent unit tangent vectors to the geodesic whereby this curve always maintains its direction. This is the necessary and sufﬁcient condition so that this curve is a geodesic.

6.2.3 Representation by Means of the Unit Tangent Vector

The previous ascertaining allows writing the geodesic’s equation in another man- ξ i dx i ner. Let the unit tangent vector to the geodesics ¼ ds , and with expression (6.2.14), it follows that 2 k ‘ m k ‘ m k d x dx dx d dx dx dx dξ ‘ þ Γ k ¼ 0 ) þ Γ k ¼ 0 ) þ Γ k ξ ξm ¼ 0 ds2 ‘m ds ds ds ds ‘m ds ds ds ‘m and with

k k ‘ k dξ ∂ξ dx ∂ξ ‘ ¼ ¼ ξ ds ∂x‘ ds ∂x‘ thus k ∂ξ ‘ þ Γ k ξm ξ ¼ 0 ∂x‘ ‘m 302 6 Geodesics and Parallelism of Vectors and the geodesic’s equation can be written as a function of the unit tangent vector and its covariant derivative as follows: ÀÁ k ‘ ∂‘ξ ξ ¼ 0 ð6:2:20Þ

6.2.4 Representation by Means of an Arbitrary Parameter

The calculation of the geodesics can be carried out considering a parameter ζ(s), and then dx m dxm dζ d2x m d2x m dζ 2 dxm d2ζ ¼ ) ¼ þ ds dζ ds ds2 dζ2 ds dζ ds2 and the substitution of these derivatives in expression (6.2.14) provides

2 2 m i j d ζ m d x dx dx 2 dx þ Γ m ¼ÀÀÁds ð6:2:21Þ dζ2 ij dζ dζ dζ 2 dζ ds which is valid for any parameter ζ(s). ζ dζ d2ζ If (s) is a linear function, it results in ds ¼ 1 and ds2 ¼ 0; then the term to the right of expression (6.2.21) is null, and the result is expression (6.2.14). Exercise 6.1 Determine the geodesic in the Riemann space E , with metric rhiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ds ¼ ðÞdx1 2 þ ðÞdx2 2ÁÁÁðÞdxN 2 given by the Cartesian coordinates. The differential equation of the geodesic is given by

d2x m dx i dxj þ Γ m ¼ 0 ds2 ij ds ds

∂ δ gij and the metric tensor of the Riemann space EN is g ¼ ij, but ∂xp ¼ 0, thus Γ m Christoffel symbols are null ij ¼ 0, so

d2x m ¼ 0 ds2

The solution of this equation is the straight line

x m ¼ ams þ bm where am and bm are constants. 6.2 Geodesics 303

Exercise 6.2 Determine the differential equations of the geodesic in the Riemann space deﬁned by the metric ds2 ¼ eÀ2ktðÞdx2 þ dy2 þ dz2 À dt2 . The coordinates are x1 ¼ x, x2 ¼ y, x3 ¼ z, and x4 ¼ t; thus, the metric tensor and the metric conjugated tensor are given, respectively, by 2 3 2 3 eÀ2kt 000 e2kt 000 6 À2kt 7 6 2kt 7 6 0e 007 ij 6 0e 007 g ¼ 4 5 g ¼ 4 5 ij 00eÀ2kt 0 00e2kt 0 0001 0001 and the non-null Christoffel symbols are

1 1 Γ1 ¼ Γ1 ¼ e2ktg ¼Àk Γ2 ¼ Γ2 ¼ e2ktg ¼Àk 14 41 2 11, 4 24 422 22, 4 1 1ÀÁ Γ3 ¼ Γ3 ¼ e2ktg ¼Àk Γ4 ¼ g44 g ¼ Γ4 ¼ Γ4 ¼ ke2kt 34 43 2 33, 4 11 2 11, 4 22 33

The ﬁrst differential equation of the geodesics

d2x dt dx dx dt þ Γ1 þ Γ1 ¼ 0 ds2 14 ds ds 41 ds ds stays

d2x dx dt À 2k ¼ 0 ds2 ds ds

In an analogous way, for the other variables, it follows that

d2y dy dt d2z dz dt À 2k ¼ 0 À 2k ¼ 0 ds2 ds ds ds2 ds ds and

d2t dx dx dy dy dz dz þ Γ4 þ Γ4 þ Γ4 ¼ 0 ) ds2 11 ds ds 22 ds ds 33 ds ds "# d2t dx 2 dy 2 dz 2 þ keÀ2kt þ þ ¼ 0 ds2 ds ds ds

Exercise 6.3 Determine the geodesics on the circular cylinder of radius r, represented by the parametric equations x1 ¼ r cos ξ1, x2 ¼ r sin ξ1, and x3 ¼ ξ2. The bidimensional space deﬁned by the surface of the cylinder which components of the metric tensor are 304 6 Geodesics and Parallelism of Vectors

∂x i ∂x j g ¼ ‘m ∂ξ‘ ∂ξm ∂x1 2 g ¼ ¼ r2 sin 2ξ1 þ r2 cos 2ξ1 ¼ r2 g ¼ g ¼ 0 11 ∂ξ1 12 21 ∂x2 2 g ¼ ¼ 1 22 ∂ξ2 and determine the metric ÀÁ ÀÁ 2 2 ds2 ¼ r2 dξ1 þ dξ2

The Christoffel symbols are all null, whereby expressions (6.2.12) and (6.2.13) stay

d2ξ2 d2ξ1 ÀÁ ÀÁ 2 ¼ 0 2 ¼ 0 dξ1 dξ2

The solutions of these differential equations are

2 1 1 2 ξ ¼ k1ξ þ k2 ξ ¼ k3ξ þ k4 where k1, k2, k3, k4 are constants and represent a circular helix. 2 1 2 For k1 ¼ 0, k3 6¼ 0 there is ξ ¼ k2, and ξ ¼ k3ξ þ k4; thus, ÀÁ ÀÁ 1 2 2 2 3 x ¼ r cos k3ξ þ k4 x ¼ r sin k3ξ þ k4 x ¼ k2 and then this curve is a circle. 1 1 2 1 For k1 6¼ 0, k3 6¼ 0 there is ξ ¼ k3ξ þ k4, and ξ ¼ k1ξ þ k2; thus, ÀÁ ÀÁ 1 2 2 2 3 1 x ¼ r cos k3ξ þ k4 x ¼ r sin k3ξ þ k4 x ¼ k1ξ þ k2 and then this curve is the generatrix of the cylinder. 1 2 2 For k1 6¼ 0, k3 ¼ 0 there is ξ ¼ k4, and ξ ¼ k1ξ þ k2; thus,

1 2 3 2 x ¼ r cos k4 x ¼ r sin k4 x ¼ k1ξ þ k2 and then the geodesics is a straight line. Exercise 6.4 Determine the geodesics on the sphere of radius r, which metric is given by ds2 ¼ r2 dφ2 þ r2 sin 2φdθ2,withx1 ¼ φ and x2 ¼ θ. The metric tensor and the conjugated metric tensor are given, respectively, by 6.2 Geodesics 305 2 3 "#1 0 r2 0 6 2 7 ij 6 r 7 g ¼ g ¼ 4 5 ij 2 2φ 1 0 r sin 0 r2 sin 2φ

Expression (6.2.9)

d2x m dxi dxj dðÞ‘nF dxm þ Γ m ¼ dt2 ij dt dt dt dt with

1 dxi dxj 2 F ¼ g ij dt dt "# 1 dðÞ‘nF d dx i dxj 2 1 d dxi dx j ¼ ‘ng ¼ ‘ng dt dt ij dt dt 2 dt ij dt dt

1 d ÀÁ1 ¼ ‘nF2 ¼ F_ 2 2 dt 2F2 stays

1 €x m þ Γ mx_ ix_ j ¼ F_ 2 ij 2F2 and with the non-null Christoffel symbols

Γ1 φ φΓ2 Γ2 φ 22 ¼Àsin cos 12 ¼ 21 ¼ cot it follows that ÀÁ ÀÁ ÀÁ 2 _ 1 2 _ 2 2 2 φ_ 2 2φ θ_ 2 F ¼ g11 x þ g22 x ¼ r þ sin Á ÀÁ F_ 2 ¼ 2r2 φ_ φ€ þ sin φ cos φ Á φ_ θ_ 2 þ sin 2φ Á θ_ θ€ 2 € m Γ m _ i _ j 2 _ m 2F x þ ij x x ¼ F x then ÀÁÀÁ φ_ 2 2φ θ_ 2 €m Γ m _ i _ j φ_ φ€ φ φ φ_ θ_ 2 2φ θ_ θ€ φ_ þ sin Á x þ ij x x À þ sin cos Á þ sin Á ¼ 0 306 6 Geodesics and Parallelism of Vectors

– m ¼ 1 ÀÁ 2 €1 Γ1 _ 2 _ 2 2 _ 1 2F x þ 22x x ¼ F x ÀÁÀÁÀÁ φ_ 2 þ sin 2φ Á θ_ 2 φ€ À sin φcos φ Á θ_ 2 À φ_ φ€ þ sin φcos φ Á φ_ θ_ 2 þ sin 2φ Á θ_ θ€ φ_ ¼ 0

cos φ 2 φ_ φ€ À 2 φ_ 2 À sin φ cos φ Á θ_ À θ€ ¼ 0 sin φ θ_

– m ¼ 2 ÀÁ 2 €2 Γ2 _ 1 _ 2 Γ2 _ 1 _ 2 2 _ 2 2F x þ 21x x þ 12x x ¼ F x ÀÁÀÁÀÁ φ_ 2 þ sin 2φ Á θ_ 2 θ€ þ 2cotφ Á φ_ θ_ À φ_ φ€ þ sin φ cos φ Á φ_ θ_ 2 þ sin 2φ Á θ_ θ€ θ_ ¼ 0

Let φ ¼ φθðÞand θ t, thus θ_ ¼ 1, θ€ ¼ 0, whereby a differential equation for m ¼ 2 stays ÀÁ φ_ 2 þ sin 2φ 2 cot φ Á φ_ À ðÞ¼φ_ φ€ þ sin φ cos φ Á φ_ 0 follows

2φ_ 3 Á cot φ þ 2φ_ Á sin 2φ cot φ À φ_ φ€ À sin φ cos φ Á φ_ ¼ 0

cos φ φ€ À 2 φ_ 2 À sin φ cos φ ¼ 0 sin φ

Putting

FðÞ¼θ cot φ it follows that

dF 1 dφ ¼À dθ sin 2φ dθ

dφ dF ¼Àsin 2φ dθ dθ d2φ dφ dF d2F dF 2 d2F ¼À2 sin φ cos φ sin 2φ ¼ 2 sin 3φ cos φ À sin 2φ dθ2 dθ dθ dθ2 dθ dθ2 then dF 2 d2F cos φ dF 2 2 sin 3φ cos φ À sin 2φ À 2 sin 4φ À sin φ cos φ ¼ 0 dθ dθ2 sin φ dθ 6.3 Geodesics with Null Length 307

The division by sin2φ provides

d2F cos φ d2F À ¼ 0 ) þ FðÞ¼θ 0 dθ2 sin φ dθ2 which solution is

F ¼ k1 cos θ þ k2 sin θ where k1, k2 are constants, then

cot φ ¼ k1 cos θ þ k2 sin θ represents the geodesics on the surfaces of the sphere, which can be rewritten in an implicit form as cos φ k r sin φ cos θ þ k r sin φ sin θ À r sin φ ¼ 0 1 2 sin φ

The relations between the spherical and Cartesian coordinates are given by

x ¼ r sin φ cos θ y ¼ r sin φ sin θ z ¼ r cos φ whereby

k1x þ k2y À z ¼ 0 represents a plane that passes through the center of the sphere. The geodesics are intersections of the sphere with the diametral planes, which normal vectors have the components ðÞk1; k2; À1 , i.e., they are the maximum circles of the sphere deﬁned by the expressions

x2 þ y2 þ z2 ¼ r2 k1x þ k2y À z ¼ 0

6.3 Geodesics with Null Length

The length of the geodesic between two points can be null, i.e., the fundamental form is undeﬁned, which makes applying expression (6.2.9) invalid for calculating this curve, for its parametric representation x i ¼ x iðÞt is not appropriate, ðB and the variational equation δ ds ¼ 0 has no meaning, and the tangent vector to A the curve is undeﬁned. 308 6 Geodesics and Parallelism of Vectors

With a new parametric representation x i ¼ x iðÞλ , which equations are continu- 2 ous and class C , and λ0 λ λ1, where λ is an invariant, so that the tangent vector dx i dλ exists for each point of the curve and has null modulus, thus yielding the condition

dx i dxj ds2 ¼ g ¼ 0 ð6:3:1Þ ij dλ dλ

dx i and the contravariant vector dλ has the direction of the displacement along the curve. Thus, the geodesics with null length has tangent vectors in all their points, so the displacement of this vector from one point to another neighboring point keeps it parallel to the null vector, then these vectors must be equipollent. The condition of parallelism between vectors for the case of the geodesic with null length is given by

d2x m dx i dxj þ Γ m ¼ 0 ð6:3:2Þ dλ2 ij dλ dλ

A geodesic is null if one of its sub-arcs is null or if it has null length. In the set of values of λ for which there are null unit tangent vectors, the geodesic generates the undeﬁned fundamental form ds2 ¼ 0. By this condition it is veriﬁed that the geodesics can be null without having null length, but if it has null length, it is necessarily null in all its points. Expressions (6.3.1) and (6.3.2) provide ðÞN þ 1 ordinary differential equations in which the unknown values are functions x i ¼ x iðÞλ that determine the condition of geodesics with null length.

Exercise 6.5 Let the metric tensor gij with constant components. Determine the equation of geodesic with null length. The metric tensor has its components constant, so the Christoffel symbols are all null and the equation of the geodesic

d2x m dx i dx j d2x m þ Γ m ¼ 0 ) ¼ 0 ds2 ij ds ds ds2 with general solution

x m ¼ ams þ bm where am, bm are constants, it follows that

dxm x m À bm ¼ am ) am ¼ ds s The geodesic with null length is determined by the equation

dxi dxj g ¼ 0 ) g aiaj ¼ 0 ij ds ds ij 6.4 Coordinate Systems 309

The substitution of expression am provides the differential equation of the geodesic with null length when gij ¼ constant: ÀÁÀÁ i i j j gij x À b x À b ¼ 0

Exercise 6.6 Calculate the geodesics with null length for the space E4 in which the 2 2 2 2 metric is ds2 ¼ ðÞdx1 þ ðÞdx2 þ ðÞdx3 À ðÞdx4 . The components of tensor gij are constants, then the equation of the geodesics with null length, in accordance with Exercise 6.5, is given by ÀÁ i i j j gij x À x0 x À x0 ¼ 0 which when developed provides ÀÁÀÁÀÁÀÁ 1 1 2 2 2 2 3 3 2 4 4 2 x À x0 þ x À x0 þ x À x0 À x À x0 ¼ 0

6.4 Coordinate Systems

In tensor calculus, the choice of the coordinate system is a function of the type of problem to be solved. For the Euclidian space E3, the simplest coordinate system is the Cartesian system. However, when analyzing a few problems of the use of curvilinear coordinates, it is more convenient. For the Riemann space EN, the Cartesian coordinate sometimes is not adequate; then it becomes necessary to search for a few coordinate systems that have special characteristics which make the solving of speciﬁc cases easier.

6.4.1 Geodesic Coordinates

For the Cartesian coordinate system, and only for this type of coordinates, the Christoffel symbols are all null in any point of the Riemann space EN, because the 2 i j coefﬁcients in a fundamental form ds ¼ gijdx dx are constants. However, it is possible to determine a coordinate system with respect to which these symbols i cancel each other in a certain point P0ðÞ2x EN, called pole. The symmetry of the Γ k Γ k Christoffel symbols ij ¼ ji allows determining a coordinate system for which all these symbols are null. i i Consider point P0(x ) be chosen as the origin of the coordinate system X having its coordinates x i ¼ 0, and the linear transformation 310 6 Geodesics and Parallelism of Vectors

1 x i ¼ xi þ C i xjxk ð6:4:1Þ 2 jk

i where the constant coefﬁcients Cjk are chosen so as to be symmetric in the indexes j and k. Differencing this expression it follows that

∂x i ∂x i 1 ∂x j 1 ∂x k 1 1 ¼ þ C i x k þ C i x j ¼ δ j þ C i δ jx k þ C i x jδ k ∂x j ∂x j 2 jk ∂x j 2 jk ∂x j i 2 jk j 2 jk j 1 1 ¼ δ j þ C i x k þ C i x k ¼ δ j þ C i x k i 2 jk 2 jk i jk

∂2x i ∂xk ¼ 0 þ C i ¼ C i ∂xk∂xj jk ∂xk jk and with the expressions

∂x i ∂x j ∂xi ¼ δ i ¼ δ i ∂xj ∂xk k ∂xk k and the transformation law of the Christoffel symbols

i 2 p i q r i ∂x ∂ x ∂x ∂x ∂x Γ ¼ þ Γ p jk ∂x p ∂x j∂x k ∂x p ∂x j ∂x k qr for the point P0

∂g i ik ¼ Γ þ Γ ) Γ ¼ C i þ Γ i ∂x j ij, k kj, i jk jk jk

Γ i i Γ i Consider jk ¼ 0 it results in Cjk ¼À jk at the point P0, i.e., the Christoffel symbols are null at the pole. This condition leads to the deﬁnition of a coordinate system called geodesic or normal coordinate system, which condition of existence is grounded in the symmetry of the Christoffel symbols. Ricci’s lemma

∂g ∂ g ¼ ij À g Γ p À g Γ p ¼ 0 ð6:4:2Þ k ij ∂xk pj ik ip kj is valid when the Christoffel symbols are null in pole, whereby the metric is constant, and the geodesic coordinates correspond to a local Euclidian coordinate system. It is highlighted that the derivatives of the Christoffel symbols do not necessarily cancel each other in this point. Pole P0 has an important relation with a derivative covariant. Let the covariant vector ui expressed in geodesic coordinates, and with its derivative covariant 6.4 Coordinate Systems 311

∂u ∂ u ¼ i À u Γ p i i ∂x j p ij Γ p in geodesic coordinates with respect to which ij ¼ 0, thus

∂u ∂ u ¼ i i i ∂x j

It is concluded that in geodesic coordinates, the covariant derivative of vector ui is equal to its partial derivative in the pole P0. The generalization of this property for the higher-order tensors is immediate. The demonstrations of the tensor relations using this type of coordinate system are simpler, and if they are valid for this special case, they will be valid for the other coordinate systems. This can be proven comparing the algebraic development carried out in item 5.2, when demonstrating the second Bianchi identity and the solution of Exercise 6.7. p Exercise 6.7 Using the geodesic coordinates, show that the curvature tensor Rijk ∂ p ∂ p ∂ p satisﬁes the second Bianchi identity ‘Rijk þ jRik‘ þ kRi‘j ¼ 0. The geodesic coordinates correspond to a local Euclidian coordinate system, and then the covariant derivatives of the curvature tensor are equal to their partial derivatives; thus, ! ∂R p ∂ ∂Γ p ∂Γ p p ijk ik ij p p q p ∂‘R ¼ ¼ À þ Γ Γ À Γ Γ ijk ∂x‘ ∂x‘ ∂x j ∂xk ik qj ij qk

As in the pole, the Christoffel symbols cancel each other, but their derivatives do not necessarily cancel each other in this point

∂R p ∂2Γ p ∂2Γ p ijk ¼ ik À ij ∂x‘ ∂x‘∂x j ∂x‘∂xk and the permutations of the indexes allow writing

p 2 p 2 p p 2 p 2 p ∂R ∂ Γ ∂ Γ ∂R ‘ ∂ Γ ∂ Γ ik‘ ¼ i‘ À ik i j ¼ ij À i‘ ∂x j ∂x j∂xk ∂x j∂x‘ ∂xk ∂xk∂x‘ ∂xk∂x j The sum of these three expressions provides

∂ p ∂ p ∂ p : : : ‘Rijk þ jRik‘ þ kRi‘j ¼ 0 Q E D

6.4.2 Riemann Coordinates

The coordinate systems in which the partial derivatives of the metric tensor gij cancel each are called Riemann coordinate systems. 312 6 Geodesics and Parallelism of Vectors

i Consider a geodesic that passes by the point P0ðÞ2x EN and the notation i dx i ξ ¼ for the unit tangent vector to this curve in P0, in which s is the arc ds 0 measured from this point. Parameter ξi represents only one geodesic that contains P0; thus, with the coordinates

yi ¼ ξ is ð6:4:3Þ there is a set of values of ξi that generates the equations which deﬁne the geodesics in new coordinates, called Riemann coordinates. The geodesics that contain point P0 are analogous to the straight lines that pass by the origin of a coordinate system in Euclidian geometry. The quadratic form of the curve in this new coordinate system is given by

2 i j : : ds ¼ gijdy dy ð6 4 4Þ

Γ i Γ and with the Christoffel symbols jk, ij, k, the geodesics are determined by

2 i i k d y i dy dy þ Γ ¼ 0 ð6:4:5Þ ds2 jk ds ds

Expression (6.4.3) must satisfy expression (6.4.5); then

Γ i ξjξk : : jk ¼ 0 ð6 4 6Þ or

Γ i j k : : jk y y ¼ 0 ð6 4 7Þ

This ascertaining translates the necessary and sufﬁcient condition so that the Γ i coordinates are valid in the Riemann space EN. It is stressed that jk ¼ 0 in point P0, Γ ‘ Γ and with g‘p jk ¼ jk, p, it is concluded by means of the expression

∂g ik ¼ Γ þ Γ ∂x j ij, k kj, i which in Riemann coordinates the partial derivatives of the metric tensor gij are null. The Riemann coordinate system is a geodesic coordinate system. 6.5 Geodesic Deviation 313

6.5 Geodesic Deviation

The deviation between two geodesics in the Riemann space EN is a generalization of the behavior of two straight lines R1 and R2 in the Euclidian space E2. Let R1 and R2 two parallel straight lines (Fig. 6.2a) on which the points A, A0, B, and B0 are located. The distance ξ between these two geodesics remains unchanged, i.e., η ¼ η0 . For the case in which R1 and R2 intersect, there are small values of the angle α which deviation between the geodesics is given by

η ﬃ α Á s ð6:5:1Þ where s is the distance of the point being considered to the point of interception of R1withR2 (Fig. 6.2b). Expression (6.5.1) shows that in this case the separation between the straight lines varies linearly with the distance from their points until the origin O; then

d2η ¼ 0 ð6:5:2Þ ds2

This behavior is not valid for geodesics in curved spaces. Consider a sphere of unit radius in the surface of which two segments of the geodesic are considered OA ¼ u and OB ¼ u, distant from each other AB ¼ η (arc of latitude), where point O is the origin of the distances u, measured on these curves (Fig. 6.3). Admitting a small value for the angle α, it follows that

η ﬃ α: sin u ð6:5:3Þ d2η 6¼ 0 ð6:5:4Þ ds2

Expression (6.5.3) shows that the deviation η between the two geodesics varies with the parameter u, measured along the same. From the origin O up to the midpoints of the geodesics, there is an increase of η; from there, this distance decreases until it cancels itself at point O0 (diametrically opposite to the point O). Expression (6.5.4) shows that the variation of η is not linear and highlights the difference between the behavior of the geodesics of a ﬂat space and a curved space.

Fig. 6.2 Geodesics in the ab Riemann space E2:(a) parallel straight lines and B′ R2 B B′ R1 (b) converging straight lines B ξ ′ ξ ξ′ ξ R2 R1 A A′ O A A′ 314 6 Geodesics and Parallelism of Vectors

Fig. 6.3 Geodesics O in spherical space E2

ξ A B ξ

O′

u=constant G2 G1 ∂xi B ξ i = ∂u i A ξ=i ()u + du dv dξ v + dv ξ i u dv

A′ B′ v=constant u +du

Fig. 6.4 Geodesics in the Riemann space EN

The generalization of this ascertaining for the Riemann space EN is carried out when admitting a family of geodesics defined by the functions x i ¼ x iðÞu, v of class C2, in which the parameter u (length of arc) varies along each curve fixing the points on them, and parameter v is constant along its length but varies when passing from one geodesics to another, i.e., it distinguishes the curves. Consider the geodesics G1 and G2 which contain the points A, A0, B, and B0, defining the distancesηiðÞu , ηiðÞu measured orthogonally to these curves. Figure 6.4 shows these parameters with the geodesics G1, G22EN. The partial derivatives

∂x i ξ i ¼ ð6:5:5Þ ∂u ∂x i ηi ¼ ð6:5:6Þ ∂v 6.5 Geodesic Deviation 315 determine the tangent vector to the geodesics deﬁned by the parameter v and the distance ηi(u) (displacement or deviation) between two nearby geodesics, the length being measured in each curve from points A and A0. For intrinsic derivative of the tangent vector, it follows that δξ i ∂ξk ∂ξ i ∂x j ¼ þ þ Γ i ξk δv ∂v ∂x j kj ∂v ∂ξk ¼ 0 ∂v δξ i ∂ξ i ∂x j ∂x j ¼ þ Γ i ξk ¼ ∂ ξ i δv ∂x j kj ∂v j ∂v

Thus, δξi ∂ξ i ∂ξ i ∂x j ∂ξ i ¼ þ Γ i ξk ηj ¼ þ Γ i ξkηj ¼ þ Γ i ξkηj δv ∂x j kj ∂x j ∂v kj ∂v kj ∂ ∂x i ¼ þ Γ i ξkηj ∂v ∂u kj

Then

δξi ∂2x i ¼ þ Γ i ξkηj δv ∂v∂u kj In an analogous way,

δηi ∂x j ¼ ∂ ηi δu j ∂u Thus,

δηi ∂2x i ¼ þ Γ i ηjξk δu ∂u∂v kj These two expressions are equal, i.e.:

δξi δηi ∂2x i ¼ ¼ þ Γ i ξkηj ð6:5:7Þ δv δu ∂v∂u kj The second-order derivative of vector ηi with respect to the parameter u is given by δ2ηi δ δηi δ δξi ¼ ¼ δu2 δu δu δu δv 316 6 Geodesics and Parallelism of Vectors and follows δ2ηi δ ∂xk δ ∂ξ i ∂xk δ ∂ξ i ∂xk ¼ ∂ ξ i ¼ þ Γ i ξj ¼ þ Γ i ξj δu2 δu k ∂v δu ∂xk jk ∂v δu ∂v jk ∂v δ ∂ξ i ¼ þ Γ i ξjηk δu ∂v jk that can be written as δ2ηi ∂ξ i ∂ ‘ i j k x ¼ ∂‘ þ Γ ξ η δu2 ∂v jk ∂u which development δ2ηi ∂ ∂ξ i ∂ ‘ ∂Γ i ∂ ‘ ÀÁ∂ ‘ ÀÁ∂ ‘ x jk j k x i j k x i j k x ¼ þ ξ η þ Γ ∂‘ξ η þ Γ ξ ∂‘η δu2 ∂x‘ ∂v ∂u ∂x‘ ∂u jk ∂u jk ∂u ∂2ξ i ∂Γ i ∂x‘ ∂ξ j ∂x‘ ∂ηk ∂x‘ ¼ þ jk ξ jηk þ Γ i þ Γ j ξ m ηk þ Γ i ξ j þ Γ k ηn ∂u∂v ∂x‘ ∂u jk ∂x‘ ‘m ∂u jk ∂x‘ ‘n ∂u with

∂ξ j ∂x‘ ∂ξ j ∂ηk ∂x‘ ∂ηk ¼ ¼ ∂x‘ ∂u ∂u ∂x‘ ∂u ∂u provides 2 i 2 i ∂Γ i j k δ η ∂ ξ ‘ ∂ξ ∂η ‘ ¼ þ jk ξ jξ ηk þ Γ i ηk þ ξ j þ Γ i Γ j ξ mξ ηk δu2 ∂u∂v ∂x‘ jk ∂u ∂u jk ‘m Γ i Γ k ξ jξ‘ηn : : þ jk ‘n ð6 5 8Þ

The variation rate of the tangent vector along the geodesics is null, and with expression (6.2.20) that deﬁnes the geodesic, this rate can be written as ÀÁ k ‘ ∂‘ξ ξ ¼ 0 or δ δξ i δ ∂ k δ ∂ k ∂ k ∂ ξ i Γ i ξ j x ∂ ξ i x Γ i ξ j x ¼ 0 ¼ k þ jk ¼ k þ jk δv δu δv ∂u δv ∂u ∂u δ ∂ξ i ¼ þ Γ i ξ jξk δv ∂u jk 6.5 Geodesic Deviation 317 thus δ ∂ξ i þ Γ i ξ jξ k ¼ 0 δv ∂u jk and with expression (6.2.20), it follows that ∂ξ i ∂ ‘ ∂ξ i i j k x i j k ‘ ∂‘ þ Γ ξ ξ ¼ ∂‘ þ Γ ξ ξ η ¼ 0 ∂u jk ∂v ∂u jk and ∂ ∂ξ i ∂ ‘ ∂Γ i ÀÁ ÀÁ x jk j k ‘ i j k ‘ i j k ‘ þ ξ ξ η þ Γ ∂‘ξ ξ η þ Γ ξ ∂‘ξ η ¼ 0 ∂x‘ ∂u ∂v ∂x‘ jk jk 2 i ∂Γ i j k ∂ ξ ‘ ∂ξ ‘ ∂ξ ‘ þ jk ξ jξ kη þ Γ i þ Γ j ξm ξ kη þ Γ i ξ j þ Γ k ξn η ¼ 0 ∂u∂v ∂x‘ jk ∂x‘ ‘m jk ∂x‘ ‘n 2 i ∂Γ i j k ∂ ξ ‘ ∂ξ ∂ξ ‘ ‘ þ jk ξ jξ kη þ Γ i ξ k þ ξ j þ Γ i Γ j ξmξ kη þ Γ i Γ k ξ jξnη ¼ 0 ∂u∂v ∂x‘ jk ∂v ∂v jk ‘m jk ‘n allows writing "# 2 i ∂Γ i j k ∂ ξ ‘ ∂ξ ∂ξ ‘ ‘ ¼À jk ξ jξ kη þ Γ i ξ k þ ξ j þ Γ i Γ j ξmξ kη þ Γ i Γ k ξ jξnη ∂u∂v ∂x‘ jk ∂v ∂v jk ‘m jk ‘n and with the substitution in expression (6.5.8) "# 2 i ∂Γ i j k δ η ‘ ∂ξ ∂ξ ‘ ‘ ¼À jk ξ jξ kη þ Γ i ξ k þ ξ j þ Γ i Γ k ξmξ kη þ Γ i Γ k ξ jξnη δu2 ∂x‘ jk ∂v ∂v jk ‘m jk ‘n ∂Γ i j k ‘ ∂ξ ∂η ‘ ‘ þ jk ξ jηkξ þ Γ i ηk þ ξ j þ Γ i Γ j ξmξ ηk þ Γ i Γ k ξ jξ ηn ∂x‘ jk ∂u ∂u jk ‘m jk ‘n

This expression will be analyzed in parts: (a) Terms that cancel each other "#ÀÁ ÀÁ ∂ξ j ∂ξ k ∂ξ j ∂ηk ∂ ξ jηk ∂ ξ jξ k A ¼ÀΓ i ξ k þ ξ j þ Γ i ηk þ ξ j ¼ Γ i À jk ∂v ∂v jk ∂u ∂u jk ∂u ∂v

and with expressions (6.5.5) and (6.5.6)

∂u ηi ∂u ¼ ) ηi ¼ ξ i ∂v ξ i ∂v 318 6 Geodesics and Parallelism of Vectors

thus "#ÀÁ "#ÀÁÀÁ ∂ ∂u ∂ ξ jξ k ∂ ξ jξ k ∂ ξ jξ k A ¼ Γ i ξ j ξ k À ¼ Γ i À ¼ 0 jk ∂u ∂v ∂v jk ∂v ∂v

(b) Terms with the derivatives of the Christoffel symbols

∂Γ i ∂Γ i ‘ ‘ B ¼À jk ξ jξ kη þ jk ξ jηkξ ∂x‘ ∂x‘

and interchanging the indexes ‘ $ k ! ∂Γ i ∂Γ i ‘ ‘ B ¼À j þ jk ξ jξ ηk ∂xk ∂x‘

Γ i Γ k ξmξ kη‘ Γ i Γ k ξ jξnη‘ Γ i Γ j ξmξ‘ηk Γ i Γ k ξ jξ‘ηn C ¼À jk ‘m À jk ‘n þ jk ‘m þ jk ‘n

With the change of the indexes n ! m, it follows that

Γ i Γ k ξmξ kη‘ Γ i Γ k ξ jξmη‘ Γ i Γ j ξmξ‘ηk Γ i Γ k ξ jξ‘ηm C ¼À jk ‘m À jk ‘m þ jk ‘m þ jk ‘m

and with the permutation of the indexes m $ ‘ in the second term, the expression is reduced to

Γ i Γ k ξmξ kη‘ Γ i Γ k ξ jξ‘ηm Γ i Γ j ξmξ‘ηk Γ i Γ k ξ jξ‘ηm C ¼À jk ‘m À jk m‘ þ jk ‘m þ jk ‘m Γ i Γ k ξmξ kη‘ Γ i Γ j ξmξ‘ηk ¼À jk ‘m þ jk ‘m

The permutation of the indexes j $ m provides

Γ i Γ k ξ jξ kη‘ Γ i Γ mξ jξ‘ηk C ¼À mk ‘j þ mk ‘j

and with the permutation of the indexes ‘ $ k in the ﬁrst term

Γ i Γ k ξ jξ‘ηk Γ i Γ mξ jξ‘ηk C ¼À mk ‘j þ m‘ kj

Joining this parcel ! 2 i ∂Γ i ∂Γ i δ η ‘ ‘ ‘ ¼À j þ jk À Γ i Γ k þ Γ i Γ mξ jξ ηk ξ jξ ηk δu2 ∂xk ∂x‘ mk ‘j m‘ kj 6.6 Parallelism of Vectors 319

and with expression (5.2.11), the result is

2 i ∂ η ‘ þ R i ξ jξ ηk ¼ 0 ð6:5:9Þ ∂u2 jk‘

This expression allows establishing N second-order ordinary differential equation for the vectors ηi that represent the deviations (distances) between the geodesics, the unit vectors ξi being tangents to these curves. The distances ηi are ηi ∂ηi dηi determined if the initial values of and ∂u (or du) are known. Exercise 6.8 Show that in a flat space the deviation of the family of geodesics defined by the function x iðÞ¼u; v uFiðÞþv GiðÞv is null. The family of geodesics is defined by

x iðÞ¼u; v uFiðÞþv GiðÞv thus

∂x i ∂x i ∂Fi ∂Gi ¼ ξ i ¼ Fi ¼ ηi ¼ u þ ∂u ∂v ∂v ∂v ∂ηi ∂Fi ∂2ηi ¼ ¼ 0 ∂u ∂v ∂u2 and expression (6.5.9)

2 i ∂ η ‘ þ R i ξ jξ ηk ¼ 0 ∂u2 jk‘ stays

i ξ jξ‘ηk 0 þ Rjk‘ ¼ 0

i and as the space is ﬂat Rjk‘ ¼ 0, which veriﬁes the previous equation and shows that the deviation of this family of geodesics is null.

6.6 Parallelism of Vectors

6.6.1 Initial Notes

In the Euclidian space, two coplanar vectors that move with the origin over a straight line AB located in the plane of these vectors are parallel and have the same norm kku ¼ kkv and maintain the same direction deﬁned by angle α, and there is no geometric difference between these varieties, i.e., the vectors are equipollent. i In Fig. 6.5 the vector u in point PxðÞ2E3 and the vector v with origin in point 320 6 Geodesics and Parallelism of Vectors

Fig. 6.5 Parallelism of vectors in the Euclidian space E2 v α B u α Q A P

Fig. 6.6 Parallelism of ab vectors in the bidimensional C spherical space: (a) parallel 2 transport of vector u along v Q different paths, (b) Q u P condition of parallelism P β C1 Q α ¼ β P u P t P t Q

QxðÞi þ dx i neighbor to point P(xi) are equipollent. This equipollence between the two vectors indicates that the same are parallels, i.e., they shift from one point to another point of the plane. The concept of displacement of a Cartesian vector can be generalized for the space EN with the deﬁnition of parallel shift of the vector along a curve (Fig. 6.5). However, in general, the space is not Euclidian, which requires a few considerations more speciﬁc for study of the parallel transport of the vector. i dxi The unit tangent vectors to the geodesic u ¼ ds represent velocities (variation rates), and the velocity variation in the measurement unit of the independent variable s is called acceleration (variation of the variation rate); thus,

∂ui d2x i duj ∂ ui ¼ þ u jΓ i ¼ þ Γ i s ∂s jk ds2 ds jk

i i and if u is constant ∂su ¼ 0, then the displacement of the unit tangent vector is a translation, i.e., it is a displacement which trajectory keeps its direction constant. The comparison between the varieties deﬁned in distinct points of the Riemann space EN is carried out from the concept of parallel transport of vector along a curve of this space. To interpret the curvature tensor geometrically, admit the parallel transport of vector u from point P(xi) to point QxðÞi þ dx i , along a curve on the spherical surface shown in Fig. 6.6a, running two different paths C1 and C2, having path C1 along the equator and path C2 along the meridian. It is observed that vector u is not 6.6 Parallelism of Vectors 321 kept constant, for in point Q will depend on the path being run, concluding that the parallel transport of u depends on the curvature of the space. Consider, for instance, in point P(xi) located on the circle of maximum diameter of the sphere shown in Fig. 6.6b the vector u(P) and the tangent vector t(P) that form an angle α and are located in the plane tangent to the sphere in this point, and the point QxðÞi þ dxi neighbor to point P(xi) and also located on the circle of maximum diameter, which tangent plane contains the vector v(Q) and the tangent vector tðÞQ that form an angle β. Let uðÞP ¼ vðÞQ . The vectors u(P) and v(Q) will be parallel if α ¼ β. The concept of parallelism between vectors is generalized for the Riemann space EN.

6.6.2 Parallel Transport of Vectors

The parallel transport of a vector is its displacement from one point to another of the space, during which the vector is kept constant. In the curved Riemann space EN, the result of the parallel transport of the vector depends on the path run between the two points. The study of the parallelism of vectors in the Riemann space EN is made according to the Levi-Civita approach by means of the elementary curved parallel- i ogram PQRSÀÁ(Fig. 6.7). The vertexesÀÁ of this parallelogram have coordinates P(x ), QxðÞi þ εi , Sxi þ δi , and Rxi þ εi þ δi , εi and δi being elementary quantities. Consider the vector u(P) embedded in the tangent space to the Riemann space EN and shifted parallel along path C1 ¼ PQR from P, with vector originating in point Q: m m pΓ m εn u ðÞ¼Q u À u np ðÞP

S xi +δ i

C2 u P R xi + ε+i δ i P xi C1

Q xi + ε i

Fig. 6.7 Levi-Civita parallelism 322 6 Geodesics and Parallelism of Vectors where the letters in parenthesis indicate the point where it refers to theÀÁ variety being analyzed. The transport of the vector from point QxðÞi þ εi to point Rxi þ εi þ δi is given by m m pΓ m εn u ðÞ¼R u À u np ðÞQ ÀÁ∂Γ m ‘ ¼ um À upΓ m εn À up À urΓ p εs Γ m þ np ε δn np sr ðÞP np ∂ ‘ ðÞP x ðÞP where vector um(R) appears in terms of the parameters deﬁned in point P; then without the lower index that indicates this point, it follows that

∂Γ m ‘ umðÞ¼R um À upΓ m εn À upΓ m δn À urΓ p Γ m εsδn þ up np ε δn À urΓ p np np sr np ∂x‘ sr ∂Γ m ‘ Â np εsε δ ∂x‘ and the last term can be disregarded on account of being of a higher order; thus,

∂Γ m ‘ umðÞﬃR um À upΓ m εn À upΓ m δn À urΓ p Γ m εsδn þ up np ε δn np np sr np ∂x‘

The permutation of the indexes r $ p and the change of the indexes s $ ‘ in the fourth term to the right allow writing ∂Γ m ‘ umðÞffiR um À upΓ m εn À upΓ m δn þ up np À Γ r Γ m ε δn np np ∂x‘ ‘p nr and with an analogous formulation for the path C2 ¼ PSR, the result is ∂Γ m ‘ ‘ u~ mðÞffiR um À upΓ m δn À upΓ m εn À up p À Γ r Γ m ε δn np np ∂xn np ‘r with umðÞR 6¼ u~ mðÞR ; thus, ∂Γ m ∂Γ m ‘ ‘ ‘ u~ mðÞÀR umðÞffiR up np À p À Γ r Γ m þ Γ r Γ m ε δn ¼ upR m ε δn ∂x‘ ∂xn ‘p nr np ‘r p‘n where the term to the left is a vector, and in the term to the right, there is the inner p m product of vector u by the variety Rp‘n, this variety being the Riemann–Christoffel curvature tensor. The deduction of the Riemann–Christoffel tensor by means of the concept of parallelism of vectors in space EN is due to Levi-Civita. The approach adopted as 6.6 Parallelism of Vectors 323

Fig. 6.8 Parallel transport ui to vector ui along the paths 1 i C1 and C2 u2 C 1 Q xi + dxi i u0

C2 P xi

m deﬁnition of this tensor, such as developed in item 5.2, where Rp‘n was obtained by simple algebraic formalism when calculating the covariant derivatives of the second-order tensor of variance (0, 2), is due to Erwin Christoffel. The geometric m interpretation of tensor Rp‘n is that the change of relative orientation between the vectors shift in parallel along different paths is measured by this tensor. Complementing the analysis of parallelism of vectors let, in Fig. 6.8, the paths of i i i i i vector u from point P(x ) where u0 exists, from point QxðÞþ dx along two i i different paths C1 and C2, which results, in general, in the condition u1 6¼ u2. i The condition for the vectors u or ui being displaced in parallel to itself along the parameterized curve xk ¼ xkðÞt is that their absolute derivatives are null:

δui δu ¼ i ¼ 0 dt dt The differential equation that represents the parallel transport of the vector ui k k along the curve deﬁned by the parametric equations x ¼ x ðÞ2t EN is given by i Γ i j k : : du þ jku dx ¼ 0 ð6 6 1Þ

In an analogous way for the covariant components of the vector, in differential form is given by

Γ j k : : dui À ikujdx ¼ 0 ð6 6 2Þ and the simple analogy with the parallelism of vectors deﬁned in Cartesian coor- i dinate systems would lead to the condition du ¼ dui ¼ 0, which would be the condition for the vector to be displaced in parallel to itself keeping its components constants. This analogy is incorrect, because these increments do not represent the components of the vector, and when the coordinate system is changed, these i differentials du , dui are not necessarily null and imply that the condition of parallelism would depend on the coordinate system. These differentials are vectors, i Γ i j k for expressions (6.6.1) and (6.6.2) provide, respectively, du ¼À jku dx and Γ j k dui ¼ ikujdx , whereby if they are null, a coordinate system will cancel in all the others. 324 6 Geodesics and Parallelism of Vectors

The substitution of the parametric equations of the curve xk ¼ xkðÞt , on which will be given the path of the vector deﬁned by expression (6.6.1), results in a system of ordinary differential equations which unknown values are the functions ui(t). The values of these functions in the ﬁnal point of the path will depend on the value of parameter t in this point. The result of solving these ordinary differential equations, in general, depends on the path.

6.6.2.1 Independence of Path

dui The independence of path is linked to the condition of the derivatives dxk of expression (6.6.1) representing exact differential

dui ¼ÀΓ i u j dxk jk and the equality ∂ Γ i j ∂ Γ i j jku j‘u ¼ ∂x‘ ∂xk allows the analytic development of this condition, whereby

i j i j ∂Γ du ∂Γ ‘ du jk u j þ Γ i ¼ j u j þ Γ i ð6:6:3Þ ∂x‘ jk dx‘ ∂xk j‘ dxk

The change of the indexes i ! j, k ! ‘, j ! p in expression (6.6.1) rewritten as

dui ¼ÀΓ i u j dxk jk provides

duj ¼ÀΓ j up dx‘ p‘ and with the change of the indexes i ! j, j ! p

du j ¼ÀΓ j up dxk pk

The substitution of these two expressions in expression (6.6.3) allows writing

i i ∂Γ ∂Γ ‘ jk u j À Γ i Γ j up ¼ j u j À Γ i Γ j up ∂x‘ jk p‘ ∂xk j‘ pk 6.6 Parallelism of Vectors 325 whereby ! i i ∂Γ ∂Γ ‘ jk À j u j þ Γ j Γ i À Γ i Γ j u p ¼ 0 ∂x‘ ∂xk j‘ pk jk p‘ and with the change of the indexes j ! p in terms between the ﬁrst parenthesis ! i i ∂Γ ∂Γ ‘ pk À p þ Γ j Γ i À Γ i Γ j u p ¼ 0 ∂x‘ ∂xk j‘ pk jk p‘ and as

i i ∂Γ ∂Γ ‘ R i ¼ pk À p þ Γ j Γ i À Γ i Γ j p‘k ∂x‘ ∂xk j‘ pk jk p‘ results in

i p : : Rp‘ku ¼ 0 ð6 6 4Þ

i p ‘ Therefore, Rp‘ku ¼ 0 8i, p, , k expresses the conditions that must be fulﬁlled so that the parallel transport of the vector ui is independent of the path. The necessary and sufﬁcient condition so that the parallel transport is independent of the path for i any vector is Rp‘k ¼ 0.

6.6.2.2 Invariance of the Modulus and the Angle Between Vectors

With expression (6.6.1) under the form

dui dxk þ Γ i u j ¼ 0 ð6:6:5Þ dt jk dt and having the vectors ui(t) and vi(t), two solutions of differential equation, the dot product between these two vectors

α i j u Á v ¼ kku kkv cos ¼ giju v is invariant; thus, d g uiv j ¼ 0 dt ij 326 6 Geodesics and Parallelism of Vectors

i j As giju v is an invariant and by Ricci’s lemma tensor gij behaves as a constant in the covariant derivation d dui dv j g uiv j ¼ g v j þ g ui dt ij ij dt ij dt

By hypothesis, the vectors are solutions of the differential equation given by expression (6.6.5); then

dui dv j ¼ 0 ¼ 0 dt dt

i j It is concluded that giju v is constant along the curve represented by the k k i j i j parametric equation x ¼ x ðÞt . If the vectors are equal u ¼ v , then giju v ¼ 2 α giju , which is a constant; therefore, the angle between the vectors is constant. With this analysis it is veriﬁed that the modulus of these vectors are invariant when they move along the parameterized curve, thence the angle α between ui and vj also remains unchanged when varying parameter t. The modulus of the vectors u and v is maintained unchanged, just as the angle between them; thus, the dot product u Á v also remains unchanged. The straight line in the Euclidian space is the only curve for which the parallel transport of a vector is the own tangent vector to this curve.

6.6.2.3 Space with Afﬁne Connections

The parallel transport of a vector is independent of the metric tensor, because Christoffel symbol of second kind can be determined by expression (2.3.9). The spaces with a parallel transport are called space with afﬁne connections. For the Riemann space EN, the afﬁne connections are the Christoffel symbols.

6.6.2.4 Integrability

The Euclidian characteristics of the Riemann space EN depend only of its metric. As the Christoffel symbols are linked to the metric tensor, the use of the contravariant and covariant coordinates of the vector is indifferent for determining the parallel displacement of a vector along a curve represented by parametric equations, which is governed by the differential equations given by expressions (6.6.1) and (6.6.2). Consider the parallel displacement that takes place along a closed curve such as Fig. 6.9a shows. In this case the vectors remain unchanged along the path, and the affine connections of this space are integratable, and then the vector in a point generates the field of parallel vectors in space EN. Figure 6.9b shows the displacement of a vector along a closed curve for non-integratable affine connections, where 6.6 Parallelism of Vectors 327

Fig. 6.9 Integrability of the affine connections: (a) integratable affine connections, (b) non-integratable affine connections the change of the vector’s direction along the path is verified. If the affine connections of the space are linked to the metric tensor, the condition of the space being Euclidian is directly related with their integrability. For the Euclidian space the affine connections are the Christoffel symbols; then i the differentials du and dui given by expressions (6.6.1) and (6.6.2), respectively, are null, and the parallel vectors have the same components in all the points of the space, whereby these affine connections are integratable. The affine connections of the Euclidian spaces are always integratable, and then it is always possible to determine a Cartesian coordinate system when the affine connections are integratable. Integrability is an invariant property of the affine connections, so it is independent of the coordinate system. Exercise 6.9 For the Riemann space in which all the pairs of points ðÞ2x1; x2 R, x2 > 0, which metric tensor is given by 2 3 1 6 0 7 4 x2 5 gij ¼ 1 0 x2

j calculate the parallel transport of a vector v along the curve of parametric coordi- 1 i x0 1 2 nates u ðÞ¼t 2 , having (x0; x0) initial values of the coordinates and x0 þ t x1 ¼ constant. The derivatives of the components of the metric tensor are

2 g11, 2 ¼ g22, 1 ¼À ðÞx2 3 and the non-null Christoffel symbols are given by

Γ 1 1 Γ 1 1 11, 2 ¼À g11, 2 ¼ 12, 1 ¼ g11, 2 ¼À 2 ðÞx2 3 2 ðÞx2 3 Γ 1 1 Γ 1 1 21, 1 ¼ g11, 2 ¼À 22, 2 ¼ g22, 2 ¼À 2 ðÞx2 3 2 ðÞx2 3 328 6 Geodesics and Parallelism of Vectors thus

1 1 Γ2 ¼ g22Γ ¼ Γ1 ¼ Γ1 ¼ Γ2 ¼À 11 11, 2 x2 12 21 22 x2

The parallel transport of the vector vj is given by

_ j Γ j _ i k v ¼À iku v and having 0 u_ iðÞ¼t 1 it follows that

v1 v1 v_ 1 ¼ÀΓ1 v1 ¼ v_ 2 ¼ÀΓ2 v1 ¼ 21 x2 21 x2

These two differential equations have as solutions

t i i 2 v ðÞ¼t v0ex

i where v0 is a constant in t ¼ 0. Exercise 6.10 Calculate the parallel displacement of vector ui along the curve ξ1 ξ2 deﬁned by the parametricÀÁ equationsÀÁ¼ c ¼ constantÀÁ and ¼ t, located on a cone 2 2 2 of parametric equation ξ3 ¼ ξ1 cos ξ2 þ ξ1 sin ξ2 , where the relations between the parametric coordinates (ξ1, ξ2) and the Cartesian coordinates are x1 ¼ ξ1 cos ξ2, x2 ¼ ξ1 sin ξ2, x3 ¼ ξ1. The fundamental form is given by

2 ξ‘ ξm ds ¼ g‘md d where

∂x i ∂x i g ¼ ‘m ∂ξ‘ ∂ξm which components are i 2 1 2 2 2 3 2 ÀÁÀÁ ∂x ∂x ∂x ∂x 2 2 g ¼ ¼ þ þ ¼ cos ξ2 þ sin ξ2 þ 1 11 ∂ξ1 ∂ξ1 ∂ξ1 ∂ξ1 ¼ 2 6.6 Parallelism of Vectors 329

∂x i ∂x i ∂x1 ∂x1 ∂x2 ∂x2 ∂x3 ∂x3 g ¼ ¼ þ þ 12 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2 ÀÁÀÁ ¼ cos ξ2 Àξ1 sin ξ2 þ sin ξ2 ξ1 cos ξ2 þ 0 ¼ 0 ∂x i 2 ∂x1 2 ∂x2 2 ∂x3 2 g ¼ ¼ þ þ 22 ∂ξ2 ∂ξ2 ∂ξ2 ∂ξ2 ÀÁÀÁÀÁ 2 2 2 ¼Àξ1 sin ξ2 þ ξ1 cos ξ2 þ 0 ¼ ξ1 then 2 3 1 6 0 7 20ÀÁ ij 6 2 7 gij ¼ ξ1 2 g ¼ 4 1 5 0 0 ÀÁ ξ1 2

For the non-null Christoffel symbols, it follows that

1 1 Γ22, 1 ¼Àξ Γ12, 2 ¼ Γ21, 2 ¼ ξ ξ1 1 g11Γ ¼À g22Γ ¼ Γ2 ¼ Γ2 ¼ 22, 1 2 12, 2 12 21 ξ1

The parametric equations of the curve are given by

dξ1 dξ2 ¼ 0 ¼ 1 dt dt and with expression (6.6.2) written under the form

‘ m du ‘ dξ þ Γ um ¼ 0 dt mn dt it follows that

du1 dξ2 du2 dξ2 þ Γ1 u2 ¼ 0 þ Γ2 u1 ¼ 0 dt 22 dt dt 12 dt whereby

du1 ξ1 du2 1 À u2 ¼ 0 þ u1 ¼ 0 dt 2 dt ξ1

Differentiating the ﬁrst differential equation and by substitution 330 6 Geodesics and Parallelism of Vectors

d2u1 u1 þ ¼ 0 dt2 2 which solution is pffiffiffi pffiffiffi 2 2 u1 ¼ k cos t þ k sin t 3 2 4 2

The derivative of this solution substituted in the differential equation

du1 ξ1 À u2 ¼ 0 dt 2 provides pffiffiffi pffiffiffi pffiffiffi 2 2 2 u2 ¼ Àk sin t þ k cos t c 3 2 4 2

For t ¼ 0 the point of coordinates (c; 0) exists; in this point writing the initial m values of the coordinates as u0 , it follows that

1 u0 ¼ k3 pffiffiffi 2 2 c 2 u ¼ k4 ∴ k4 ¼ pffiffiffi u 0 c 2 0 whereby pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 c 2 2 2 2 2 u1 ¼ u1 cos t þ u2 sin tu2 ¼À u1 sin t þ u2 cos t 0 2 2 0 2 c 0 2 0 2

For t ¼ 2π: pffiffiffi pffiffiffi pffiffiffi c 2 pffiffiffi 2 pffiffiffi pffiffiffi u1 ¼ u1 cos 2π þ u2 sin 2π u2 ¼À u1 sin 2π þ u2 cos 2π 0 2 0 c 0 0

These expressions show that u1 6¼ u2 in the interval [0; 2π], so the direction of this vector varies along the curve. Exercise 6.11 For the parallelÀÁ displacementÀÁ of a vector along a deﬁned path in the 2 ξ1 2 ξ2 2 space with metric ds ¼ d þ g22 d , show that upon reaching theðð ﬁnal point of the path this vector forms with its initial position an angle equal to KdS,

S where dS ¼ dξ1dξ2. 6.6 Parallelism of Vectors 331

The metric tensor linked to the fundamental form is 10 gij ¼ 0 g22 and the unit vector that moves along the path has components

sin α ξ1 ¼ cos αξ2 ¼ pffiffiffiffiffiffi g22

As this unit vector moves in parallel to itself

δξ1 ¼ 0 δs follows

1 2 2 δξ d ‘ dξ dα dξ ¼ ðÞþcos α Γ1 ξ ¼Àsin α þ Γ1 ξ2 δs ds ‘m ds ds 22 ds 1 ∂g Γ1 ¼ g11Γ ¼À 22 22 22, 1 2 ∂ξ1 δξ1 dα 1 ∂g sin α dξ2 ¼Àsin α À 22 ffiffiffiffiffiffi ¼ 0 δ 1 p s ds 2 ∂ξ g22 ds then pffiffiffiffiffiffi ∂ g dξ2 dα ¼À 22 ds ∂ξ1 ds

The integration provides ð pffiffiffiffiffiffi ∂ g dξ2 α ¼ 22 ds ∂ξ1 ds but ! ð pffiffiffiffiffiffi ð pffiffiffiffiffiffi ðð 2pffiffiffiffiffiffi ∂ g dξ2 ∂ g ∂ g α ¼ 22 ds ¼ 22 dξ2 ¼ 22 dξ1 dξ2 ∂ξ1 ds ∂ξ1 ∂ξ1∂ξ1 S The non-null component of the Riemann–Christoffel tensor of the first kind is

1 ∂2g R ¼À 22 1212 2 ∂ξ1∂ξ1 332 6 Geodesics and Parallelism of Vectors

and as detg ¼ g22 ffiffiffiffiffiffi ∂2 ∂2p 1 g22 g22 K ¼À 1 1 ¼À 1 1 2g22 ∂ξ ∂ξ ∂ξ ∂ξ

It is veriﬁed that ðð α ¼ KdS Q:E:D: S

Exercise 6.12 Given vector ui of constant modulus that moves in parallel along a i i i i curve x ¼ x ðÞs in the Riemann space EN, and the vector v ¼ αu parallel to vector ui, where α is a scalar, show that the condition that vector vi must obey is δ vi dðÞ‘n α i δ s ¼ ds v . The condition of parallel displacement is given by

δui ¼ 0 δs and with the expression of vector vi ¼ αui, it follows that

δvi δui dα dα vi dα ¼ α þ ui ¼ ui ¼ δs δs ds ds α ds whereby

δvi dðÞ‘nα ¼ vi Q:E:D: δs ds

6.6.3 Torsion

The parallel transport of a vector along different ways can lead to two coincident i points. Let pointPxðÞ2EN and the points Q and S be located in the neighborhood of i this point, and determined by means of the translations of vectors du ¼ dε ei and i dv ¼ dλ ei, respectively (Fig. 6.10). AdmittingÀÁ the parallel transport along PS vector SR1 is obtained with compo- εi Γ m ε‘ λn ÀÁnents d À n‘d d ; then PR1 ¼ PS þ SR1, which components are δi εi Γ m ε‘ λn λi d þ d À n‘d d . In a similar way, the parallel transport of along the ÀÁsegment PQ provides the vector PR2 ¼ PQ þ QR2, with components εi λi Γ m ε‘ λn d þ d À ‘nd d . 6.6 Parallelism of Vectors 333

Fig. 6.10 Curved space E N i i R with torsion S x + dλ 1

dλi R2 dε i P Q

For the space EN with afﬁne connections not necessarily symmetric in the indexes n and ‘, vector R2R1 ¼ PR2 À PR1 is given by ÀÁ Γ m Γ m ε λ : : R2R1 ¼ n‘ À ‘n d d em 6¼ 0 ð6 6 6Þ which allows deﬁning the tensor

m Γ m Γ m : : Tn‘ ¼ n‘ À ‘n ð6 6 7Þ

Expression (6.6.7) deﬁnes the torsion tensor of space EN. This tensor measures the difference of closing the “elementary parallelogram” formed by the vectors and their transport parallels. If the connections are the Christoffel symbols, then m Tn‘ ¼ 0 due to the symmetry. In this space for the vectors em results the rotation when having the parallel transport between nearby points. The Christoffel symbols are symmetric connections of the Riemann space EN and form the Levi-Civita connections of this space, i.e., the torsion tensor of a Levi- Civita connection is null. Γ m Γ m If n‘ 6¼ ‘n, the vectors em are submitting a torsion; thus,

m m n‘ Ω ¼ÀT ‘ds ÀÁn Ωm Γ m Γ m n‘ : : ¼À n‘ À ‘n ds ð6 6 8Þ

The antisymmetric pseudotensor of the second order is dsn‘ and is obtained by i i means of the dyadic product of the vectors du ¼ dε ei and dv ¼ dλ ei, whereby

ÀÁ n ‘ n‘ 1 n ‘ ‘ n 1 dε dε ds ¼ dε dλ À dε dλ ¼ ‘ ð6:6:9Þ 2 2 dλn dλ

1 has 2 NNðÞÀ 1 independent components. Putting ÀÁ ÀÁ 1 ‘ 1 ‘ Ωm ¼À Γ m À Γ m dεndλ þ Γ m À Γ m dε dλn 2 n‘ ‘n 2 n‘ ‘n 334 6 Geodesics and Parallelism of Vectors and with the permutation of the dummy indexes n $ ‘ in the ﬁrst term to the right ÀÁ ÀÁ 1 ‘ 1 ‘ Ωm ¼À Γ m À Γ m dεndλ þ Γ m À Γ m dε dλn 2 ‘n n‘ 2 n‘ ‘n then ÀÁ Ωm Γ m Γ m ε‘ λn ¼ n‘ À ‘n d d

It is concluded that

m R2R1 ¼ Ω em ð6:6:10Þ where Ωm are the N components of the vector that evaluates the torsion of the space m EN.IfΩ ¼ 0, then EN is a Riemann space. Γ m Γ m ∂ If there is no symmetry of the afﬁne connections, i.e., n‘ 6¼ ‘n, then kgij 6¼ 0. ∂ Exercise 6.13 Show that kgij ¼ 0 requires that the torsion tensor is null. The nullity of the covariant derivative of the metric tensor allows writing

∂ Γ ‘ Γ ‘ kgij ¼ gij, k À gi‘ jk À gj‘ ki ¼ 0 Γ ‘ Γ ‘ gi‘ jk ¼ gij, k À gj‘ ki and with the cyclic permutation of the indexes

Γ ‘ Γ ‘ gj‘ ki ¼ gjk, i À gk‘ ij thus

Γ ‘ Γ ‘ gi‘ jk ¼ gij, k À gjk, i þ gk‘ ij and in an analogous way

Γ ‘ Γ ‘ gk‘ ij ¼ gki, j À gi‘ jk then ‘ ‘ ‘ 1 g Γ ¼ g À g þ g À g Γ ) Γ ¼ gij g À g þ g i‘ jk ij, k jk, i ki, j i‘ jk jk 2 ij, k jk, i ki, j that defines the Christoffel symbol of the first kind that is symmetrical, whereby by the definition of the torsion tensor

‘ Γ ‘ Γ ‘ : : : Tjk ¼ jk À kj ¼ 0 Q E D 6.6 Parallelism of Vectors 335

Problems

2 2 2 2 2 6.1. Show that in the space with metric ds ð¼ dx þ dy þ dz À cdt ð, the curve with parametric representation x ¼ c r cos φ Á r cos θ ds, y ¼ c r cos φ Á r ð ð sin θ ds, z ¼ c r sin φds, t ¼ c rds has null length. ÀÁ d ∂F ∂F 6.2. Deduce the Euler–Lagrange equation dt ∂x_ p À ∂xp ¼ 0. 6.3. Using the Euler–Lagrange equation, determine the geodesics on the sphere of radius r. i i 6.4. Show that the distance L betweensffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi two points P(x ) and QðÞx in the Riemann XN ÀÁ i i 2 space EN is given by L ¼ x À x . i¼1 6.5. Demonstrate that Pythagoras theorem is valid for the Riemann space EN. Bibliography

Ahsan, Zafar. 2008. Tensor Analysis with Applications. New Delhi: Anamaya Publishers. Akivis, Maks A., and Vladislav V. Golderg. 1977. An Introduction to Linear Algebra & Tensors. New York: Dover. Akivis, Maks A., and Vladislav V. Golderg. 2003. Tensor Calculus with Applications. Singapore: World Scientific. Aracil, Carlos Mataix. 1951. Calculo Vectorial Intrıńseco, 3rd ed. Madrid: Editorial Dossat. Aris, Rutherford. 1989. Vectors, Tensors and the Basic Equations of Fluid Mechanics. New York: Dover. Barbotte, Jean. 1948. Le Calcul Tensoriel. Paris: Bordas. Bergamann, Peter Gabriel. 1976. Introduction to the Theory of Relativity. New York: Dover. Betten, J. 1987. Tensorrechnung fur€ Ingenieure. Stuttgart: B.G. Teubner. Bishop, Richard L., and Samuel I. Goldemberg. 1980. Tensor Analysis on Manifolds. New York: Dover. Blaschke, Wilhelm, and Hans Reichardt. 1960. Einfuhrung€ in die Differentialgeometrie. Berlin: Springer. Block, H.D. 1962. Introduction to Tensor Analysis. Columbus: Charles E. Merrill Books. Bonola, R. 1955. Non-Euclidian Geometry. New York: Dover. Borishenko, A.I., and I.E. Tarapov. 1979. Vector and Tensor Analysis with Applications. New York: Dover. Bourne, D.E., and P.C. Kendall. 1996. Vector Analysis and Cartesians Tensor, 3rd ed. London: Chapman & Hall. Bowen, Ray M., and C.C. Wang. 2008. Vectors & Tensors. New York: Dover. Brand, Louis. 1947. Vector and Tensor Analysis. New York: Wiley. Brillouin, L. 1946. Les Tenseurs en Mećanique et en E´lasticie´. New York: Dover. Charon, Jean E. 1963. Relativite´Geńe´rale. Paris: E´ ditions Rene´ Kister. Cisoti, Umberto. 1928. Lezioni di Calcolo Tensoriale. Milano: Libreria Editrice Politecnica. Coburn, Nathaniel. 1955. Vector and Tensor Analysis. New York: The MacMillan Company. Craig, Homer Vicent. 1943. Vector and Tensor Analysis. New York: McGraw-Hill. Creanga, Ioan, and Tudora Luchian. 1963. Introducere in Calculul Tensorial. Bucuresti: Editura Didactica si Pedagogica. Danielson, D.A. 1992. Vector and Tensors in Engineering and Physics. Boston: Addison-Wesley Publishing Company. Das, Anadijiban. 2007. Tensors: The Mathematics of Relativity Theory and Continuum Mechanics. New York: Springer.

De, U. Chand. 2007. Differential Geometry of Curves and Surfaces in E3: Tensor Approach. New Delhi: Anamay Publisher. De, U.C., Absos Ali Shaikh, and Joydeep Sengupta. 2005. Tensor Calculus. Oxford: Alpha Science International Ltd. Deckert, Adalbert. 1958. Vectoren und Tensoren, eine Einfuhrung€ . Leipzig: C. F. Winter’sche Velargshandlung. Delachet, A. 1955. Calcul Vectoriel et Calcul Tensoriel. Paris: Press Universitaires de France. Denis-Papin, M., R. Faure, and A. Kaufmann. 1962. Calcul Matriciel et de Calcul Tensoriel. Paris: Editions Eyrolles. Dirac, P.A.M. 1981. General Theory of Relativity. New York: Dover. Dodson, C.T.J., and T. Poston. 1979. Tensor Geometry: The Geometric Viewpoint and its Uses. London: Pitman. Dube, K.K. 2009. Differential Geometry and Tensors. Unnao: I.K. International Publishing House Pvt. Ltd. Duschek, Adalbert, and August Hochrainer. 1946. Grundzuge€ der Tensorrechnung in Analytischer Darstellung: Tensoralgebra. Teil 1. Berlin: Springer. Edwards Jr., C.H. 1994. Advanced Calculus of Several Variables. New York: Dover. Eisele, John A., and Robert M. Mason. 1970. Applied Matrix and Tensor Analysis. New York: Wiley-Interscience. Eisenhart, Luther Pfahler. 1940. An Introduction to Differential Geometry with the Use of Tensors Calculus. Princeton: Princeton University Press. Eisenhart, Luther Pfahler. 1949. Riemann Geometry. Princeton: Princeton University Press. Eisenhart, Luther Pfahler. 2005a. Coordinate Geometry. New York: Dover. Eisenhart, Luther Pfahler. 2005b. Non-Riemann Geometry. New York: Dover. Finzi, Bruno, and Pastori Maria. 1949. Calcolo Tensoriale e Applicazioni. Bologna: Nicola Zanichelli Editores. Flugge,€ Wilhelm. 1986. Tensor Analysis and Continuum Mechanics. Berlin: Springer. Frankel, Theodore. 2007. The Geometry of Physics. Cambridge: Cambridge University Press. Gallot, Sylvester, Domique Hulin, and Jacques Lafontaine. 2004. Riemann Geometry, 3rd ed. Berlin: Springer. Gerretsen, Johan C.H. 1962. Lectures on Tensor Calculus and Differential Geometry. Groningen: P. Noordhoff N. V. Gibbs, J. Willard. 1929. Vector Analysis. New Haven: Yale University Press. Golab, Stanislaw. 1974. Tensor Calculus. Warszawa: Elsevier PWN-Polish Scientiﬁc Publishers. Grinfeld, Pavel. 2015. Introduction of Tensor Analysis and the Calculus of Moving Surfaces. New York: Springer. Guggenheimer, Heinrich W. 1963. Differential Geometry. New York: Dover. Guille´n, Franscisco Javier. 1957. Calculo Tensorial. Mexico: Editora Iglesias. Hausner, M.A. 1998. A Vector Space Approach to Geometry. New York: Dover. Hawkins, G.A. 1963. Multilinear Analysis for Students in Engineering and Science. New York: Wiley. Hay, G.E. Vector and Tensor Analysis. New York: Dover. Reprinting of the 1953 edition. Hestenes, David. 1996. Grassmann’s Vision. In: Hermann Gunther Grassmann (1804–1877)— Visionary Mathematician, Scientist and Neohumanist Scholar, 191–201. Dordrecht: Kluwer Academic Publisher. Hladik, Jean, and Pierre-Emmanuel Hladik. 1999. Le Calcul Tensoriel en Physique. Paris: Dunod. Hodge, W.V.D. 1952. Invariants of Quadratic Differential Forms. Cambridge: Cambridge University Press. Hoffman, Banesh. About Vectors. New York: Dover. Reprinting of the 1953 edition. Horst, Teichmann. 1973. Physikalische Anwendugen der Vektor-und Tensorrechnung. Großkrotzenburg: Hain-Druck KG. Iben, Hans Karl. 1999. Tensorrechnung. Leipzig: B.G. Teubner. Itskov, Mikhail. 2007. Tensor Algebra and Tensor Analysis for Engineers. Berlin: Springer. Bibliography 339

Jaeger, L.G. 1966. Cartesian Tensors in Engineering Science. Oxford: Pergamon Press. Jeanperrin, Claude. 1999. Initiation Progressive au Calcul Tensoriel. Paris: Ellipses. Jeanperrin, Claude. 2000. Utilisation du Calcul Tensoriel dans les Geóme´tries Riemmaniennes. Paris: Ellipses. Kay, David C. 1988. Theory and Problems of Tensor Calculus. Schaum’s Outline Series. New York: McGraw-Hill. Khan, Quaddrus. 2015. Tensor Analysis and Its Applications. New Delhi: Partridge. Kilcevski, N.A. 1956. Elemente de Calcul Tensorial si Aplicatiile lui in Mecanica. Bucuresti: Editura Tehnica˘. Klingbeil, Eberhard. 1993. Tensorrechnung fur€ Ingenieure. Berlin: Wissenschaftsverlag. Knowless, James K. 1998. Linear Vectors Spaces and Cartesian Tensors. Oxford: Oxford University Press. Korn, Granino A., and Theresa M. Korn. 2000. Mathematical Handbook for Scientists and Engineers. New York: Dover. Kreyszig, Erwin. 1993. Advanced Engineering Mathematics. New York: Wiley. Lagrange, Rene´. 1922. Sur le Calcul Diffe´rentiel Absolut. Annales de la Faculte´ des Sciences de Toulouse. 3a Se´rie, Tome 14, 1–69. Lang, Serge. 1987. Calculus of Several Variables. London: Springer. Lass, Harry. 1950. Vector and Tensor Analysis. New York: McGraw-Hill. Lawden, D.F. 2002. Introduction to Tensor Calculus, Relativity and Cosmology. New York: Dover. Lebedev, Leonid P., and Michael J. Cloud. 2003. Tensor Analysis. Singapore: World Scientific Publishing. Lebedev, Leonid P., Michael J. Cloud, and Victor A. Eremeyev. 2010. Tensor Analysis with Applications in Mechanics. Singapore: World Scientific Publishing. Lee, John M. 1997. Riemann Manifolds. An Introduction to Curvature. New York: Springer. Levi-Civita, Tullio. The Absolute Differential Calculus (Calculus of Tensors). New York: Dover. Reprinting of the 1926 edition. Linchnerowicz, Andre´. 1987. E´le´ments de Calcul Tensoriel. Paris: E´ ditions Jacques. Loewner, Charles. 2008. Theory of Continuous Groups. New York: Dover. Lovelock, David, and Hanno Rund. 1988. Tensors, Differential Forms and Variational Principles. New York: Dover. Mattews, P.C. 1998. Vector Calculus. London: Springer. McConnell, A.J. 1957. Applications of Tensor Analysis. New York: Dover. Mercier, Jacques L. 1971. An Introduction to Tensor Calculus. Groningen: Wolters-Noordhoff Publishing. Michal, Aristotle D. 1947. Matrix and Tensor Calculus. New York: Wiley. Mital, P.K. 1995. Tensor Analysis for Scientists. New Delhi: Har-Anand Publications. Nayak, Prasun Kumar. 2012. Textbook of Tensor Calculus and Differential Geometry. New Delhi: PHI Learning Private Ltd. Nelson, Edward. 1967. Tensor Analysis. Princeton: Princeton University Press. Neuenschwander, Dwight E. 2015. Tensor Calculus of Physics: A Concise Guide. Baltimore: Johns Hopkins University Press. Oeijord, Nils K. 2003. The Very Basics of Tensors. Bloomington: iUniverse. Ollendorff, F. 1950. Die Welt der Vektoren.O¨ sterreich: Springer. Ollof, Rainer. 2004. Geometrie der Raumzeit. Wiesbaden: Vieweg. 3 Auflage. Pathria, R.K. 2003. Theory of Relativity. New York: Dover. Pauli, Wolfang. 1976. Theory of Relativity. New York: Dover. Sańchez, Emil. 2007. Tensores. Rio de Janeiro: Editora Intercieˆncia (in Portuguese). Sańchez, Emil. 2011. Calculo Tensorial. Rio de Janeiro: Editora Intercieˆncia (in Portuguese). Santalo´, Luis A. 1977. Vectores y Tensores con sus Aplicaciones. Buenos Aires: Editorial Universitaria Buenos Aires. Schade, Heinz. 1997. Tensoranalysis. Berlin: Walter Gruyter. 340 Bibliography

Schey, H.M. 1997. Div, Grad, Curl and All That: An Informal Text on Vector Calculus. New York: W. W. Norton & Company. Schmidt, Harry. 1953. Einfuhrung€ in die Vektor-und Tensorrechnung unter besonderer Berucksichtigung€ ihrer Physikalischer Bedeutung. Berlin: VEB Verlag Techinik. Schouten, J.A. 1989. Tensor Analysis for Physicists. New York: Dover. Schouten, J.A. Ricci-Calculus. Berlin: Springer. Reprinting of the 1954 edition. Schroeder, Dieter. 2006. Vektor-und Tensorpraxis. Frankfurt am Main: Verlag Harri Deutsch. Schutz, Bernard. 2009. A First Course in General Relativity. Cambridge: Cambridge University Press. Semay, Claude, and Bernard Silvester-Brac. 2009. Introduction au Calcul Tensoriel. Applications a la Physique. Paris: Dunod. Sharma, J.N., and A.R. Vasishita. 1987. Vector Calculus. Meerut: Krishna Prakashan Mandir. Simonds, G. James. 1994. A Brief on Tensor Analysis. New York: Springer. Singh, Shalini. 2007. Tensor Calculus. New Delhi: Sarup & Sons. Sokolnikoff, Ivan S. 1951. Tensor Analysis: Theory and Applications. New York: Wiley. Solkonikoff, I.S., and R.M. Redheffer. 1966. Mathematics of Physics and Modern Engineering, 3rd ed. New York: McGraw-Hill Book Company. Spain, Barry. 1958. Tensor Calculus. Edinburgh: Holland Oliver and Boyd. Spiegel, Murray R. 1971. Calculo Avanc¸ado (in Portuguese). S~ao Paulo: Colec¸~ao Schaum, Editora McGraw-Hill do Brasil. Stephani, Hans. 2004. Relativity.An Introduction to Special and General Relativity. Cambridge: Cambridge University Press. Struik, Dirk J. 1988. Lectures on Classical Differential Geometry. New York: Dover. Synge, J.L., and A. Schild. 1978. Tensor Calculus. New York: Dover. Thomas, Tracy Y. 1931. The Elementary Theory of Tensors. New York: McGraw-Hill. Thomas, Tracy Y. 1965. Concepts from Tensor Analysis and Differential Geometry. New York: Academic Press. Weatherburn, C.E. 2008. An Introduction to Riemann Geometry and the Tensor Calculus. Cambridge: Cambridge University Press. Weinreich, Gabriel. 1998. Geometrical Vectors. Chicago: The University of Chicago Press. Weyl, Hermann. 1950. Space-Time-Matter. New York: Dover. Wrede, R.C. 1963. Introduction to Vector and Tensor Analysis. New York: Dover. Index

A C Absolute tensor, 52–58, 123 Calculus, 1, 7, 48, 76, 92, 101, 116, 117, 125, Analysis, 14, 18, 19, 32, 44, 47, 59, 60, 130, 137, 142, 148, 155, 158, 159, 163, 67, 68, 90, 94, 100, 129, 131, 168, 227–229, 295, 296, 309 174, 176, 177, 180, 210, 227, 228, Capacity 231, 233, 266, 267, 323, 326 scalar, 58–59 dimensional, 291–293 tensorial, 60–61 Angle, 7, 8, 16, 17, 39–41, 50, 140, 163, Cartan, 260 166, 249, 281, 292, 313, 319, 321, Cartesian, 7, 79, 84, 197, 227, 292, 320 325–326, 330 coordinate, 8, 17, 26, 28–31, 62, 77–79, 84, Anti-symmetric tensor, 43, 51, 56, 69, 100, 105, 109, 112, 116, 140, 156, 165, 70, 99, 100, 201–203 171, 172, 174, 177, 181, 183, 190, 196, Antisymmetry, 43, 58, 59, 100, 233, 204, 205, 211, 214, 216, 225, 302, 307, 236, 240, 242–244, 248, 250, 309, 323, 327, 328 252, 258, 259, 262, 288 tensor, 74–78, 131 Aristotle, ix Christoffel, E.B., 91 Associate tensor, 71 symbol, 82–102, 105–109, 112, 113, 116, 119, 122, 126, 133–135, 180, 215, 230, 232, 234, 236, 243, 252, 269, B 276, 279–284, 300, 302–305, 308–312, Basis 318, 326, 327, 329, 333, 334 contravariant, 9, 34 Circulation, 159–160, 197 covariant, 4, 9, 34, 65 Cofactor, 24, 35, 57, 90, 265 orthonormal, 6–7 Comma notation, 161 reciprocal, 3–6, 23, 63 Components Beltrami, 160, 164, 214 Cartesian, 101 Bianchi contravariant, 8, 9, 11, 31, 35–37, 44–46, ﬁrst identity, 234–238, 240, 242, 288 62, 79, 101, 107, 122, 195, 263 second identity, 235–238, 253, 258, covariant, 8–12, 22, 31, 32, 34, 36, 45–47, 262, 273, 311 63, 101, 109, 121, 129, 132, 191, 226, Bolyai, xiii 238, 254, 260, 263, 272, 323

Components (cont.) Derivative physical, 62–66, 101, 166, 168, 170, 183, absolute, 73–135, 323 201, 218 contravariant, 73–135, 214 Conjugated covariant, 73–135, 167–169, 172, 179–181, metric tensor, 23, 26, 35, 36, 134, 270, 183, 185, 201, 202, 214, 215, 227–229, 278, 282, 292, 304 231, 234, 235, 258, 259, 273, 275, 302, tensor, 35, 81 311, 323, 334 Contraction, 14, 15, 32, 47, 48, 54, 55, 89–90, directional, 166, 168, 170, 173, 174, 226 121, 124, 126, 180, 181, 250–253, 255, intrinsic, 315 258, 262, 264, 278, 290, 291 Descartes, x, xi Contravariant components, 8, 9, 11, 31, 35–37, Determinant, 7, 23–25, 34, 35, 44–46, 49, 41, 44–46, 62, 64, 79, 101, 107, 121, 50, 52, 53, 57, 90, 196, 201, 205, 191, 195, 263 234, 239, 251, 276 Coordinates, x, xi, xiii, xv Dextrorotatory, 5, 6, 24 Cartesian, 8, 26, 28, 29, 31, 77–79, 84, 100, Direction cosine, 139, 140, 147–151, 163 105, 109, 112, 116, 140, 156, 165, 171, Divergence, 125, 137, 155, 174–194, 200, 172, 174, 177, 181, 183, 190, 196, 204, 207, 209, 214, 217, 220, 253, 205, 211, 214, 216, 292, 302, 309, 323, 263, 264 327, 328 theorem, 177–179, 183, 187, 200 geodesic, 310, 311 Domain polar, 94, 100, 101, 107 multiply connected, 138 spherical, 173, 191, 205, 221 simply connected, 137, 138 Counterclockwise, 138 Dot product, 4, 5, 8, 15, 17, 22, 28, 30–38, product, 139 139, 141, 162, 166, 171, 175, 176, Covariant components, 8–12, 22, 31, 32, 34, 181, 183, 187, 201, 203, 249, 282, 36, 46, 47, 63, 101, 109, 121, 129, 180, 325, 326 192, 226, 238, 254, 260, 263, 272, 323 Dual, 3–7 Curl, 137, 155, 194–197, 199, 201–204, 206, Dummy index, 33, 49, 87, 102, 103, 106, 207, 209, 219, 226 108, 109, 111–113, 120, 122, 125, Curvature 172, 230, 238, 241, 258, 262, 334 constant, 264, 273–275, 290 Gauss, 267–268, 270, 275–277, 293 Riemann, 246–250, 254–256, 266, 273, E 289, 292 Eddington, 260 scalar, 251, 271, 286 Einstein Curve, 16, 17, 19, 21, 39–41, 73, 76, 80, 130, space, 275, 278, 290 131, 137, 138, 140, 142–145, 147, 148, tensor, 262–264, 271–272 150, 159, 160, 165, 199, 281, 295, 296, Equation 299, 301, 304, 307, 308, 311–315, 319, biharmonic, 217 320, 323, 324, 326–330, 332, 335 harmonic, 217 oriented, 138 Einstein, xiii, xvii Euclid, 2, 3, 7, 156, 227, 228, 245, 246, 267, 280, 292, 295, 309–313, D 319, 320, 326, 327 da Vinci, ix Euler, 296 D’Alembert, J.L.R., 225 operator, 225 D’Alembertian, 225 F Delta, 46 Field Kronecker, 3, 16, 46–48, 119, 127 conservative, 209 inverter, 161 homogeneous, 156 Density irrotational, 196, 203, 209 scalar, 53, 60, 61, 123, 124 lamellar, 210 tensorial, 60, 61 line of, 156–159, 162 Index 343

potential, 209 I pseudoscalar, 156 Inner product, 15, 32, 33, 49, 54, 56, scalar, 155–160, 209, 210 58, 116, 117, 130, 135, 230, 250 solenoidal, 177, 178, 187, 209 Integer tensorial, 155–160, 169, 170 line, 73, 143–147, 149, 150, 159, vectorial, 155–160, 168, 170, 174, 177, 160, 170, 197 196, 209 surface, 138, 141, 148, 150, 152, 175 vorticular, 177 volume, 141, 153 Flat space, 228, 280, 281, 313, 319 Integrability, 326–332 Flow, 139–141, 174, 181, 184 Intrinsic, 7, 17, 128–131, 227 Form, 3, 4, 8, 17–20, 22, 23, 32, 41, 44, 56, Invariant, 10, 18, 19, 30, 53, 54, 56, 101, 68, 73, 88, 90, 91, 95–99, 106, 113, 122, 128, 131, 139, 162, 187, 248, 250, 120, 125, 126, 129, 145, 146, 155, 254, 268, 286, 288, 308, 325–327 180, 183, 195, 196, 201, 210, 225, Inverse, 10, 13 227, 234, 239, 246, 248–252, 263, Isotropy, 273–280 265, 280, 295, 300, 307–309, 312, 321, 325, 328, 329, 331, 333 normalized, 248–250 J Function Jacobian, 10, 18, 52, 53, 56, 69, 123 scalar, 74, 75, 77, 101, 122, 123, 156, 161–165, 170, 173, 178, 188, 203, 210, 212, 214–218, 220, 221, K 224–226, 258, 264, 281 Kronecker delta, 46–50 tensorial, 74, 77, 168, 183, 184 vectorial, 10, 76, 79, 139, 140, 156, 159, 163, 167, 171, 174, 176–179, 181, 183, L 189, 193, 195, 199, 200, 202, 203, 206, Levorotatory, 5, 24 207, 209, 218, 225 Lagrange, 296 multiplier, 298 Lame´, 165 G Lamellar, 210 Gauss, J.C.L., 152 Laplace Geodesics, 227, 295–334 equation, 217 with null length, 307–309 operator, 161, 214 Gibbs, xiv Laplacian, 214–221 Gradient, 74, 101, 124, 137, 155, Law, 7, 9–16, 39, 54, 55, 57, 59, 61, 66, 160–164, 207, 209–211, 214, 68, 74, 77, 78, 86, 87, 92, 103, 104, 216, 218, 220, 226 110, 111, 113, 124, 161, 162, 173, Gram, 46 230, 231, 246, 310 Graßmann Length, 16–18, 21, 39, 40, 50, 197, 280, Hermann Gunther,€ xii 291, 295, 296, 314, 315, 335 Grassmann, xiv Levi-Civita, T., 45, 85, 228, 259, 260, Marcel, xvii 321, 322, 333 Line, 32, 50, 146, 147, 149, 150, 152, 159, 162, 166, 227, 243, 295, 302, 304, 319, 326 H Lobachevsky, xiii Hamilton, 161 Heaviside, xiv Hesse, L.O., 225 M operator, 224–225 Mass, 60, 156 Hessian, 225 Matrix, 7 Homogeneous, 2, 7, 10, 12, 16, 19, 43, inverse, 10 53, 57, 156, 166 rotation, 7, 10, 194 344 Index

Metric, 2, 19, 20, 26, 30–38, 45, 246, outer, 14, 54, 55, 61 280, 303, 309 scalar, 131, 139 tensor, 16–38, 45, 75, 81–85, 88–90, 92–94, tensorial, 12, 169, 224 96, 97, 117–119, 122, 130, 133, 165, vectorial, 50 182, 184, 187, 195, 218, 231, 234, Pseudoscalar, 123, 156 238, 241, 245, 250, 254, 263, 265–267, Pseudotensor, 43–47, 49–51, 56–60, 125–127, 270–272, 275, 276, 278–283, 286, 291, 195, 201–203, 268, 333 292, 297, 302–304, 308, 311, 312, 326, Pseudovector, 124, 125, 194 327, 331, 334 Minkowski, 264, 280–281 space, 264, 280–281 Q Mixed product, 4, 5, 46 Quaternions, xi, xii Multiplication Quadratic, 19, 228, 312 of tensor, 13–14, 170 Quotient law, 15–16, 66, 68, 230 of vectors, 2, 34

R N Radius, 73, 176, 277, 295, 303, 304, 313, 335 Nabla, 161–163, 167, 169, 176, 181, 183, Relative tensors, 52–54, 123–125 194, 202, 207, 224 Ricci-Curbastro, B., 101, 228 Norm identity, 92–94, 118, 254 tensor, 319 lemma, 133 of vector, 16, 31–32, 41, 50, 65, 249, 282 pseudotensor, 45–47, 49–51, 126, Normal, 40, 137–140, 147, 148, 150–152, 163, 201–203, 268 166, 167, 174, 175, 178, 182, 184, 187, tensor, 250–263, 265–267, 270–272, 275, 197, 199, 307, 310 276, 285, 286, 288, 289, 291, 292 Riemann, B. curvature, 246–250, 254–256, 266, 273, O 289, 292 Orthogonal, 6, 13, 16, 20, 21, 40–43, 63, 64, 84, geometry, 17, 19, 228 88–89, 96, 97, 140, 164–166, 181–182, space, 295, 296, 298, 302, 303, 309, 197, 201, 216, 218, 243, 246, 247, 250, 313, 314, 320, 321, 326, 332–335 255, 256, 269–270 Rotation, 7, 10, 194, 333 Outer product, 14, 54, 55, 61

S P Scalar Parallelepiped, 4, 5, 64, 174–177, 181, 182 curvature, 250–262, 267, 271, 272, Parallel transport, 228, 320–328, 332, 333 274–278, 285–292 Permutation density, 123 symbol, 3, 44–47, 56, 71 Schouten, A., 228 tensor, 241, 250 Schur, 273, 274 Plato, ix Shift, 320, 323 Poisson, 218 Space equation, 218 conformal, 264, 281–291 Pole, 309–311 of constant curvature, 264, 273–275, 290 Positive, 40, 41, 149, 198, 228, 245, 280 homogeneous, 16 Potential, 101, 162 isotropic, 16, 289, 290, 292 Product plane, 8, 227, 246, 249, 250, 281 dyadic, 167–168, 333 Sphere, 73, 137, 176, 295, 304, 307, 313, inner, 15, 32, 33, 49, 54, 56, 58, 116, 321, 335 117, 130, 135, 230, 250, 322 Spherical, 28, 97, 156, 173, 191, 205, 221, mixed, 4, 5, 46, 59 276, 314, 320 Index 345

Stevin, x null, 253 Stokes, 147–150, 196–201 orthogonal, 16 Subspace, 2 relative, 52–54, 123–125 Subtraction, 14, 54, 55, 116–117, 120 of Riemann-Christoffel, 231, 233–236, Sum 238–240, 243, 245, 246, 250, 251, of tensors, 14, 16 253, 259, 260, 262, 265, 267, 268, of vectors, 2 278–281, 283–284, 286, 288, 290–293, Surface 322, 331 level, 156, 161, 166, 210 symmetric, 18, 38, 42, 92, 120, 203, 252 oriented, 140–141, 153 Theorem smooth, 137–141, 148, 150–153, 184 Gauss-Ostrogradsky, 152, 153, 163, 177 symmetry, 267 Green, 142–147, 150 Symmetric tensor, 18, 38, 42, 92, 120, 203, Pythagoras, 335 252, 263 Stokes, 148, 196–201 Symmetry, 18, 22, 43, 68, 84, 90, 93, 99, 100, Torsion, 332–335 108, 118–120, 126, 230, 232, 241, 242, Trace, 254, 291 244, 250, 252, 259, 265, 267, 271, 288, Transformation 297, 309, 310, 333, 334 homogeneous, 7, 10, 12, 43, 53 System inverse, 7, 12, 52, 91 Cartesian, 102 linear, 7, 13, 16, 18, 68, 69, 77, 88, 245, 309 cylindrical, 26, 29, 62, 96 spherical, 28, 62, 97 U Unit T tangent vector, 301–302, 308, 311, 320 Tangent, 39–41, 76, 80, 130, 131, 142, 156, vector, 3, 5–7, 17, 18, 20, 31, 36, 39–43, 62, 158, 165, 249, 281, 298, 300–302, 307, 79–81, 101, 102, 141, 161–163, 165, 308, 311, 315, 316, 319–321, 326 166, 169, 174, 175, 177, 178, 182, 184, Tensor 187, 199, 246–248, 255–257, 319, 331 anti-symmetric, 43, 51, 56, 99, 100, 201, 203 associated, 32, 35–36, 51, 190 V Cartesian, 74–75, 77–78, 131 Vector conjugated, 22–24, 26–30, 32, 34, 35, 38, contravariant, 8, 9, 31–32, 34, 36, 37, 41, 81, 96, 97, 234, 271, 276, 280, 303 64, 65, 78, 79, 81, 102–104, 107, 121, contravariant, 15, 32, 101–106, 108, 122 129, 167, 172, 179, 180, 191, 231, 234, covariant, 32, 52, 109–113, 117, 119, 184 308 curvature, 227–246, 248, 250–253, 259, coplanar, 246, 247, 319 260, 262, 265, 268, 275, 276, 278, 279, covariant, 9, 10, 32, 100, 109–111, 281, 288–290, 292, 293, 311, 320, 322 116, 120, 122, 133, 168, 172, 180, homogeneous, 16, 53 228, 229, 310 isotropic, 16, 54, 248 mixed, 4, 5, 12, 46, 104 metric, 16–38, 45, 75, 81–85, 88–90, null, 2, 40, 210, 308 92–94, 97, 117–119, 122, 130, 133, Volume, 4, 5, 59, 141, 150–153, 163, 164, 165, 182, 184, 187, 195, 218, 231, 174, 176–178 234, 238, 241, 245, 250, 254, 263, 265–267, 270–272, 275, 276, 278–283, 286, 292, 297, 302–304, 308, 311, W 312, 326, 327, 331, 334 Weyl, H., 85, 286 mixed, 13, 14, 52, 55, 105, 113–116, tensor, 286–292 119, 231 Whirl, 194