Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers. Hung Nguyen-Schäfer Jan-Philip Schmidt.

Transcription

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

2 Mathematical Engineering Series editors Claus Hillermeier, Neubiberg, Germany Jörg Schröder, Essen, Germany Bernhard Weigand, Stuttgart, Germany

3 More information about this series at

4 Hung Nguyen-Schäfer Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers Second Edition

5 Hung Nguyen-Schäfer EM-motive GmbH A Joint Company of Daimler and Bosch Ludwigsburg, Germany Jan-Philip Schmidt Interdisciplinary Center for Scientific Computing (IWR) University of Heidelberg Heidelberg, Germany ISSN ISSN (electronic) Mathematical Engineering ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer-Verlag Berlin Heidelberg 2014, 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg

6 In memory of Gregorio Ricci-Curbastro ( ) and Tullio Levi-Civita ( ), who invented Tensor Calculus, for which Elwin Bruno Christoffel ( ) had prepared the ground; Carl Friedrich Gauss ( ) and Bernhard Riemann ( ), who invented Differential Geometry

7 ThiS is a FM Blank Page

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

9 ThiS is a FM Blank Page

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

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

12 Contents 1 General Basis and Bra-Ket Notation Introduction to General Basis and Tensor Types General Basis in Curvilinear Coordinates Orthogonal Cylindrical Coordinates Orthogonal Spherical Coordinates Eigenvalue Problem of a Linear Coupled Oscillator Notation of Bra and Ket Properties of Kets Analysis of Bra and Ket Bra and Ket Bases Gram-Schmidt Scheme of Basis Orthonormalization Cauchy-Schwarz and Triangle Inequalities Computing Ket and Bra Components Inner Product of Bra and Ket Outer Product of Bra and Ket Ket and Bra Projection Components on the Bases Linear Transformation of Kets Coordinate Transformations Hermitian Operator Applying Bra and Ket Analysis to Eigenvalue Problems References Tensor Analysis Introduction to Tensors Definition of Tensors An Example of a Second-Order Covariant Tensor Tensor Algebra General Bases in General Curvilinear Coordinates Metric Coefficients in General Curvilinear Coordinates Tensors of Second Order and Higher Orders xi

13 xii Contents Tensor and Cross Products of Two Vectors in General Bases Rules of Tensor Calculations Coordinate Transformations Transformation in the Orthonormal Coordinates Transformation of Curvilinear Coordinates in E N Examples of Coordinate Transformations Transformation of Curvilinear Coordinates in R N Tensor Calculus in General Curvilinear Coordinates Physical Component of Tensors Derivatives of Covariant Bases Christoffel Symbols of First and Second Kind Prove That the Christoffel Symbols Are Symmetric Examples of Computing the Christoffel Symbols Coordinate Transformations of the Christoffel Symbols Derivatives of Contravariant Bases Derivatives of Covariant Metric Coefficients Covariant Derivatives of Tensors Riemann-Christoffel Tensor Ricci s Lemma Derivative of the Jacobian Ricci Tensor Einstein Tensor References Elementary Differential Geometry Introduction Arc Length and Surface in Curvilinear Coordinates Unit Tangent and Normal Vector to Surface The First Fundamental Form The Second Fundamental Form Gaussian and Mean Curvatures Riemann Curvature Gauss-Bonnet Theorem Gauss Derivative Equations Weingarten s Equations Gauss-Codazzi Equations Lie Derivatives Vector Fields in Riemannian Manifold Lie Bracket Lie Dragging Lie Derivatives Torsion and Curvature in a Distorted and Curved Manifold

14 Contents xiii Killing Vector Fields Invariant Time Derivatives on Moving Surfaces Invariant Time Derivative of an Invariant Field Invariant Time Derivative of Tensors Tangent, Cotangent Bundles and Manifolds Levi-Civita Connection on Manifolds References Differential Forms Introduction Definitions of Spaces on the Manifold Differential k-forms The Notation ω X Exterior Derivatives Interior Product Pullback Operator of Differential Forms Pushforward Operator of Differential Forms The Hodge Star Operator Star Operator in Vector Calculus and Differential Forms Star Operator and Inner Product Star Operator in the Minkowski Spacetime References Applications of Tensors and Differential Geometry Nabla Operator in Curvilinear Coordinates Gradient, Divergence, and Curl Gradient of an Invariant Gradient of a Vector Divergence of a Vector Divergence of a Second-Order Tensor Curl of a Covariant Vector Laplacian Operator Laplacian of an Invariant Laplacian of a Contravariant Vector Applying Nabla Operators in Spherical Coordinates Gradient of an Invariant Divergence of a Vector Curl of a Vector The Divergence Theorem Gauss and Stokes Theorems Green s Identities First Green s Identity Second Green s Identity Differentials of Area and Volume

15 xiv Contents Calculating the Differential of Area Calculating the Differential of Volume Governing Equations of Computational Fluid Dynamics Continuity Equation Navier-Stokes Equations Energy (Rothalpy) Equation Basic Equations of Continuum Mechanics Cauchy s LawofMotion Principal Stresses of Cauchy s Stress Tensor Cauchy s Strain Tensor Constitutive Equations of Elasticity Laws Maxwell s Equations of Electrodynamics Maxwell s Equations in Curvilinear Coordinate Systems Maxwell s Equations in the Four-Dimensional Spacetime The Maxwell s Stress Tensor The Poynting s Theorem Einstein Field Equations Schwarzschild s Solution of the Einstein Field Equations Schwarzschild Black Hole References Tensors and Bra-Ket Notation in Quantum Mechanics Introduction Quantum Entanglement and Nonlocality Alternative Interpretation of Quantum Entanglement The Hilbert Space State Vectors and Basis Kets The Pauli Matrices Combined State Vectors Expectation Value of an Observable Probability Density Operator Density Operator of a Pure Subsystem Density Operator of an Entangled Composite System Heisenberg s Uncertainty Principle The Wave-Particle Duality De Broglie Wavelength Formula The Compton Effect Double-Slit Experiments with Electrons The Schr odinger Equation Time Evolution in Quantum Mechanics The Schr odinger and Heisenberg Pictures Time-Dependent Schr odinger Equation (TDSE) Discussions of the Schr odinger Wave Functions

16 Contents xv 6.13 The Klein-Gordon Equation The Dirac Equation References Appendix A: Relations Between Covariant and Contravariant Bases Appendix B: Physical Components of Tensors Appendix C: Nabla Operators Appendix D: Essential Tensors Appendix E: Euclidean and Riemannian Manifolds Appendix F: Probability Function for the Quantum Interference Appendix G: Lorentz and Minkowski Transformations in Spacetime Appendix H: The Law of Large Numbers in Statistical Mechanics Mathematical Symbols in This Book Further Reading Index

17 ThiS is a FM Blank Page

18 About the Authors Hung Nguyen-Schäfer is a senior technical manager in development of electric machines for hybrid and electric vehicles at EM-motive GmbH, a joint company of Daimler and Bosch in Germany. He received B.Sc. and M.Sc. in mechanical engineering with nonlinear vibrations in fluid mechanics from the University of Karlsruhe (KIT), Germany, in 1985 and a Ph.D. degree in nonlinear thermo- and fluid dynamics from the same university in He joined Bosch Company and worked as a technical manager on many development projects. Between 2007 and 2013, he was in charge of rotordynamics, bearings, and design platforms of automotive turbochargers at Bosch Mahle Turbo Systems in Stuttgart. He is also the author of three other professional engineering books Aero and Vibroacoustics of Automotive Turbochargers, Springer (2013), Rotordynamics of Automotive Turbochargers in Springer Tracts in Mechanical Engineering, Second Edition, Springer (2015), and Computational Design of Rolling Bearings, Springer (2016). Jan-Philip Schmidt is a mathematician. He studied mathematics, physics, and economics at the University of Heidelberg, Germany. He received a Ph.D. degree in mathematics from the University of Heidelberg in His doctoral thesis was funded by a research fellowship from the Heidelberg Academy of Sciences, in collaboration with the Interdisciplinary Center for Scientific Computing (IWR) at the University of Heidelberg. His academic working experience comprises several research visits in France and Israel, as well as project works at the Max-Planck- Institute for Mathematics in the Sciences (MPIMIS) in Leipzig and at the Max- Planck-Institute for Molecular Genetics (MPIMG) in Berlin. He also worked as a research associate in the AVACS program at Saarland University, Cluster of Excellence (MMCI). xvii

19 Chapter 1 General Basis and Bra-Ket Notation We begin this chapter by reviewing some mathematical backgrounds dealing with coordinate transformations and general basis vectors in general curvilinear coordinates. Some of these aspects will be informally discussed for the sake of simplicity. Therefore, those readers interested in more in-depth coverage should consult the literature recommended under Further Reading. To simplify notation, we will denote a basis vector simply as basis in the following section. We assume that the reader has already had fundamental backgrounds about vector analysis in finite N-dimensional spaces with the general bases of curvilinear coordinates. However, this topic is briefly recapitulated in Appendix E. 1.1 Introduction to General Basis and Tensor Types A physical state generally depending on N different variables is defined as a point P(u 1,...,u N ) that has N independent coordinates of u i. At changing the variables, such as time, locations, and physical characteristics, the physical state P moves from one position to other positions. All relating positions generate a set of points in an N-dimension space. This is the point space with N dimensions (N-point space). Additionally, the state change between two points could be described by a vector r connecting them that obviously consists of N vector components. All state changes are displayed by the vector field that belongs to the vector space with N dimensions (N-vector space). Generally, a differentiable hypersurface in an N-dimensional space with general curvilinear coordinates {u i } for i ¼ 1, 2,...,N is defined as a differentiable (N 1)-dimensional subspace with a codimension of one. Subspaces with any codimension are called manifolds of an N-dimensional space (cf. Appendix E). Physically, vectors are invariant under coordinate transformations and therefore do not change in any coordinate system. However, their components change and Springer-Verlag Berlin Heidelberg 2017 H. Nguyen-Schäfer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI / _1 1

20 2 1 General Basis and Bra-Ket Notation depend on the coordinate system. That means the vector components vary as the coordinate system changes. Generally, tensors are a very useful tool applied to the coordinate transformations between two general curvilinear coordinate systems in finite N-dimensional real spaces. The exemplary second-order tensor can be defined as a multilinear functional T that maps an arbitrary vector in a vector space into the image vector in another vector space. Like vectors, tensors are also invariant under coordinate transformations; however, the tensor components change and depend on the relating transformed coordinate system. Therefore, the tensor components change as the coordinate system varies. Scalars, vectors, and matrices are special types of tensors: scalar (invariant) is a zero-order tensor, vector is a first-order tensor, matrix is arranged by a second-order tensor, bra and ket are first- and second-order tensors, Levi-Civita permutation symbols in a three-dimensional space are third-order pseudo-tensors (Table 1.1). We consider two important spaces in tensor analysis: first, Euclidean N-spaces with orthogonal and curvilinear coordinate systems; second, general curvilinear Riemannian manifolds of dimension N (cf. Appendix E). 1.2 General Basis in Curvilinear Coordinates We consider three covariant basis vectors g 1, g 2, and g 3 to the general curvilinear coordinates (u 1,u 2,u 3 ) at the point P in Euclidean space E 3. The non-orthonormal basis g i can be calculated from the orthonormal bases (e 1,e 2,e 3 ) in Cartesian coordinates x j ¼ x j (u i ) using Einstein summation convention (cf. Sect. 2.1). Table 1.1 Different types of tensors Tensors of 0-order T 1-order T i 2-order T ij 3-order T ijk X (X) Scalar a 2 R X Vector v 2 R N X Matrix M 2 R N R N X Bra hb and ket Ai 2R N, X X R N R N Levi-Civita symbols 2 R N R N R N Higher-order tensors 2 R N...R N X Higher-order T ij...pk

21 1.2 General Basis in Curvilinear Coordinates 3 g i r u i ¼ X3 j¼1 ¼ e j x j u i for j ¼ 1, 2, 3 r x j : xj r xj ui xj u i ð1:1þ The metric coefficients can be calculated by the scalar products of the covariant and contravariant bases in general curvilinear coordinates with non-orthonormal bases (i.e. non-orthogonal and non-unitary). There are the covariant, contravariant, and mixed metric coefficients g ij, g ij, and g i j, respectively. g ij ¼ g ji ¼ g i g j ¼ g j g i 6¼ δ j i g ij ¼ g ji ¼ g i g j ¼ g j g i 6¼ δ j i g j i ¼ g i g j ¼ g j g i ¼ δ j i ð1:2þ where the Kronecker delta δ j i is defined as δ j i 0 for i 6¼ j 1 fori ¼ j: Similarly, the bases of the orthonormal coordinates can be written in the non-orthonormal bases of the curvilinear coordinates u i ¼ u i (x j ). e j r x j ¼ X3 i¼1 ¼ g i u i x j for i ¼ 1, 2, 3 r ui ui x j r ui ui x j ð1:3þ The covariant and contravariant bases of the orthonormal coordinates (orthogonal and unitary bases) have the following properties: e i e j ¼ e j e i ¼ δ j i ; e i e j ¼ e j e i ¼ δ j i ; e i e j ¼ e j e i ¼ δ j i : ð1:4þ The contravariant basis g k of the curvilinear coordinate u k is perpendicular to the covariant bases g i and g j at the given point P, as shown in Fig The contravariant basis g k can be defined as αg k g i g j r r ui u j ð1:5þ where α is a scalar factor;

22 4 1 General Basis and Bra-Ket Notation g 3 u 3 e 3 g 3 x 3 g 1 u 1 g u e e 1 2 x 1 x 2 Fig. 1.1 Covariant and contravariant bases of curvilinear coordinates P g 2 g 2 g k is the contravariant basis of the curvilinear coordinate of u k. Multiplying Eq. (1.5) by the covariant basis g k, the scalar factor α results in g i g j g k ¼ αg k g k ¼ αδk k h i ¼ α ð1:6þ g i ; g j ; g k The expression in the square brackets is called the scalar triple product. Therefore, the contravariant bases of the curvilinear coordinates result from Eqs. (1.5) and (1.6). g i ¼ g j g k h i; g j ¼ h g k g i i; g k ¼ g i g j h i ð1:7þ g i ; g j ; g k g i ; g j ; g k g i ; g j ; g k Obviously, the relation of the covariant and contravariant bases results from Eq. (1.7). g k g i ¼ g i g j g i h i ¼ δi k g i ; g j ; g k ð1:8þ where δ i k is the Kronecker delta. The scalar triple product is an invariant under cyclic permutation; therefore, it has the following properties:

23 1.2 General Basis in Curvilinear Coordinates 5 g i g j g k ¼ ðg k g i Þg j ¼ g j g k g i ð1:9þ Furthermore, the scalar triple product of the covariant bases of the curvilinear coordinates can be calculated [1]. x i x j x k ½g 1 ; g 2 ; g 3 ¼ ε ijk u 1 u 2 u 3 ¼ x 1 x 1 x 1 u 1 u 2 u 3 x 2 x 2 x 2 J u 1 u 2 u 3 x 3 x 3 x 3 u 1 u 2 u 3 ð1:10þ where J is the Jacobian, determinant of the covariant basis tensor; ε ijk is the Levi- Civita symbols in Eq. (A.5), cf. Appendix A. Squaring the scalar triple product in Eq. (1.10), one obtains x 1 x 1 x 1 2 ½g 1 ; g 2 ; g 3 2 u 1 u 2 u 3 g ¼ x 2 x 2 x 2 11 g 12 g 13 ¼ g u 1 u 2 u 3 21 g 22 g 23 x 3 x 3 x 3 g 31 g 32 g 33 u 1 u 2 u 3 ¼ g ij g ¼ J 2 p ) J ¼ ffiffiffi g ð1:11þ where g ij ¼ g i g j are the covariant metric coefficients. Thus, the scalar triple product of the covariant bases results in a right-handed rule coordinate system in which the Jacobian is always positive. J ¼ ½g 1 ; g 2 ; g 3 ¼ ðg 1 g 2 Þ g p 3 ¼ ffiffiffi ð1:12þ g > 0 The covariant and contravariant bases of the orthogonal cylindrical and spherical coordinates will be studied in the following subsections Orthogonal Cylindrical Coordinates Cylindrical coordinates (r,θ,z) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 1.2 shows a point P in the cylindrical coordinates (r,θ,z) which is embedded in the orthonormal Cartesian coordinates (x 1,x 2,x 3 ). However, the cylindrical coordinates change as the point P varies.

24 6 1 General Basis and Bra-Ket Notation x 3 (r,θ,z) (u 1,u 2,u 3 ): u 1 r ; u 2 θ ; u 3 z g 3 z P g 2 e 1 e 3 R e2 0 θ r g 1 x 2 x 1 Fig. 1.2 Covariant bases of orthogonal cylindrical coordinates The vector OP can be written in Cartesian coordinates (x 1,x 2,x 3 ): R ¼ ðr cos θþ e 1 þ ðr sin θþ e 2 þ z e 3 x 1 e 1 þ x 2 e 2 þ x 3 e 3 ð1:13þ where e 1, e 2, and e 3 are the orthonormal bases of Cartesian coordinates; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u 1, u 2, and u 3 are used for r, θ, and z, respectively. Therefore, the coordinates of P(u 1,u 2,u 3 ) can be expressed in Cartesian coordinates: 8 9 Pu 1 ; u 2 ; u 3 < x 1 ¼ r cos θ u 1 cos u 2 = ¼ x 2 ¼ r sin θ u 1 sin u : 2 x 3 ¼ z u 3 ; The covariant bases of the curvilinear coordinates can be computed from ð1:14þ g i ¼ R u i ¼ R x j : xj u i ¼ e j x j u i for j ¼ 1, 2, 3 ð1:15þ The covariant basis matrix G can be calculated from Eq. (1.15).

25 1.2 General Basis in Curvilinear Coordinates 7 G ¼ ½g 1 g 2 g 3 0 x 1 x 1 x 1 1 u 1 u 2 u x ¼ 2 x 2 x 2 cos θ r sin θ 0 sin θ r cos θ 0 A u 1 u 2 u 3 C x 3 x 3 x 3 A u 1 u 2 u 3 The determinant of the covariant basis matrix G is called the Jacobian J. jgj J ¼ x 1 x 1 x 1 u 1 u 2 u 3 x 2 x 2 x 2 cos θ r sin θ 0 ¼ sin θ r cos θ 0 u 1 u 2 u 3 x 3 x 3 x ¼ r u 1 u 2 u 3 ð1:16þ ð1:17þ The inversion of the matrix G yields the contravariant basis matrix G 1. The relation between the covariant and contravariant bases results from Eq. (1.8). g i g j ¼ δ i j ðkronecker deltaþ ð1:18aþ At det (G) 6¼ 0 given from Eq. (1.17), Eq. (1.18a) is equivalent to G 1 G ¼ I ð1:18bþ Thus, the contravariant basis matrix G 1 can be calculated from the inversion of the covariant basis matrix G, as given in Eq. (1.16). 0 u 1 u 1 u g 1 x 1 x 2 x G 1 ¼ 4 g 2 5 u ¼ 2 u 2 u 2 ¼ 1 r cos θ r sin θ sin θ cos θ 0 A g 3 x 1 x 2 x 3 C r u 3 u 3 u 3 A 0 0 r x 1 x 2 x 3 The contravariant bases of the curvilinear coordinates can be written as ð1:19aþ g i ¼ ui x j e j for j ¼ 1, 2, 3 ð1:19bþ The calculation of the determinant and inversion matrix of G will be discussed in the following section. According to Eq. (1.16), the covariant bases can be rewritten as

26 8 1 General Basis and Bra-Ket Notation 8 < g 1 ¼ ðcos θþ e 1 þ ðsin θþe 2 þ 0 e 3 ) jg 1 j ¼ 1 g 2 ¼ ðr sin θþe 1 þ ðr cos θþ e 2 þ 0 e 3 ) jg 2 j ¼ r : g 3 ¼ 0 e 1 þ 0 e 2 þ 1 e 3 ) jg 3 j ¼ 1 The contravariant bases result from Eq. (1.19b). 8 g 1 ¼ ðcos θþ e 1 þ ðsin θþ e 2 þ 0 e 3 ) g 1 ¼ 1 >< g 2 ¼ sin θ e 1 þ cos θ e 2 þ 0 e 3 ) g 2 1 ¼ r r r >: g 3 ¼ 0 e 1 þ 0 e 2 þ 1 e 3 ) g 3 ¼ 1 ð1:20þ ð1:21þ Not only the covariant bases but also the contravariant bases of the cylindrical coordinates are orthogonal due to g i g j ¼ g j g i ¼ δ j i ; g i g j ¼ 0 for i 6¼ j; g i g j ¼ 0 for i 6¼ j: Orthogonal Spherical Coordinates Spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 1.3 shows a point P in the spherical coordinates (ρ,φ,θ) which is embedded in the orthonormal Cartesian coordinates (x 1,x 2,x 3 ). However, the spherical coordinates change as the point P varies. The vector OP can be rewritten in Cartesian coordinates (x 1,x 2,x 3 ): R ¼ ðρ sin φ cos θþ e 1 þ ðρ sin φ sin θþ e 2 þ ρ cos φ e 3 x 1 e 1 þ x 2 e 2 þ x 3 e 3 ð1:22þ where e 1, e 2, and e 3 are the orthonormal bases of Cartesian coordinates; φ is the equatorial angle; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u 1, u 2, and u 3 are used for ρ, φ, and θ, respectively. Therefore, the coordinates of P(u 1,u 2,u 3 ) can be expressed in Cartesian coordinates:

27 1.2 General Basis in Curvilinear Coordinates 9 x 3 ρ sinϕ g 1 (ρ,ϕ,θ) (u 1,u 2,u 3 ): u 1 ρ ; u 2 ϕ ; u 3 θ P g 3 e 1 e 3 ϕ ρ e 0 2 θ g 2 ρ cosϕ x 2 x 1 Fig. 1.3 Covariant bases of orthogonal spherical coordinates 8 9 Pu 1 ; u 2 ; u 3 < x 1 ¼ ρ sin φ cos θ u 1 sin u 2 cos u 3 = ¼ x 2 ¼ ρ sin φ sin θ u 1 sin u 2 cos u : 3 x 3 ¼ ρ cos φ u 1 cos u 2 ; ð1:23þ The covariant bases of the curvilinear coordinates can be computed by means of g i ¼ R u i ¼ R xj xj u i x j ¼ e j u i for j ¼ 1, 2, 3 Thus, the covariant basis matrix G can be calculated from Eq. (1.24). G ¼ ½g 1 g 2 g 3 ¼ 0 x 1 x 1 0 x 1 1 u 1 u 2 u 3 x 2 x 2 x 2 B u 1 u 2 u 3 x 3 x 3 x 3 A sin φ cos θ u 1 u 2 u 3 ρcos φ cos θ ρ sin φ sin θ sin φ sin θ ρcos φ sin θ ρsin φ cos θ cos φ ρ sin φ 0 1 A ð1:24þ ð1:25þ

28 10 1 General Basis and Bra-Ket Notation jgj J ¼ ¼ ρ 2 sin φ sin φ cos θ ρcos φ cos θ ρ sin φ sin θ sin φ sin θ ρcos φ sin θ ρsin φ cos θ cos φ ρ sin φ 0 ð1:26þ The determinant of the covariant basis matrix G is called the Jacobian J. Similarly, the contravariant basis matrix G 1 is the inversion of the covariant basis matrix. 0 u 1 u 1 u g 1 x 1 x 2 x 3 G 1 ¼ 4 g 2 5 u ¼ 2 u 2 u 2 g 3 B x 1 x 2 x 3 u 3 u 3 u 3 A ð1:27aþ 0 x 1 x 2 x 3 1 ρ sin φ cos θ ρsin φ sin θ ρcos φ ¼ 1 cos φ cos θ cos φ sin θ sin φ B ρ sin θ cos θ A 0 sin φ sin φ The contravariant bases of the curvilinear coordinates can be written as g i ¼ ui x j e j for j ¼ 1, 2, 3 ð1:27bþ The matrix product GG 1 must be an identity matrix according to Eq. (1.18b). 0 1 ρ sin φ cos θ ρsin φ sin θ ρcos φ G 1 G ¼ 1 cos φ cos θ cos φ sin θ sin φ B ρ sin θ cos θ A 0 0 sin φ sin φ sin φ cos θ ρcos φ cos θ ρ sin φ sin θ 1 0 sin φ sin θ ρcos φ sin θ ρsin φ cos θ A 0 1 0A I cos φ ρ sin φ According to Eq. (1.25), the covariant bases can be written as ð1:28þ g 1 ¼ ðsin φ cos θþ e 1 þ ðsin φ sin θþ e 2 þ cos φ e 3 ) jg 1 j ¼ 1 g 2 ¼ ðρ cos φ cos θþ e 1 þ ðρ cos φ sin θþ e 2 ðρ sin φþ e 3 ) jg 2 j ¼ ρ g 3 ¼ ðρ sin φ sin θþ e 1 þ ðρ sin φ cos θþe 2 þ 0 e 3 ) jg 3 j ¼ ρ sin φ ð1:29þ The contravariant bases result from Eq. (1.27b).

29 1.3 Eigenvalue Problem of a Linear Coupled Oscillator 11 g 1 ¼ ðsin φ cos θþe 1 þ ðsin φ sin θþ e 2 þ cos φ e 3 ) g 1 ¼ 1 g 2 ¼ 1 cos φ cos θ e 1 þ 1 cos φ sin θ e 2 1 ρ ρ ρ sin φ e 3 ) g 2 1 ¼ ρ g 3 ¼ 1 sin θ e 1 þ 1 cos θ e 2 þ 0 e 3 ) g 3 1 ¼ ρ sin φ ρ sin φ ρ sin φ ð1:30þ Not only the covariant bases but also the contravariant bases of the spherical coordinates are orthogonal due to g i g j ¼ g j g i ¼ δ j i ; g i g j ¼ 0for i 6¼ j; g i g j ¼ 0for i 6¼ j: 1.3 Eigenvalue Problem of a Linear Coupled Oscillator In the following subsection, we will give an example of the application of vector and matrix analysis to the eigenvalue problems in mechanical vibration. Figure 1.4 shows the free vibrations without damping of a three-mass system with the masses m 1, m 2, and m 3 connected by the springs with the constant stiffness k 1, k 2, and k 3.In case of the small vibration amplitudes and constant spring stiffnesses, the vibrations can be considered linear. Otherwise, the vibrations are nonlinear for that the bifurcation theory must be used to compute the responses [2]. Using Newton s second law, the homogenous vibration equations (free vibration equations) of the three-mass system can be written as [2 6]: m 1 x 1 þ k 1 x 1 þ k 2 ðx 1 x 2 Þ ¼ 0 m 2 x 2 þ k 2 ðx 2 x 1 Þþk 3 ðx 2 x 3 Þ ¼ 0 m 3 x 3 þ k 3 ðx 3 x 2 Þ ¼ 0 ð1:31þ Thus, m 1 x 1 þ ðk 1 þ k 2 Þx 1 k 2 x 2 ¼ 0 m 2 x 2 k 2 x 1 þ ðk 2 þ k 3 Þx 2 k 3 x 3 ¼ 0 m 3 x 3 k 3 x 2 þ k 3 x 3 ¼ 0 Using the abbreviations of k 12 k 1 + k 2 and k 23 k 2 + k 3, one obtains

30 12 1 General Basis and Bra-Ket Notation Fig. 1.4 Free vibrations of a three-mass system x1 x2 x3 k 1 k2 k3 m x 2 2 m 3 m1x 1 k k = k + k = k + k 3x m m 1 2 m3 3 k 1x 1 k2( x1 x2) k ( x 1) 2 2 x k ( x 3) 3 2 x k ( x 2) 3 3 x m 1 x 1 þ k 12 x 1 k 2 x 2 ¼ 0 m 2 x k 2 x 1 þ k 23 x 2 k 3 x 3 ¼ 0 m 3 x k 3 x 2 þ k 3 x 3 ¼ 0 The vibration equations can be rewritten in the matrix formulation: 2 4 m m m x 1 k 12 k 2 0 x 2 A þ 4 k 2 k 23 k 3 x 3 0 k 3 k x 1 x 2 x A 0 A 0 ð1:32þ Thus, x þ M 1 K x ¼ 0, x þ Ax ¼ 0 ð1:33þ where 2 A M 1 K ¼ 6 4 k 12 m 1 k 2 m 1 0 k 2 m 2 k 23 m 2 k 3 m 2 0 k 3 m 3 k 3 m The free vibration response of Eq. (1.33) can be assumed as x ¼ Xe λt ) _x ¼ λ Xe λt ¼ λx ) x ¼ λ 2 Xe λt ¼ λ 2 x ð1:34þ where λ is the complex eigenvalue that is defined by

31 1.3 Eigenvalue Problem of a Linear Coupled Oscillator 13 λ ¼ α jω 2 C ð1:35þ in which ω is the eigenfrequency; α is the growth/decay rate [2]. Substituting Eq. (1.34) into Eq. (1.33) one obtains the eigenvalue problem A þ λ 2 I X e λt ¼ 0 ð1:36þ where X is the eigenvector relating to its eigenvalue λ; I is the identity matrix. For any non-trivial solution of x, the determinant of (A + λ 2 I) must vanish. det A þ λ 2 I ¼ 0 ð1:37þ Equation (1.37) is called the characteristic equation of the eigenvalue. Obviously, this characteristic equation is a polynomial of λ 2N where N is the degrees of freedom (DOF) of the vibration system. In this case, the DOF equals 3 for a three-mass system in the translational vibration. Solving the characteristic equation (1.37), one obtains 6 eigenvalues (¼2*DOF) for the vibration equations of the three-mass system. In the case without damping where the real parts are equal to zero, there are six different eigenvalues, three of them for forward whirls and three for backward whirls. λ 1,2 ¼jω 1 ; λ 3,4 ¼jω 2 ; λ 5,6 ¼jω 3 ð1:38þ Substituting the eigenvalue λ i into Eq. (1.36), one obtains the corresponding eigenvector X i. A þ λ 2 i I Xi ¼ 0 for i ¼ 1, 2,...,6: ð1:39þ The eigenvectors in Eq. (1.39) relating to the eigenvalues show the vibration modes of the system. It is well known that N ordinary differential equations (ODEs) of second order can be transformed into 2N ODEs of first order using the simple trick of adding N identical ODEs of first order to the original ODEs. _x ¼ _x _x ¼ _x, M x þ Kx ¼ 0 x ¼M 1 Kx, _x ð1:40þ 0 I x ¼ x M 1 K 0 _x Substituting a new (2N 1) vector of

32 14 1 General Basis and Bra-Ket Notation z ¼ x ) _z ¼ _x _x x ð1:41þ into Eq. (1.40), the vibration equations of first order can be rewritten down _x 0 I x ¼ x M 1 K 0 _x ð1:42þ, _z ¼ Bz The free vibration response of Eq. (1.42) can be assumed as z ¼ Ze λt ) _z ¼ λ Ze λt ¼ λz ð1:43þ where λ is the complex eigenvalue given in λ ¼ α jω 2 C ð1:44þ Within ω is the eigenfrequency; α is the growth/decay rate. Substituting Eq. (1.43) into Eq. (1.42) one obtains the eigenvalue problem ðb λiþze λt ¼ 0 ð1:45þ where Z is the eigenvector relating to its eigenvalue λ; I is the identity matrix. For any non-trivial solution of z, the determinant of (B λi) must vanish: detðb λiþ ¼ 0 ð1:46þ Equation (1.46) is called the characteristic equation of the eigenvalue that is identical to Eq. (1.37). Obviously, this characteristic equation is a polynomial of λ 2N where N is the degrees of freedom (DOF) of the vibration system. In this case, the DOF equals 3 because of the three-mass system. Solving the characteristic equation (1.46), one obtains 6 eigenvalues (¼ 2*DOF) for the vibration equations of the three-mass system. In the case without damping where the real parts are equal to zero, there are six different eigenvalues, three of them for forward whirls and three for backward whirls. λ 1,2 ¼jω 1 ; λ 3,4 ¼jω 2 ; λ 5,6 ¼jω 3 ð1:47þ Substituting the eigenvalue λ i into Eq. (1.45), it gives the corresponding eigenvector Z i. ðb λ i IÞZ i ¼ 0 for i ¼ 1, 2,...,6: ð1:48þ The eigenvectors in Eq. (1.48) relating to the eigenvalues describe the vibration modes of the system.

33 1.5 Properties of Kets 15 Fig. 1.5 Notation of bra and ket Bra BracKet Ket 1.4 Notation of Bra and Ket The notation of bra and ket was defined by Dirac for applications in quantum mechanics and statistical thermodynamics [7]. Bra and ket are tuples of independent coordinates in a finite N-dimensional space in Riemannian manifold (cf. Appendix E). The name of bra and ket comes from the angle bracket <>,as shown in Fig Dividing the bracket into two parts, one obtains the left one called bra and the right one named ket. In general, bra and ket can be considered as vectors, matrices, and high-order tensors. In contrast to vectors (first-order tensors), bra and ket have generally neither direction nor vector length in the point space. They are only a tuple of N coordinates (dimensions), such as of time, position, momentum, velocity, etc. Bra and ket are independent of any coordinate system, but their components depend on the relating basis of the coordinate system; i.e., they are changed at the new basis by coordinate transformations. Therefore, the bra and ket notation is a powerful tool mostly used in quantum mechanics and statistical thermodynamics in order to describe a physical state as a point of N dimensions in a finite N-dimensional complex space. Some examples of bra and ket can be written in different types: Ket vector jki ¼ þ i 1 2 i 1 Ket matrix jmi ¼ 1 þ i i 3 7 5! Bra vector K ð 1 i Þ 1 ð 2þ i 1! Bra matrix hmj ¼ 1 i 1 2 2þ i. 1.5 Properties of Kets We denote the finite N-dimensional complex vector space by C N.Aket Ki can be defined as an N-tuple of the coordinates u 1,...,u N : K(u 1,...,u N ) 2 C N. Given three arbitrary kets Ai, Bi, and Ci 2 C N and two scalars α, β 2 C, the following properties of kets result in [8]: Commutative property of ket addition: AiþBi ¼ Bi þai

34 16 1 General Basis and Bra-Ket Notation Distributive property of ket multiplication by a scalar addition: ðα þ βþ Ai ¼ αai þβai Distributive property of multiplication of ket addition by a scalar: α jai þ B Þ ¼ αai þαbi Associative property of ket addition: Aiþ jbi þ Ci ¼ jai þ Bi Associative property of ket multiplication by scalars: αβ Ai ¼ βαai ¼ αβai Property of ket addition to the null ket 0 i: Aiþ0i ¼ Ai Property of ket multiplication by the null scalar: 0 Ai ¼ 0i Property of ket addition to an inverse ket Ai: Aiþ Ai ¼AiAi ¼0i: þ Ci 1.6 Analysis of Bra and Ket Bra and Ket Bases Ket vector Ai of the coordinates (u 1,...,u N ): A(a 1,...,a N ) 2 V N is the sum of its components in the orthonormal ket bases and can be written as 0 1 where a 1 a 2 : a N C A ¼ XN i ¼ 1 a i ji; i a i ðα i þ jβ i Þ 2 C ð1:49þ

35 1.6 Analysis of Bra and Ket 17 a i is the ket component in the basis ii; a i is a complex number, a i 2 C; ii is the orthonormal basis of the coordinates (u 1,...,u N ). The ket bases of { 1i, 2i,..., Ni} in the coordinates (u 1,...,u N ) are column vectors, as given in ji 1 B : A ; j2i 1 B : A ;...; ji i B 0 1 A ;...; jni 0 B : A 0 1 ð1:50þ Bra ha is defined as the transpose conjugate (also adjoint) Ai * of the ket Ai. Therefore, the bra is a row vector and its elements are the conjugates of the ket elements. To formulate bra of Ai, at first ket Ai must be transposed; then, its complex elements are conjugated into bra elements. 0 1 α 1 þ jβ 1 α jai 2 þ jβ 2 B A ) α N þ jβ N A T ¼ ð α1 þ jβ 1 Þ ðα 2 þ jβ 2 Þ... αn þ jβ N ) jai * ½ðα 1 jβ 1 Þ ðα 2 jβ 2 Þ... ðα N jβ N ÞhAj ð1:51þ The ket vector Ai * is called the transpose conjugate (adjoint) of the ket vector Ai and equals bra ha. Thus, bra ha can be written in the bra bases haj jai * ¼ XN hj: j a * j ; a* j α j jβ j 2 C ð1:52þ j ¼ 1 where the component a* is the complex conjugate of its component a. Analogously, the bra bases result from Eq. (1.50). h1j ½1 0 : 0 ; h2j ½0 1 : 0 ; hj j ½ ; hnj½ : ð1:53þ Due to orthonormality, the product of bra and ket is a scalar and obviously equals the Kronecker delta. hj i ji j hijji ¼ δ j i ¼ 0; i 6¼ j 1; i ¼ j ð1:54þ

36 18 1 General Basis and Bra-Ket Notation The combined symbol h i j i of bra hi and ket ji in Eq. (1.54) is defined as the inner product (scalar product) of bra hi and ket ji Gram-Schmidt Scheme of Basis Orthonormalization The basis { g i i} is non-orthonormal in the curvilinear coordinates in the space R 3. Using the Gram-Schmidt scheme [8, 9], the orthonormal bases ( e 1 i, e 2 i, e 3 i) are created from the non-orthogonal bases ( g 1 i, g 2 i, g 3 i ). The orthonormalization procedure of the basis is discussed in Appendix E.2.6. The first orthonormal ket basis is generated by j je 1 i j1i ¼ g 1i j j The second orthonormal ket basis results from g 1 j je 2 i ji¼ 2 g 2i he 1 jg 2 i je 1 i jjg 2 i he 1 jg 2 i je 1 ij The third orthonormal ket basis is similarly calculated in j je 3 i j3i ¼ g 3i he 1 jg 3 i je 1 i he 2 jg 3 i je 2 i jjg 3 i he 1 jg 3 i je 1 i he 2 jg 3 i je 2 ij Generally, the orthonormal ket basis e j i can be rewritten in the N-dimensional space. e j ji¼ j g j g j E E Xj 1 i ¼1 Xj 1 i ¼1 D E e i jg j je i i D E e i jg j je i i for j ¼ 1, 2,..., N ð1:55þ Using the Gram-Schmidt procedure, the ket orthonormal bases of { 1i, 2i,..., Ni} in the coordinates (u 1,...,u N ) are generated from any non-orthonormal bases, as given in Eq. (1.50). The bra orthonormal bases of {h1, h2,...,hn } in the coordinates (u 1,...,u N ) are the adjoint of the ket orthonormal bases.

37 1.6 Analysis of Bra and Ket Cauchy-Schwarz and Triangle Inequalities The Cauchy-Schwarz and triangle inequalities immediately apply to the Bra-Ket notation: 1) Cauchy-Schwarz Inequality The well-known Cauchy-Schwarz inequality provides the relation between the inner product of bra and ket, and their norms. hajbi jjaij jjbij; jai, jbi 2 V N ð1:56þ 2) Triangle Inequality The triangle inequality formulates the inequality between the sum of two kets and the ket norms. jjaiþ jbij jjaij þ jjbij; jai, jbi 2 V N ð1:57þ Computing Ket and Bra Components The component of a ket results from multiplying the ket by a bra basis according to Eqs. (1.49) and (1.54): hj j jaihjjai ¼ XN i ¼ 1 hj j a i ji i ¼ XN hjjii a i ¼ XN δj i a i i ¼ 1 i ¼ 1 ¼ a j α j þ jβ j ð1:58þ Equation (1.58) indicates that the ket component in the orthogonal bases equals the scalar product between the ket and its relating basis. Similarly, the bra component can be computed by multiplying the bra by a ket basis.

38 20 1 General Basis and Bra-Ket Notation haj ji j hajji ¼ XN ¼ XN i ¼ 1 hja i * i ji j hijji a * i ¼ XN δ j i a* i i ¼ 1 i ¼ 1 ¼ a * j α j jβ j ð1:59þ It is straightforward that the bra component is equal to the complex conjugate of the relating ket component a j, as given in Eq. (1.52) Inner Product of Bra and Ket The inner product of bra ha and ket Bi is defined as! hajbi ¼ XN hja i * i i ¼ 1 ¼ XN X N i ¼ 1j ¼1 ¼ XN i ¼1 a * i b i X N j ¼ 1 a * i b jhijji ¼ XN b j ji j X N! i ¼ 1j ¼1 a * i b jδ j i ð1:60aþ It is obvious that the inner product of bra and ket is a complex number according to Eqs. (1.59) and (1.60a). In case of Ai¼ Bi, the inner product in Eq. (1.60a) becomes! hajai ¼ XN hja i * i i ¼ 1 ¼ XN X N i ¼ 1j ¼1 ¼ XN i ¼ 1 X N j ¼ 1 a * i a jhijji ¼ XN a * i a i ¼ XN i ¼ 1 a j ji j X N! i ¼ 1j ¼ 1 a * i a jδ j i α 2 i þ β 2 i ¼ jjaij 2 ð1:60bþ Thus, the norm (length) of the ket Ai is given in p jjaij ¼ ffiffiffiffiffiffiffiffiffiffiffiffi hajai ð1:60cþ The inner product in Eq. (1.60a) can be rewritten using Eq. (1.52).

39 1.6 Analysis of Bra and Ket b 1 hajbi ¼ XN a * i b i ¼ a * 1 a * 2 a * i a * b N b i ¼ 1 i 5 b N ¼ jai * jbi ¼ haj jbi hajbi ð1:61þ Similarly, the inner product of bra hb and ket Ai can be calculated as follows:! hbjai ¼ XN hjb j * j j ¼ 1 ¼ XN X N i ¼ 1j ¼1 ¼ XN i ¼1 b * i a i X N i ¼ 1 b * j a ihjjii ¼ XN a i ji i X N! i ¼ 1j ¼1 b * j a iδ i j Conjugating Eq. (1.62), one obtains the transpose conjugate of hb Ai: ð1:62þ! * hbjai * ¼ XN b * i a i i ¼1 * X N ¼ ¼ XN a * i b * i i ¼1 ¼ hajbi i ¼1 a * i b i ð1:63þ Thus, the inner product is skew-symmetric (anti-symmetric) contrary to the inner product of two regular vectors. Some properties of the inner product (scalar product) are valid: Skew-symmetry: ha Bi¼hB Ai * ; Positive definiteness: ha Ai¼ Ai 2 0; Distributive property: h A (αb+βc) i¼αha Bi + βha Ci for α, β 2 C. Furthermore, the linear adjoint operator has the following properties: 1. (αβγ) * ¼ [α(βγ)] * ¼ (βγ) * α * ¼ γ * β * α * for α, β, γ 2 C Note that the product order is changed in the adjoint operation of the scalar product. 2. (α Ai) * ¼ αai * ¼ Ai * α * ¼hA α * for α 2 C 3. (ha α) * ¼ α * ha * ¼ α * Aifor α 2 C 4. ha α ** ¼ ha α for α 2 C 5. hb Ai * ¼ Ai *.hb * ¼ ha. BihA Bi: skew-symmetric (anti-symmetric) 6. ha α * Bi * ¼ Bi *.α.ha * ¼ hb.α. AihB α Ai for α 2 C 7. ( AihB ) * ¼ hb *. Ai * ¼ BihA : the outer product of ket and bra.

40 22 1 General Basis and Bra-Ket Notation Outer Product of Bra and Ket The outer product of ket Ai and bra hb is defined as jaihbj ¼ XN i ¼ 1 ¼ XN X N i ¼ 1j ¼1! a i ji i a i b * j ji i hj j X N j ¼ 1 hjb j * j! ð1:64þ where the product term ii hj is called the outer product of the bases ii and hj. Contrary to the inner product resulting a scalar of (1 1) matrix 2 V, the outer product is an operator of (N N) matrix 2 V NN because the ket is an (N 1) column vector 2 V N and the bra is a (1 N) row vector 2 V N. Now, ket Ai can be expressed in ket bases: jai ¼ XN jia i i i ¼ 1 ð1:65þ According to Eq. (1.58), the ket component is a i ¼ hijai Substituting a i into Eq. (1.65), one obtains the ket jai ¼ XN ji i hija i ¼ 1 i XN i ¼ 1 I i jai ð1:66þ where I i is the projection operator (outer product) according to [8], as defined by I i jii i hj¼ ½ ¼ ð1:67þ The element I ij of the projection operator (matrix) is 1 at the i row and j column, as shown in Eq. (1.67); otherwise, other elements are equal to zero. Obviously, the sum of all projection operators is the identity matrix. I XN I i ¼ XN ji i hj i i ¼1 i ¼1 ð1:68þ According to Eq. (1.66), the identity property of the ket is proved by

41 1.6 Analysis of Bra and Ket 23 jai ¼ XN I i jai ¼ IA j i: i ¼ 1 ð1:69þ Ket and Bra Projection Components on the Bases The projection component of ket Ai on the basis i i can be calculated as jai i ¼ I i jai ¼ ji i hj i jai ¼ ji i hijai ¼ jia i i ð1:70þ Thus, ket Ai can be expressed, as shown in Eq. (1.49): jai ¼ XN jai i ¼ XN jia i i i ¼ 1 i ¼ 1 ð1:71þ Similarly, the projection component of bra ha on the basis h i is computed as haj i ¼ haji i ¼ haj ji i hj¼ i hajii hj i ¼ hja i * i ð1:72þ Bra ha can be expressed in its projection components, as given in Eq. (1.52). haj ¼ XN haj i ¼ XN hja i * i i ¼ 1 i ¼ 1 ¼ jai * ¼ XN i ¼ 1! * jia i i ¼ XN a * i i ¼ 1 hj i ð1:73þ Linear Transformation of Kets We consider the complex vector spaces V and V. Each of them belongs to the finite N-dimensional complex space C N. The linear transformation T maps the ket Ai in V into the image ket A i in V, as shown in Fig The image ket A i can be written in the bases ii [8, 9] as

42 24 1 General Basis and Bra-Ket Notation Fig. 1.6 Linear transformation T of a ket Ai T A A = T A N V C N V C T : jai! A 0 ¼ TjAi X N A 0 ¼ T jia i i ¼ XN Tjia i i i ¼ 1 i ¼ 1 ð1:74þ In this case, the ket basis ii is also mapped into the image ket basis i i. The transformation operator T for the basis can be written as T : ji! i E i 0 ) E i 0 ¼ Tji i ð1:75þ The image ket basis i i is formulated as a linear combination of the old ket bases ii. E i 0 ¼ Tji¼ i XN T ji ji; j i ¼ 1, 2,..., N j ¼ 1 ð1:76þ where the operator element T ji is in the j row and i column of the transformation matrix T of the transformation operator T. Multiplying both sides of Eq. (1.76) by the bra basis hk, one obtains D E kji 0 ¼ hkjtji¼ i hkj XN T ji ji j j ¼ 1 ¼ XN j ¼ 1 T ji hkjji ¼ XN T ji δ j k ¼ T ki j ¼ 1 Thus, the operator element results from Eq. (1.77): D E T ki ¼ kji 0 ¼ hkjtji i ð1:77þ ð1:78þ Substituting Eq. (1.76) into Eq. (1.74), one obtains the image ket.

43 1.6 Analysis of Bra and Ket 25 X N A 0 ¼ TjAi ¼ T jia i i i ¼ 1 ¼ XN Tjia i i ¼ XN i ¼ 1 ¼ XN j ¼ 1 X N i ¼ 1 i ¼ 1 X N j ¼ 1 T ji ji j!! T ji a i ji j XN a 0 j ji j j ¼ 1 a i ð1:79þ The component of the image ket in the basis ji is given from Eqs. (1.78) and (1.79). a 0 j ¼ XN T ji a i ¼ XN i ¼ 1 A 0 ¼ TNN jai i ¼ 1 hjt j jia i i, ð1:80þ The image ket in Eq. (1.80) can be rewritten in the transformation matrix T NN a 0 1 T 11 T 12 : T 1i T 1N a 1 a 0 2 T 21 T 22 : T 2i T 2N 6 : 7 4 a 0 5 ¼ a 2 : : : : : j T j1 : : T ji T jn 5 : 6 7 ð1:81þ 4 a i 5 a 0 T N N1 T N2 : T Ni T NN a N where the matrix element T ji is computed by the ket transformation T, as given in Eq. (1.78). T ji ¼ hjt j ji i Coordinate Transformations The ket basis e i i is transformed into the new ket basis g i i by the transformation S 1, as shown in Fig The transformed ket basis can be written as [9]. S 1 : je i i! jg i i ) jg i i ¼ S 1 je i i, S : jg i i! je i i ) je i i ¼ Sjg i i ð1:82þ Analogous to the ket transformation, the old ket basis can be written in a linear combination of the new bases:

44 26 1 General Basis and Bra-Ket Notation Fig. 1.7 Coordinate transformation of bases S A e A g = S A e e i R N 1 S g i = S 1 e i g i R N je i i ¼ XN j ¼ 1 g j E S ji for i ¼ 1, 2,..., N ð1:83þ in which S ji is the matrix element of the transformation matrix S. Multiplying both sides of Eq. (1.83) by the bra basis hg k, one obtains the matrix element S ki. hg k je i j XN g j j ¼ 1 ¼ XN i ¼ hg k j ¼ 1 δ j k S ji ¼ S ki ES ji ¼ XN j ¼ 1 D E g k jg j S ji ð1:84þ Thus, using Eq. (1.82) it gives S ki ¼ hg k je i i ¼ hg k jsjg i i ð1:85þ An arbitrary ket Ai can be expressed linearly in terms of the old ket basis: jai ¼ XN je i i ai e i ¼ 1 where a e i is the ket component in the old basis e i i. Substituting Eq. (1.83) into Eq. (1.86), one obtains! jai ¼ XN je i iai e ¼ XN X N jg k is ki ai e i ¼ 1 i ¼! 1 k ¼ 1 ¼ XN X N S ki ai e jg k i XN a g k j g ki k ¼ 1 i ¼ 1 k ¼ 1 ð1:86þ ð1:87þ Therefore, the ket component in the transformed basis g k i can be calculated by Eq. (1.87).

45 1.6 Analysis of Bra and Ket 27 a g k ¼ XN i ¼ 1 S ki a e i ¼ XN i ¼ 1 h g k jsjg i i a e i, jai g ¼ SjAi e ð1:88þ The transformed ket in Eq. (1.88) can be rewritten in the transformation matrix S NN a 1 g a 2 : a j 5 a N S 11 S 12 : S 1i S 1N S 21 S 22 : S 2i S 2N ¼ : : : : : S j1 S j2 : S ji S jn 5 S N1 S N2 : S Ni S NN where the matrix element is given in Eq. (1.89): S ki ¼ hg k je i i ¼ hg k a 1 a 2 6 : 4 a i a N e 7 5 ð1:89þ jsjg i i ð1:90þ The components of the transformed ket a g k in the new basis g k i are derived from Eq. (1.89). In the following section, a combined transformation of kets consisting of three transformations is carried out [9, 10], as shown in Fig. 1.8: 1. Basis transformation S 1 from Ai g in the basis g i i to Ai e in the basis e i i; 2. Ket transformation T from Ai e to A i e in the basis e i i; 3. Basis transformation S from A i e in the basis e i i to A i g in the basis g i i. The first transformation yields the first transformed ket: jai e ¼ S 1 jai g ð1:91þ The second transformation yields the second transformed ket: A g 1 1 S g i R N U A g 1 g = (STS ) A U A g A e e = S i 1 R A N g T A e e i = T R A 2 3 N e S g A = S A g i R N e Fig. 1.8 The combined transformation of kets

46 28 1 General Basis and Bra-Ket Notation E e A 0 ¼ TjAi e The third transformation leads to the third transformed ket: A 0 E g E 0 e ¼ S A ð1:92þ ð1:93þ Finally, the combined transformed ket of three transformations results from Eqs. (1.91), (1.92), and (1.93). E g E A 0 0 e ¼ S A ¼ STjAi e ¼ STS 1 jai g UjAi g ð1:94þ where the combined transformation U is defined as (STS 1 ). The component of the product of many operators is computed [8, 9] according to Eq. (1.78). U ij ¼ hju i ji¼ j hj i STS 1 ji j ¼ XN X N k ¼1l ¼ 1 ¼ XN X N k ¼1l ¼ 1 hjs i jki hkjtji l hjs l 1 ji j S ik T kl S 1 lj ð1:95þ The ket transformed component of A i g can be obtained from Eqs. (1.94) and (1.95). A 0 g ¼ UjAi g, a 0 i g ¼ XN U ij a g j j ¼ 1 ¼ XN j ¼ 1 X N X N k ¼1l ¼ 1 S ik T kl S 1 lj! a g j ð1:96þ The transformed ket in Eq. (1.96) can be rewritten in the transformation matrix U NN. 2 3 a 0 1 a 0 2 : a 0 5 j a 0 N g U 11 U 12 : U 1i U 1N a 1 U 21 U 22 : U 2i U 2N a 2 ¼ : : : : : U j1 : : U ji U jn 5 : a i 5 U N1 U N2 : U Ni U NN a N g ð1:97þ

47 1.6 Analysis of Bra and Ket Hermitian Operator Hermitian operator plays a key role in eigenvalue problems using in quantum mechanics. Let T be a matrix; the adjoint T { (spoken T dagger) is defined as the transpose conjugate (adjoint) of the matrix T. In quantum mechanics, the operator T { of the matrix T is called a Hermitian operator (or self-adjoint operator) if it equals the operator T of the matrix T (i.e., T { ¼ T or T { ¼ T), cf. Chap. 6. Let T be an operator (matrix or second-order tensor). Its transpose is written as T ¼ T ij ) T T ¼ T ji The complex conjugate of T is defined as T * ¼ T * ij The transpose conjugate of T is called the adjoint T { that is calculated as T T * * ¼ Tji ¼ T * ji T { ¼ T { ij * *, T * ji ¼ Tji ¼ T { ij, T T ¼ T { * If T is hermitian, then T is self-adjoint; i.e., T { ij ¼ T ij, T { ¼ T: An arbitrary ket Bi can be transformed by the linear operator T into a ket: T : jbi 2 R N! TjBi 2 R N ð1:98þ The inner product of bra ha and transformed ket T Bi can be written using the hermitian matrix T { ¼ T * : hajtbi ¼ A * TB ¼ A * T B ¼ T * * A B ¼ T { * A B ¼ T { A jbi ð1:99þ ¼ T { AjB This result shows that the inner product between the transformed bra ht { A and ket Bi is the same inner product of the bra ha and transformed ket T Bi. As a rule of thumb, the inner product does not change when moving the operator T from the second ket into the first bra and changing T into T {. There are some properties of the inner product with a complex number α and its conjugate α * :

48 30 1 General Basis and Bra-Ket Notation hajα Bi ¼ α hajbi ¼ α * AjB for α 2 C hα AjBi ¼ α * hajbi ¼ Ajα * ð1:100þ B for α 2 C The eigenvalue problem derives from the characteristic equation: TjAi ¼ λjai; jai 6¼ 0, for λ 2 C ð1:101þ The inner product between the bra ha and its transformed ket T Ai results as hajta i ¼ hajλai ¼ λhajai ð1:102þ Some characteristics of the eigenvalue problem are discussed in the following section [8, 9]: 1) Eigenvalue of the Hermitian operator is a real number. According to Eq. (1.99), the inner product in Eq. (1.102) with the Hermitian operator T { (¼ T) becomes hajtai ¼ T { AjA htajai ¼ hλajai ¼ λ * ð1:103þ hajai Comparing Eq. (1.102) to Eq. (1.103), one obtains λ * ¼ λ ð1:104þ This result proves that the eigenvalue λ must be a real number. 2) Eigenkets of the Hermitian operator are orthogonal. Given two eigenkets with their different eigenvalues λ and μ, the eigenvalue problems can be formulated: TjAi ¼ λjai; TjBi ¼ μjbi; jai 6¼ 0, jbi 6¼ 0; ð1:105þ The inner product between the kets Ai and T Bi can be given according to Eq. (1.100). hajtb i ¼ hajμ Bi ¼ μhajbi ð1:106þ Similarly, the inner product between the kets T Ai and Bi can be rewritten according to Eq. (1.100). htajbi ¼ hλajbi ¼ λ * hajbi ð1:107þ Comparing Eq. (1.106)toEq.(1.107), one obtains the Hermitian operator T { (¼ T) according to Eq. (1.99):

49 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems 31 hajtbi ¼ T { AjB ¼ htajbi, μhajbi ¼ λ * hajbi ð1:108þ Therefore, μ λ * hajbi ¼ 0 ð1:109þ Because the eigenvalues μ and λ * are different, the inner product ha Bi must be zero according to Eq. (1.109). This result indicates that the eigenkets Ai and Bi are orthogonal. 3) Hermitian matrix is diagonalizable in the normalized basis. For the eigenvalue problem in Eq. (1.101), there exists an eigenket (eigenvector) relating to its eigenvalue. Instead of the formulation given in Eq. (1.105), the Hermitian matrix can be easily written in the orthonormal basis e i i for the eigenvalue λ i : Te j i i ¼ λ i je i i, λ λ λ i ¼ λ 0 i6 7 for i ¼ 1, 2,..., N λ N 0 0 ð1:110þ Therefore, the Hermitian matrix T is obviously diagonalizable at changing the eigenvector basis X i i into the orthonormal basis e i i. In this case, the eigenvalues locate on the main diagonal and other matrix elements are zero in the Hermitian matrix, as shown in Eq. (1.110) [9]. 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems Many problems in physics and engineering can be formulated similar to _X ¼ TX j i ð1:111þ The solution of Eq. (1.111) can be assumed as jxi ¼ jeie λt ð1:112þ where Ei is the eigenvector, λ is the complex eigenvalue, λ ¼ α +jω 2 C in which ω is the eigenfrequency and α is the growth/decay rate. Calculating the first derivative of the solution, one obtains

50 32 1 General Basis and Bra-Ket Notation _X ¼ λjei e λt ¼ λjxi ð1:113þ Substituting Eq. (1.113) into Eq. (1.111), the eigenvalue problem is given as TX j i ¼ λjxi ð1:114þ Equation (1.114) can be rewritten in the matrix form with the identity matrix I. ðt λiþjxi ¼ ji 0 ð1:115þ For nontrivial solutions of Eq. (1.115), the eigenvalue-related determinant must be zero. detðt λiþjt λij ¼ 0 ð1:116þ Equation (1.116) is called the characteristic equation whose solutions are the eigenvalues. The characteristic equation is the polynomial of λ n ; n equals two times of the degrees of freedom (DOF) of the system. Pðλ n Þa n λ n þ a n1 λ n1 þ :::: þ a 1 λ þ a 0 ¼ 0 ð1:117þ There exists an eigenvector (eigenket) for each eigenvalue. The eigenvector results from Eq. (1.115). ðt λ i IÞjX i i ¼ ðt λ i IÞjE i ie λit ¼ j0i 8λ i 2 C ) ðt λ i IÞjE i i ¼ j0i ð1:118þ An example for the eigenvalue problem will be given in the following subsection. Let T be the system matrix; it can be written as T ¼ The characteristic equation of the eigenvalues λ yields

51 1.7 Applying Bra and Ket Analysis to Eigenvalue Problems 33 ð1 λþ 0 0 jt λij ¼ 0 λ ð1 λþ λ 1 ¼ ð1 λþ 1 ð1 λþ ¼ ð1 λþ½λλ ð 1Þþ1 ¼ ð1 λþ λ 2 λ þ 1 ¼ ð1 λþ λ 1 þ j pffiffiffi 3 λ 1 j pffiffi 3 ¼ Thus, there are three eigenvalues as follows: λ 1 ¼ 1; λ 2 ¼ 1 pffiffiffi 2 þ j 3 ; 2 λ 3 ¼ 1 pffiffiffi 2 j 3 2 : Using Eq. (1.118), one obtains the eigenvectors of the eigenvalue problem: For λ ¼ λ 1 ¼ 1: x : 4 x 2 5 ¼ x 3 0 Therefore, 8 < 0x 1 ¼ 0! x 1 1 x 2 x 3 ¼ 0! x 3 ¼x 2 ¼ 0 : x 2 ¼ 0! x 2 ¼ 0 ) je 1 i ¼ A 0 For λ ¼ λ 2 ¼ 1 2 þ j pffiffi 3 2 : 2 pffiffiffi j pffiffi 2 þ j 3 x x 2 5 ¼ pffiffiffi j 3 5 x 3 0 2

52 34 1 General Basis and Bra-Ket Notation Thus, 8 pffiffi 1 2 j 3 x 1 ¼ 0! x 1 ¼ 0 2 >< 1 pffiffi 2 þ j 3 x 2 x 3 ¼ 0! x 2 2j 2 x 2 þ 1 pffiffi 2 j 3 >: x 3 ¼ 0! x 3 ¼ 2 p ffiffiffi 3 j ) je 2 i ¼ 0 1 pffiffiffi 2j A 3 j For λ ¼ λ 3 ¼ 1 2 j pffiffi 3 2 : 2 pffiffiffi þ j pffiffi 2 j 3 x x 2 5 ¼ pffiffiffi þ j 3 5 x Therefore, 8 pffiffi 1 2 þ j 3 x 1 ¼ 0! x 1 ¼ 0 2 >< 1 pffiffi 2 j 3 x 2 x 3 ¼ 0! x 2 2j 2 x 2 þ 1 pffiffi 2 þ j 3 p >: x 3 ¼ 0! x 3 ¼ ffiffiffi 3 þ j: 2 ) je 3 i ¼ 0 1 p2j A ffiffiffi 3 þ j References 1. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 2. Nguyen-Schäfer, H.: Rotordynamics of Automotive Turbochargers. Springer, Berlin (2012) 3. Kraemer, E.: Rotordynamics of Rotors and Foundations. Springer, Heidelberg (1993) 4. Muszýnska, A.: Rotordynamics. CRC, Taylor and Francis, London (2005) 5. Vance, J.: Rotordynamics of Turbomachinery. Wiley, New York, NY (1988) 6. Yamamoto, T., Ishida, Y.: Linear and Nonlinear Rotordynamics. Wiley, New York, NY (2001) 7. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford Science Publications, Oxford (1958) 8. Shankar, R.: Principles of Quantum Mechanics, 2nd edn. Springer, Berlin (1980) 9. Griffiths, D.J.: Introduction to Quantum Mechanics, 2nd edn. Pearson Prentice Hall, Upper Saddle River, NJ (2005) 10. Longair, M.: Quantum Concepts in Physics. Cambridge University Press, Cambridge (2013)

53 Chapter 2 Tensor Analysis 2.1 Introduction to Tensors Tensors are a powerful mathematical tool that is used in many areas in engineering and physics including general relativity theory, quantum mechanics, statistical thermodynamics, classical mechanics, electrodynamics, solid mechanics, and fluid dynamics. Laws of physics and physical invariants must be independent of any arbitrarily chosen coordinate system. Tensors describing these characteristics are invariant under coordinate transformations; however, their tensor components heavily depend on the coordinate bases. Therefore, the tensor components change as the coordinate system varies in the considered spaces. Before going into details, we provide less experienced readers with some examples. Different tensors are listed in Table 2.1, which can be expressed in different chosen bases for any curvilinear coordinate. Using Einstein summation convention, the notation can be shortened. Note that Einstein summation convention is only valid for the same indices in the lower and upper positions. The relating contravariant or covariant tensor components can be expressed in the covariant or contravariant bases (cf. Appendix E). The tensor order is determined by the number of the coordinate basis. Thus, the component of a first-order tensor has only one dummy index i relating to a single basis. In case of a second-order tensor, its component contains two dummy indices i and j that relate to the double bases. Similarly, the component of an N-order tensor has N dummy indices relating to N bases. The dummy indices (inner indices) are the repeated indices running from the values from 1 to N in Einstein summation convention. The free index (outer index) can be independently chosen for any value from 1 to N; i.e., for any tensor component in the particular coordinate, as shown in the below example. Note that the dimensions of the dummy and free indices must be the same value of the space dimensions. Springer-Verlag Berlin Heidelberg 2017 H. Nguyen-Schäfer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI / _2 35

54 36 2 Tensor Analysis Table 2.1 Tensors in general curvilinear coordinates Type Component Basis Tensor First-order tensors 2 R N T i, T i g i, g i ð T 1Þ ¼ T i g i, T i g i Second-order T ij, T ij, T i j, g i, g i, T ¼ T ij g tensors T j i g j, g j i g j,t ij g i g j,t i jg i g j, T j ig i g j 2 R N R N Third-order tensors T ik. j, T j ik. g j, g j i g j g k, T ijk g i g j g k, T ik :jg i g k g j, T j ik:g i g k g j T ijk, T ijk, g i, g i, T ¼ T ijk g, 2 R N R N R N g k, g k N-order tensors T ijk...n, 2 R N... R N T ijk...n, T ik...n. j, T j ik...n. g i, g i, g j, g j, g k, g k,... g n, g n T ðnþ ¼ T ijk...n g i g j g k...g n, T ik...n :jg i g k...g n g j T ¼ T ij g i g j XN X N j¼1 i¼1 T ij g i g j ; i, j : dummy indices T j ¼ T ij g i XN T ij g i ; i : dummy, j : free index i¼1 2.2 Definition of Tensors The definition of tensors is based on multilinear algebra by a multilinear map. We consider the linear vector space L and its dual vector space L*. Each of the vector spaces belongs to the finite N-dimensional space R N, the image space W, to the real space R. A mixed tensor of type (m, n) is a multilinear functional T which maps an (m + n) tuple of vectors of the vector spaces L and L* into the real space W [1] (cf. Fig. 2.1): T m n T : ð L L fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} Þ L * L * fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} n copies m copies R N R fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} N R N R fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} N n copies m copies! W! R T : ðu 1 ;...; u n ; v 1 ;...; v m Þ! Tu ð 1 ;...; u n ; v 1 ;...; v m Þ 2 R ð2:1þ Mapping the covariant basis {g jn }2 L and contravariant basis {g im }2 L* by the multilinear functional T of the tensor type (m, n), one obtains its images in the real

55 2.2 Definition of Tensors 37 T ( u1,..., un; v1,..., vm ) T u,..., u ; v,..., v ( 1 n 1 m ) * T : ( L... L) ( L... L ) N N N N R... R R... R n copies m copies * W R Fig. 2.1 Multilinear mapping functional T space W R. The real images are called the components of the (m + n)-order mixed tensor T with respect to the relating bases: T g j1 ;...; g jn ; g i1 ;...; g im Tj1...jn i1...im 2 W R ð2:2þ Thus, the (m + n)-order mixed tensor T of type (m, n) 2 T n m (L) can be expressed in the covariant and contravariant bases. In total, the (m + n)-order tensor T has N m+n components. T ¼ Tj1...jn i1...im g i1... g im g j1... g jn 2 Tn m ðlþ ð2:3þ where T n m (L) is called the tensor space that consists of all tensors T of type (m, n): T m n ðlþ ¼ ð fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} L LÞ m copies L * L * fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} n copies In case of covariant and contravariant tensors T, only the basis of the dual vector space L* or real vector space L are respectively considered in Eq. (2.3). n-order covariant tensors of type (0, n) in the tensor space T n (L): T ¼ T j1...jn g j1... g jn 2 T n ðlþ ¼ L * L * ð2:4þ fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} n copies m-order contravariant tensors of type (m, 0) in the tensor space T m (L): T ¼ T i1...im g i1... g im 2 T m ðlþ ¼ ðl LÞ fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} m copies ð2:5þ

56 38 2 Tensor Analysis An Example of a Second-Order Covariant Tensor An arbitrary vector v can be expressed in the covariant basis g k in the N-dimensional vector space V as v ¼ v k g k for k ¼ 1, 2,..., N ð2:6þ Applying the bilinear mapping T to the vector v and using the Kronecker delta, one obtains its mapping image Tv. Straightforwardly, this is a tensor of one lower order compared to the mapping tensor T. Tv T ij g i g j v k g k ¼ T ij v k ðg i g k Þg j ¼ T ij v k δk ig j for i ¼ k ¼ T kj v k g j for j, k ¼ 1, 2,..., N T * j g j for j ¼ 1, 2,..., N whereas the second-order covariant tensor T can be expressed as T ¼ T ij g i g j for i, j ¼ 1, 2,..., N; T 2 R N R N ð2:7aþ Note that in case of a three-dimensional vector space R 3 (N ¼ 3), there are nine covariant components T ij. The number of the tensor components can be calculated by N n (3 2 ¼ 9), in which n is the number of indices i and j (n ¼ 2). Obviously, that the mapping image Tv is also a tensor of one lower order compared to the tensor T. The covariant tensor component T j * can be calculated by T * j T kj v k ¼ g j Tv: ð2:7bþ 2.3 Tensor Algebra General Bases in General Curvilinear Coordinates The vector r can be written in Cartesian coordinates of Euclidean space E 3,as displayed in Fig. 2.2.

57 2.3 Tensor Algebra 39 g 3 u 3 e 3 g 3 x 3 g 1 g 2 g 2 u u 1 g e e 1 2 x x 1 2 Fig. 2.2 Bases of general curvilinear coordinates in the space E 3 P r ¼ x i e i ð2:8þ The differential dr results from Eq. (2.8) in dr ¼ e i dx i ¼ r x i dxi ð2:9þ Using the chain rule of differentiation, the orthonormal bases e i of the coordinates x i are defined by e i r x i ¼ r u j u j x i u j ð2:10þ g j for j ¼ 1, 2,..., N xi Analogously, the bases of the curvilinear coordinates u i can be calculated in the curvilinear coordinate system of E N g j r u j ¼ r x k x k u j ¼ e k x k u j for k ¼ 1, 2,..., N ð2:11þ The curvilinear coordinates u i are functions of the coordinates x i, the covariant bases in Eq. (2.11) can be calculated using the chain rule of differentiation.

58 40 2 Tensor Analysis g j ¼ r u j ¼ r x i x i u j ¼ e i x i u j e i x i j for i ¼ 1, 2,..., N ð2:12þ Thus, the curvilinear basis g j can be written in a linear combination of the orthonormal basis e i according to Eq. (2.12). The derivative x j i is called the shift tensor between the orthonormal and curvilinear coordinates. Generally, the basis g i of the curvilinear coordinate u i can be rewritten in a linear combination of the basis g j of other curvilinear coordinate u j. The derivative u i j is defined as the shift tensor between both curvilinear coordinates. g i ¼ r 0 j u u 0 j u i u 0 j ¼ g 0 j u i g 0 j j u0 i for j ¼ 1, 2,..., N ð2:13þ In the curvilinear coordinate system (u 1,u 2,u 3 ) of Euclidean space E 3, its basis is generally non-orthogonal and non-unitary (non-orthonormal basis); i.e., the bases are not mutually perpendicular and their vector lengths are not equal to one [2 4]. In this case, the curvilinear coordinate system (u 1,u 2,u 3 ) has three covariant bases g 1, g 2, and g 3 and three contravariant bases g 1, g 2, and g 3 at the origin P, as shown in Fig Generally, the origin P of the curvilinear coordinates could move everywhere in Euclidean space. Therefore, the bases of the curvilinear coordinates only depend on the respective origin P. For this reason, the curvilinear bases are not fixed in the whole curvilinear coordinates like in Cartesian coordinates. The vector r of the point P(u 1,u 2,u 3 ) can be written in covariant and contravariant bases. r ¼ u 1 g 1 þ u 2 g 2 þ u 3 g 3 ¼ u 1 g 1 þ u 2 g 2 þ u 3 g 3 ð2:14þ where u 1, u 2, u 3 are the contravariant vector components of the coordinates (u 1,u 2,u 3 ); g 1, g 2, g 3 are the covariant bases of the coordinate system (u 1,u 2,u 3 ); u 1, u 2, u 3 are the covariant vector components of the coordinates (u 1,u 2,u 3 ); g 1, g 2, g 3 are the contravariant bases of the coordinate system (u 1,u 2,u 3 ). The covariant base g i is defined by the tangential vector to the corresponding curvilinear coordinate u i for i ¼ 1, 2, 3. Both bases g 1 and g 2 generates a tangential surface to the curvilinear surface (u 1 u 2 ) at the considered origin P. Note that the basis g 1 is not perpendicular to the bases g 2 and g 3. However, the contravariant basis g 3 is perpendicular to the tangential surface (g 1 g 2 ) at the origin P. Generally, the

59 2.3 Tensor Algebra 41 contravariant basis (g k ) results from the cross product of the other covariant bases (g i g j ). α g k ¼ g i g j for i, j, k ¼ 1, 2, 3 ð2:15þ where α is a scalar factor. Multiplying Eq. (2.15) by the covariant basis g k, the scalar factor α can be calculated as α g k g k ¼ αδ k k ¼ α ¼ g i g j g k h i ð2:16þ ) α ¼ g i g j g k g i ; g j ; g k The scalar factor α equals the scalar triple product that is given in [3]: α ½g 1 ; g 2 ; g 3 ¼ g i g j g k ¼ ðg k g i Þg j ¼ g j g k g i g 11 g 12 g 13 2 g 31 g 32 g 33 2 g 21 g 22 g 23 2 ¼ g 21 g 22 g 23 ¼ g 11 g 12 g 13 ¼ g 31 g 32 g 33 g 31 g 32 g 33 g 21 g 22 g 23 g 11 g 12 g rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 p ¼ det ffiffi g ¼ J > 0 g ij ð2:17þ where J is defined as the Jacobian, as given in J ε ijk x i u 1 x j u 2 x k u 3 ¼ x 1 x 1 x 1 u 1 u 2 u 3 x 2 x 2 x 2 u 1 u 2 u 3 x 3 x 3 x 3 u 1 u 2 u 3 ð2:18þ Thus, g k ¼ ε ijk p ffiffiffi g g i g j ¼ ε ijk J g i g j ð2:19þ where ε ijk is the Levi-Civita permutation symbols in Eq. (A.5), cf. Appendix A. According to Eq. (2.19), the contravariant basis g k is perpendicular to both covariant bases g i and g j. Additionally, the contravariant basis g k is chosen such that the vector length of the contravariant basis equals the inversed vector length of its relating covariant basis. Therefore,

60 42 2 Tensor Analysis g i g j g i g k g i ¼ h i ¼ δi k g i ; g j ; g k ð2:20þ As a result, the relation between the contravariant and covariant bases is given in the general curvilinear coordinate system (u 1,...,u N ). g i g k ¼ g k g i ¼ δi k for i, k ¼ 1, 2,..., N g i g k ¼ g k g i 6¼ δi k ð2:21þ for i, k ¼ 1, 2,..., N The basis g i is called dual to the basis g j [5] if g i g j ¼ δ j i for i, j ¼ 1, 2,..., N ð2:22þ where δ j i is the Kronecker delta. Let {g 1, g 2,...,g N } be a covariant basis of the curvilinear coordinates {u i }. The contravariant basis {g 1, g 2,...,g N }, the dual basis to the covariant basis, can be written in the matrix formulation as 2 3 G ¼ ½g 1 g 2 : g i g N ; G 1 ¼ 6 4 g 1 g 2 : g j g N 7 5 ) G1 G ¼ I ð2:23þ where g j is the j row vector of G 1 ; g i is the i column vector of G. The covariant and contravariant bases (dual bases) of the orthogonal cylindrical and spherical coordinates are computed in the following section Orthogonal Cylindrical Coordinates The cylindrical coordinates (r,θ,z) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 2.3 shows a point P in the cylindrical coordinates (r,θ,z) embedded in the orthonormal Cartesian coordinates (x 1,x 2,x 3 ). However, the cylindrical coordinates change as the point P varies. The vector OP can be written in Cartesian coordinates (x 1,x 2,x 3 ): R ¼ ðr cos θþ e 1 þ ðr sin θþ e 2 þ z e 3 x 1 e 1 þ x 2 e 2 þ x 3 e 3 ð2:24þ where e 1, e 2, and e 3 are the orthonormal bases of Cartesian coordinates; θ is the polar angle.

61 2.3 Tensor Algebra 43 x 3 (r,θ,z) (u 1,u 2,u 3 ): u 1 r ; u 2 θ ; u 3 z g 3 z P g 2 e 1 e 3 0 θ R e2 r g 1 x 2 x 1 Fig. 2.3 Covariant bases of orthogonal cylindrical coordinates To simplify the formulation with Einstein symbol, the coordinates of u 1, u 2, and u 3 are used for r, θ, and z, respectively. Therefore, the coordinates of P(u 1,u 2,u 3 ) are given in Cartesian coordinates: 8 9 < x 1 ¼ r cos θ u 1 cos u 2 = Pu 1 ; u 2 ; u 3 ¼ x 2 ¼ r sin θ u 1 sin u : 2 x 3 ¼ z u 3 ; The covariant bases of the curvilinear coordinates are computed from ð2:25þ g i ¼ R u i ¼ R xj xj u i ¼ e x j j u i for j ¼ 1, 2, 3 ð2:26þ The covariant basis matrix G yields from Eq. (2.26): G ¼ ½g 1 g 2 g 3 0 x 1 x 1 x 1 1 u 1 u 2 u x ¼ 2 x 2 x 2 cos θ r sin θ 0 sin θ r cos θ 0 A u 1 u 2 u 3 C x 3 x 3 x 3 A u 1 u 2 u 3 ð2:27þ The determinant of G is called the Jacobian J.

62 44 2 Tensor Analysis jgj J ¼ x 1 x 1 x 1 u 1 u 2 u 3 x 2 x 2 x 2 cos θ r sin θ 0 ¼ sin θ r cos θ 0 u 1 u 2 u 3 x 3 x 3 x ¼ r u 1 u 2 u 3 ð2:28þ The relation between the covariant and contravariant bases yields from Eq. (2.22): g i g j ¼ δ i j ðkronecker deltaþ ð2:29þ Thus, the contravariant basis matrix G 1 results from the inversion of the covariant basis matrix G, as given in Eq. (2.27) g 1 G 1 ¼ 4 g 2 5 ¼ 1 r cos θ r sin θ sin θ cos θ 0 A g 3 r 0 0 r ð2:30þ The calculation of the determinant and inversion matrix of G will be discussed in the following section. According to Eq. (2.27), the covariant bases can be denoted as 8 < g 1 ¼ ðcos θþ e 1 þ ðsin θþe 2 þ 0 e 3 ) jg 1 j ¼ 1 g 2 ¼r ð sin θþe 1 þ ðr cos θþ e 2 þ 0 e 3 ) jg 2 j ¼ r ð2:31þ : g 3 ¼ 0 e 1 þ 0 e 2 þ 1 e 3 ) jg 3 j ¼ 1 The contravariant bases result from Eq. (2.30). 8 g 1 ¼ ðcos θþ e 1 þ ðsin θþ e 2 þ 0 e 3 ) g 1 ¼ 1 >< g 2 ¼ sin θ e 1 þ cos θ e 2 þ 0 e 3 ) g 2 1 ¼ r r r >: g 3 ¼ 0 e 1 þ 0 e 2 þ 1 e 3 ) g 3 ¼ 1 ð2:32þ Not only the covariant bases but also the contravariant bases of the cylindrical coordinates are orthogonal due to g i g j ¼ g j g i ¼ δ j i g i g j ¼ 0for i 6¼ j; g i g j ¼ 0for i 6¼ j:

63 2.3 Tensor Algebra 45 x 3 ρ sinϕ (ρ,ϕ,θ ) (u 1,u 2,u 3 ): u 1 ρ ; u 2 ϕ ; u 3 θ g 1 P g 3 e 1 e 3 ϕ ρ e 0 2 θ g 2 x 2 ρ cosϕ x 1 Fig. 2.4 Covariant bases of orthogonal spherical coordinates Orthogonal Spherical Coordinates The spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. The spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 2.4 shows a point P in the spherical coordinates (r,θ,z) embedded in the orthonormal Cartesian coordinates (x 1,x 2,x 3 ). However, the spherical coordinates change as the point P varies. The vector OP can be written in Cartesian coordinates (x 1,x 2,x 3 ): R ¼ ðρ sin φ cos θþ e 1 þ ðρ sin φ sin θþ e 2 þ ρ cos φ e 3 x 1 e 1 þ x 2 e 2 þ x 3 e 3 ð2:33þ where e 1, e 2, and e 3 are the orthonormal bases of Cartesian coordinates; φ is the equatorial angle; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u 1, u 2, and u 3 are used for ρ, φ, and θ, respectively. Therefore, the coordinates of P(u 1,u 2,u 3 ) are given in Cartesian coordinates:

64 46 2 Tensor Analysis 8 9 Pu 1 ; u 2 ; u 3 < x 1 ¼ ρ sin φ cos θ u 1 sin u 2 cos u 3 = ¼ x 2 ¼ ρ sin φ sin θ u 1 sin u 2 cos u : 3 x 3 ¼ ρ cos φ u 1 cos u 2 ; ð2:34þ The covariant bases of the curvilinear coordinates are computed from g i ¼ R u i ¼ R xj xj u i ¼ e x j j u i for j ¼ 1, 2, 3 Thus, the covariant basis matrix G can be calculated from Eq. (2.35). G ¼ ½g 1 g 2 g 3 ¼ 0 x 1 x 1 0 x 1 1 u 1 u 2 u 3 x 2 x 2 x 2 B u 1 u 2 u 3 x 3 x 3 x 3 A sin φ cos θ u 1 u 2 u 3 ρcos φ cos θ ρ sin φ sin θ sin φ sin θ ρcos φ sin θ ρsin φ cos θ cos φ ρ sin φ 0 The determinant of the covariant basis matrix G is called the Jacobian J. ¼ ρ 2 sin φ jgj J ¼ 1 A sin φ cos θ ρcos φ cos θ ρ sin φ sin θ sin φ sin θ ρcos φ sin θ ρsin φ cos θ cos φ ρ sin φ 0 ð2:35þ ð2:36þ ð2:37þ Similarly, the contravariant basis matrix G 1 is the inversion of the covariant basis matrix G g 1 ρ sin φ cos θ ρsin φ sin θ ρcos φ G 1 ¼ 4 g 2 5 ¼ 1 cos φ cos θ cos φ sin θ sin φ B g 3 ρ sin θ cos θ A ð2:38þ 0 sin φ sin φ The matrix product GG 1 must be an identity matrix according to Eq. (2.23).

65 2.3 Tensor Algebra ρ sin φ cos θ ρsin φ sin θ ρcos φ G 1 G ¼ 1 cos φ cos θ cos φ sin θ sin φ B ρ sin θ cos θ A 0 0 sin φ sin φ sin φ cos θ ρcos φ cos θ ρ sin φ sin θ 1 0 sin φ sin θ ρcos φ sin θ ρsin φ cos θ A 0 1 0A I cos φ ρ sin φ ð2:39þ According to Eq. (2.36), the covariant bases can be written as g 1 ¼ ðsin φ cos θþ e 1 þ ðsin φ sin θþ e 2 þ cos φ e 3 ) jg 1 j ¼ 1 g 2 ¼ ðρ cos φ cos θþ e 1 þ ðρ cos φ sin θþ e 2 ðρ sin φþ e 3 ) jg 2 j ¼ ρ g 3 ¼ ðρ sin φ sin θþ e 1 þ ðρ sin φ cos θþe 2 þ 0 e 3 ) jg 3 j ¼ ρ sin φ ð2:40þ The contravariant bases result from Eq. (2.38). g 1 ¼ ðsin φ cos θþe 1 þ ðsin φ sin θþ e 2 þ cos φ e 3 ) g 1 ¼ 1 g 2 ¼ 1 cos φ cos θ e 1 þ 1 cos φ sin θ e 2 1 ρ ρ ρ sin φ e 3 ) g 2 1 ¼ ρ g 3 ¼ 1 sin θ e 1 þ 1 cos θ e 2 þ 0 e 3 ) g 3 1 ¼ ρ sin φ ρ sin φ ρ sin φ ð2:41þ Not only the covariant bases but also the contravariant bases of the spherical coordinates are orthogonal due to g i g j ¼ g j g i ¼ δ j i ; g i g j ¼ 0 for i 6¼ j; g i g j ¼ 0 for i 6¼ j: Metric Coefficients in General Curvilinear Coordinates The covariant basis vectors g 1, g 2, and g 3 to the general curvilinear coordinates (u 1, u 2,u 3 ) at the point P can be calculated from the orthonormal bases (e 1,e 2,e 3 )in Cartesian coordinates x j ¼ x j (u i ), as shown in Fig. 2.2.

66 48 2 Tensor Analysis g i r u i ¼ r x k x k u i ¼ e k x k u i for k ¼ 1, 2, 3 ð2:42þ The covariant metric coefficients g ij are defined as g ij g i g j ¼ r r u i u j ¼ g j g i g ji ¼ xk x l u i u j e ke l ¼ xk x l u i u j δ kl ¼ xk x k u i u j ð2:43þ Similarly, the contravariant metric coefficients g ij can be denoted as g ij g i g j ¼ g j g i ¼ g ji ð2:44þ Furthermore, the contravariant basis can be rewritten as a linear combination of the covariant bases. g i ¼ A ij g j ð2:45þ According to Eq. (2.44) and using Eqs. (2.21) and (2.45), the contravariant metric coefficients can be expressed as Thus, g ik g i g k ¼ A ij g j g k ¼ A ij δ k j ¼ A ik ð2:46þ g i ¼ g ij g j for j ¼ 1, 2, 3 ð2:47þ Analogously, one obtains the covariant basis g k ¼ g kl g l for l ¼ 1, 2, 3 ð2:48þ The mixed metric coefficients can be defined by

67 2.3 Tensor Algebra 49 gk i gi g k ¼ g ij g j g k ¼ g ij g j g kl g l ¼ g ij g kl g j g l ¼ g ij g kl δj l ¼ g ij g kj ¼ δk i ð2:49aþ Thus, g ij g kj ¼ g kj g ij ¼ δ i k ð2:49bþ Therefore, the contravariant metric tensor is the inverse of the covariant metric tensor. g ij g kj ¼ g kj g ij ¼ δ i k, M1 M ¼ MM 1 ¼ I ð2:50þ where M 1 and M are the contravariant and covariant metric tensors. Thus, g 11 g 12 : g 1N g 11 g 12 : g 1N M 1 g M ¼ 21 g 22 : g 2N : : g ij : 5 g 21 g 22 : g 2N : : g ij : 5 g N1 g N2 : g 2 NN g N1 3 g N2 : g NN 1 0 : 0 ¼ g ij g kj ¼ δk i 0 1 : 0 ¼ : 1 05 ¼ I 0 0 : 1 ð2:51þ According to Eqs. (2.42) and (2.49a), the contravariant bases of the curvilinear coordinates can be derived as δi k g k g i ¼ g k xj u i e j ) δi k e j ¼ g k xj u i e j e j ¼ g k xj u i ) g k ¼ δi k u i x j e j ¼ uk x j e j ð2:52þ Thus, g j ¼ uj x i e i for i ¼ 1, 2, 3 ð2:53þ

68 50 2 Tensor Analysis Generally, the covariant and contravariant metric coefficients of the general curvilinear coordinates have the following properties: 8 >< g ji ¼ g ij ¼ g i g j 6¼ δ j i ðcov: metric coefficientþ g ji ¼ g ij ¼ g i g j 6¼ δ j i ðcontrav: metric coefficientþ >: g j i ¼ g i g j ¼ δ j i ðmixed metric coefficientþ Using Eqs. (2.42) and (2.53), one obtains the Kronecker delta ð2:54þ δ j i ¼ g i g j ¼ xk u i u j x l e k e l ¼ xk u i u j x l δ l k ¼ xk u j u i x k ¼ uj u i The Kronecker delta is defined by δ j 0 for i 6¼ j i 1 for i ¼ j ð2:55aþ ð2:55bþ As an example, the covariant metric tensor M in the cylindrical coordinates results from Eq. (2.31) g 11 g 12 g M ¼ 4 g 21 g 22 g 23 5 ¼ 4 0 r ð2:56aþ g 31 g 32 g The contravariant metric coefficients in the contravariant metric tensor M 1 are calculated from inverting the covariant metric tensor M g 11 g 12 g M 1 ¼ 4 g 21 g 22 g 23 5 ¼ 4 0 r ð2:56bþ g 31 g 32 g Analogously, the covariant metric tensor M in the spherical coordinates results from Eq. (2.40) g 11 g 12 g M ¼ 4 g 21 g 22 g 23 5 ¼ 4 0 ρ ð2:57aþ g 31 g 32 g ðρ sin φþ 2 The contravariant metric coefficients in the contravariant metric tensor M 1 are calculated from inverting the covariant metric tensor M.

69 2.3 Tensor Algebra g 11 g 12 g M 1 ¼ 4 g 21 g 22 g 23 5 ¼ 4 0 ρ 2 0 g 31 g 32 g ðρ sin φ Þ ð2:57bþ Tensors of Second Order and Higher Orders Mapping an arbitrary vector x 2 R N by a linear functional T, one obtains its image vector y ¼ Tx [2, 3]. T : R N! T : x! y ¼ Tx T x j g j ¼ x j T g j ¼ x j T ik g i g k g j ¼ x j T ik δj kgi ¼ T ij x j g i R N ð2:58þ where T is a second-order tensor 2 R N R N. It is obvious that the image vector y ¼ Tx is a tensor of one lower order compared to the tensor T that can be considered as a linear operator. The second-order tensor T can be generated from the tensor product (dyadic product) of two vectors u and v, as denoted in Eq. (2.60). Let u and v be two arbitrary vectors. They can be written in the covariant and contravariant bases as u ¼ u i g i ¼ u i g i 2 R N ; v ¼ v j g j ¼ v j g j 2 R N ð2:59þ The tensor product of two vectors u and v results in the second-order tensor T. T : u, v 2 R N! T u v 2 R N R N T ¼ u i v j g i g j u i v j g i g j T ij g i g j T ¼ u i v j g i g j u i v j g i g j T ij g i g j ð2:60þ Note that the terms g i g j and g i g j are called the covariant and contravariant basis tensors. Hence, they are not the same notations as the covariant and contravariant metric coefficients g ij and g ij, respectively. g i g j 6¼ g i g j g ij g i g j 6¼ g i g j g ij Similarly, one obtains the properties of the mixed basis tensors:

70 52 2 Tensor Analysis g i g j 6¼ g i g j gj i ¼ δj i g i g j 6¼ g i g j g j i ¼ δ j i Each of the covariant and contravariant tensor components T ij and T ij contains nine independent elements (N 2 ¼ 9) 2 R 3 R 3 in a three-dimensional space (N ¼ 3). T ij ¼ u i v j ; T ij ¼ u i v j ð2:61þ An example of the tensor product of two contravariant vectors u and v in a three-dimensional space R 3 is given. 0 T ¼ u v u 1 u 2 u v 1 v 2 v 3 1 A u 1 v 1 u 1 v 2 u 1 v 3 T 11 T 12 T 13 u 2 v 1 u 2 v 2 u 2 v 3 A T 21 T 22 T 23 A; 8u i, v j, T ij 2 R u 3 v 1 u 3 v 2 u 3 v 3 T 31 T 32 T 33 The second-order tensor T consists of nine tensor components (N 2 ¼ 3 2 ) 2 R 3 R 3 in a three-dimensional space R 3 (N ¼ 3) with two indices i and j. The basis g j of the general curvilinear coordinates is mapped by the linear functional T in Eq. (2.58) into the image vector T j that can be written according to Eq. (2.2) as T j T g j ð2:62þ Each vector T j can be expressed in a linear combination of the contravariant basis g i as T j ¼ T ij g i ð2:63þ where T ij is the covariant tensor component of the second-order tensor T. Multiplying Eq. (2.63) by the covariant basis g i and using Eq. (2.62), the covariant tensor component T ij results in T ij g i gi ¼ T g j g i ¼ g i T g j ð2:64þ ) T ij δi i ¼ T ij ¼ g i T g j Equation (2.64) can be written in the contravariant bases g i and g j as follows:

71 2.3 Tensor Algebra 53 T ij ðg i g j Þ ¼ g i T g j ðg i g j Þ ¼ ðg i g i ÞT g j g j ¼ δi itδ j j ¼ T ð2:65þ ) T ¼ T ij g i g j Similarly, the vector T j is formulated in a linear combination of the covariant basis g i. T j ¼ T g j ¼ T ij g i ð2:66þ in which T ij is the contravariant component of the second-order tensor T. Multiplying Eq. (2.66) by the contravariant basis g i, the contravariant tensor component T ij can be computed as T ij g i g i ¼ ðt g j Þg i ¼ g i T g j ) T ij δi i ¼ T ij ¼ g i T g j ð2:67þ Similarly, Eq. (2.67) can be written in the covariant bases g i and g j T ij g i g j ¼ g i :T:g j g i g j ¼ ðg i :g i ÞT g j :g j ¼ δi itδ j j ¼ T ð2:68þ ) T ¼ T ij g i g j Alternatively, the vector component can be rewritten in a linear combination of the mixed tensor component. T j ¼ T g j ¼ T j i: gi ð2:69þ Multiplying Eq. (2.69) by the covariant basis g i, the mixed tensor component T i. j can be calculated as T j i: gi g i ¼ ðt g j Þg i ¼ g i T g j ) T j i: δ i i ¼ T j i: ¼ g i T g j ð2:70þ Note that in Eq. (2.70), the dot after the lower index indicates the position of the basis of the upper index locating after the tensor T. In this case, the tensor T is located between the lower basis g i and upper basis g j [4 6]. Equation (2.70) can be written in the covariant and contravariant bases g j and g i as follows:

72 54 2 Tensor Analysis T j i: g i g j ¼ g i T g j g i g j ¼ ðg i g i ÞT g j g j ¼ δi itδ j j ¼ T ) T ¼ T j i: gi g j Analogously, one obtains the mixed tensor component T i.j. T:j i g j g j ¼ ðt g i Þg j ¼ T g i g j ¼ T g j g i ¼ g i T g j ) T:j i δ j j ¼ T:j i ¼ gi T g j ð2:71þ ð2:72þ Note that in Eq. (2.72), the dot before the lower index indicates the position of the basis of the upper index locating in front of the tensor T. In this case, the tensor T is located between the upper basis g i and lower basis g j [4 6]. Equation (2.72) can be written in the covariant and contravariant bases g i and g j as follows: T i :jðg i g j Þ ¼ g i T g j ðg i g j Þ ¼ ðg i g i ÞT g j g j ¼ δi itδ j j ¼ T ð2:73þ ) T ¼ T:j i g i g j Briefly, the second-order tensor can be written in different expressions according to the covariant, contravariant, and mixed components. T ð2þ ¼ 8 T ij g i g j ; T ij ¼ g i T g j >< T ij g i g j ; T ij ¼ g i T g j T j i: gi g j ; T j i: ¼ g j i T g >: T:j i g i g j ; T:j i ¼ gi T g j Note that if the second-order tensor T is symmetric, then ð2:74þ T ij ¼ T ji ; T ij ¼ T ji ; T j i: ¼ T j :i ; T i :j ¼ T i j: ð2:75þ Compared to the second-order tensors, the first-order tensor T (1) has only one dummy index, as shown in T ð1þ ¼ T ig i ; T i ¼ T g i T i g i ; T i ¼ T g i ð2:76þ An N-order tensor T (N ) is the tensor product of the N covariant, contravariant, and mixed bases of the coordinates:

73 2.3 Tensor Algebra 55 T ðnþ ¼ 8 < : T ij...n g i g j...g n T ij...n g i g j...g n T ij:: l...n g i g j...gl...g n ð2:77þ The N-order tensors contain the 2 N expressions in total. Two of them are in respect of the covariant and contravariant tensor components; and (2 N 2) expressions, in respect of the mixed tensor components [3]. In case of a second-order tensor T (2) for N ¼ 2, there are four expressions: two with the covariant and contravariant tensor components and two with the mixed tensor components, as displayed in Eq. (2.74) Tensor and Cross Products of Two Vectors in General Bases Tensor Product Let u and v be two arbitrary vectors in the finite N-dimensional vector space R N. They can be written in the covariant and contravariant bases as u ¼ u i g i ¼ u i g i 2 R N ; v ¼ v j g j ¼ v j g j 2 R N ð2:78þ The tensor product T of two vectors generates a second-order tensor that can be defined by the linear functional T. In the covariant bases: T : u, v 2 R N! T u v ¼ u i v j g i g j ¼ u i v j g i g j 2 R N R N T ¼ u i v j g i g j T ij g i g j ¼ u i v j g i g j T ij g i g j ð2:79þ where T ij is the contravariant component of the second-order tensor T; T ij is the covariant component of the second-order tensor T; In the covariant and contravariant bases: T : u, v 2 R N! T u v ¼ u i v j g i g j ¼ u i v j g i g j 2 R N R N T ¼ u i v j g i g j T i j g i g j ¼ u i v j g i g j T j i gi g j ð2:80þ where T j i and T i j are the mixed components of the second-order tensor T.

74 56 2 Tensor Analysis The tensor product T of two vectors in an orthonormal basis (e.g. Euclidean coordinate system) is an invariant (scalar). The invariant is independent of the coordinate system and has an intrinsic value in any coordinate transformations. In Newtonian mechanics, the mechanical work W that is created by the force vector F and path vector x does not change in any chosen coordinate system. This mechanical work W ¼ F x is called an invariant and has an intrinsic value of energy. Given three arbitrary vectors u, v, and w in R N and a scalar α in R, the tensor product of two vectors has the following properties [3]: Distributive property uvþ ð wþ ¼ uv þ uw ðu þ vþw ¼ uw þ vw Associative property ðα u Þv ¼ uðα vþ ¼ α uv Cross Product The cross product of two vectors u and v can be defined by a linear functional T. T : u, v 2 R N! T ¼ u v ¼ u i v j g i g j 2 R N ð2:81þ Obviously, the cross product T of two vectors is a vector (first-order tensor) of which the direction is perpendicular to the bases of g i and g j. Using the scalar triple product in Eq. (1.10), the cross product of the bases can be written as g i g j pffiffiffiffi ¼ ε ijk g g k ¼ ε ijk J g k ð2:82þ where ε ijk is the Levi-Civita permutation symbol; J is the Jacobian. Thus, the cross product in Eq. (2.81) can be expressed as T ¼ u v T k g pffiffi k ¼ ε ijk g u i v j g k ¼ ε ijk Ju i v j g k ð2:83þ The covariant component T k of the first-order tensor T results from Eqs. (2.81) to (2.83).

75 2.3 Tensor Algebra 57 T ¼ u i v j g i g j ¼ u i v j g i g j g k g k T k g k ) T k ¼ u i v j pffiffi ð2:84þ g i g j g k ¼ ε ijk g u i v j ¼ ε ijk Ju i v j The Levi-Civita permutation symbol (pseudo-tensor) can be defined as 8 < þ1 ifði, j, kþ is an even permutation; ε ijk ¼ 1 if ði, j, kþ is an odd permutation; : 0ifi¼j, ori ¼ k; or j ¼ k ð2:85þ Therefore, ε ε ijk ¼ ijk ¼ ε jki ¼ ε kij ðeven permutationþ; ε ikj ¼ε kji ¼ε jik ðodd permutationþ ð2:86þ The permutation symbol ε ijk contains totally 27 elements (N n ¼ 3 3 ) for i, j, k (n ¼ 3) in a 27-dimensional tensor space R 3 R 3 R 3. Note that the permutation symbol is used in Eq. (2.83) because the direction of the cross-product vector is opposite if the dummy indices are interchanged with each other in Einstein summation convention (cf. Appendix A). where J denotes the Jacobian. pffiffiffi g g k ¼ J g k ¼ g i g j ¼ g j g i ε ijk g i g j ε ijk g i g j ) g k ¼ pffiffi ¼ g J pffiffiffi ) g i g j ¼ ε ijk g g k ¼ ε ijk Jg k ð2:87þ Rules of Tensor Calculations In order to carry out the tensor calculations, some fundamental rules must be taken into account in tensor calculus [11 13] Calculation of Tensor Components Let T be a second-order tensor; it can be written in different tensor forms:

76 58 2 Tensor Analysis T ¼ T ij g i g j ¼ T i j g i g j ¼ T ij g i g j ¼ T j i g j gi ð2:88þ Multiplying the first row of Eq. (2.88) by the covariant basis g k, one obtains T ij g k g j g i ¼ T ij g kj g i ¼ Tk ig i ) Tk i ¼ ð2:89aþ Tij g kj for j ¼ 1, 2,..., N Analogously, multiplying the second row of Eq. (2.88) by the contravariant basis g k, one obtains T ij g k g j g i ¼ T ij g kj g i ¼ Ti kgi ) Ti k ¼ T ij g kj ð2:89bþ for j ¼ 1, 2,..., N Multiplying Eq. (2.89a) byg kj, the contravariant tensor components result in T ij ¼ T i k gkj for k ¼ 1, 2,...,N ð2:90aþ Multiplying Eq. (2.89b) byg kj, one obtains the covariant tensor components T ij ¼ Ti k g kj for k ¼ 1, 2,..., N ð2:90bþ Substituting Eqs. (2.89a) and (2.90a), one obtains the contraction rules between the contravariant tensor components. T ij ¼ T ip g pk g kj for k, p ¼ 1, 2,..., N ð2:91aþ Similarly, the contraction rules between the covariant tensor components result from substituting Eqs. (2.89b) and (2.90b). T ij ¼ T ip g pk g kj for k, p ¼ 1, 2,..., N ð2:91bþ Analogously, the contraction rules between the mixed tensor components can be derived as T i j ¼ T i p gpk g kj for k, p ¼ 1, 2,..., N ð2:92aþ Similarly, one obtains T j i ¼ T p i g pk gkj for k, p ¼ 1, 2,..., N ð2:92bþ

77 2.3 Tensor Algebra Addition Law Tensors of the same orders and types can be added together. The resulting tensor has the same order and type of the initial tensors. The tensor resulted from the addition of two covariant or contravariant tensors A and B can be calculated as C ¼ A þ B ¼ A ijk þ B ijk g i g j g k ¼ C ijk g i g j g k ¼ B þ A ) C ijk ¼ A ijk þ B ijk ¼ B ijk þ A ijk ; C ¼ A þ B ¼ A ijk þ B ijk gi g j g k ¼ C ijk ð2:93þ g i g j g k ¼ B þ A ) C ijk ¼ A ijk þ B ijk ¼ B ijk þ A ijk Similarly, the tensor resulted from the addition of two mixed tensors A and B can be written as C ¼ A þ B ¼ A pq ijk þ Bpq ijk g p g q g i g j g k ¼ C pq ijk g p g q gi g j g k ¼ B þ A ð2:94þ ) C pq ijk ¼ Apq ijk þ Bpq ijk ¼ Bpq ijk þ Apq ijk Straightforwardly, the addition of tensors is commutative, as proved in Eqs. (2.93) and (2.94) Outer Product On the contrary, the outer product can be carried out at tensors of different orders and types. The tensor components resulted from the outer product of two mixed tensors A and B can be calculated as AB ¼ A pq ij g p g q gi g j B rts kl gk g l g r g t g s ¼ C pqrst ijkl g p g q g i g j g k g l g r g t g s 6¼ BA ) C pqrst ijkl ¼ A pq ij Brts kl ¼ B rts kl Apq ij ð2:95þ The outer product of two tensors results a tensor with the order that equals the sum of the covariant and contravariant indices. The outer product is not commutative, but their tensor components are commutative, as shown in Eq. (2.95). In this example, the resulting ninth-order tensor is generated from the outer product of the mixed fourth-order tensor A and mixed fifth-order tensor B. Obviously, the outer product of tensors A, B, and C is associative, i.e. A(BC) ¼ (AB)C.

78 60 2 Tensor Analysis Contraction Law The contraction operation can be only carried out at the mixed tensor types of different orders. The tensor contraction is operated in many contracting steps where the tensor order is shortened by eliminating the same covariant and contravariant indices of the tensor components. We consider a mixed tensor of high orders. In this example, the mixed fifth-order tensor A of type (2, 3) can be transformed from the coordinates {u i } into the barred coordinates { u i }. The transformed tensor components can be calculated according to the transformation law in Eq. (2.144). A ij klm ¼ ui u j u r u s u t u p u q u k u l u m Apq rst ð2:96þ Carrying out the first contraction of Ā in Eq. (2.96) atl ¼ i, one obtains the tensor components A ij kim ¼ ui u j u r u s u t u p u q u k u l ¼ uj u q u r u k u t u m u m Apq rst u i u p u s u i ¼ uj u r u t u q u k u m δ p s Apq rst ¼ uj u r u t u q u k u m Apq rpt, B j km ¼ uj u r u t u q u k u m B rt q A pq rst ð2:97aþ As a result, the resulting tensor components B are the third-order tensor type after the first contraction of A at s ¼ p: A pq rst δ p s ¼ Apq rpt B q rt ð2:97bþ Further contracting the tensor components B in Eq. (2.97a) at k ¼ j, one obtains B j jm ¼ ut u j u r u m u q u j ¼ ut u m δ r q B q rt ¼ ut u m B q qt, C m ¼ ut u m C t B q rt ð2:98aþ

79 2.3 Tensor Algebra 61 As a result, the resulting tensor components C are the first-order tensor type after the second contraction of B in Eq. (2.98a) atr ¼ q: B q rt δ q r ¼ B q qt C t ð2:98bþ Inner Product The inner product of tensors comprises two basic operations of the outer product and at least one contraction of tensors. As an example, the outer product of two third-order tensors A and B results in a sixth-order tensor. AB ¼ A mp q g m g p gq Bst r g r gs g t ¼ A mp q B ð2:99þ st r gq g r g m g p g s g t Using the first tensor contraction in Eq. (2.99) atr ¼ q, one obtains the resulting fourth-order tensor components of the inner product. A mp q B st r δ q r ¼ Amp q B st q Cmp st ð2:100þ Similarly, using the second tensor contraction law in Eq. (2.100) at p ¼ s, the resulting second-order tensor components result in C mp st δ p s ¼ C ms st D m t ð2:101þ Finally, applying the third tensor contraction law to Eq. (2.101) at m ¼ t, the resulting tensor component is an invariant (zeroth-order tensor). D m t δ m t ¼ D t t D ð2:102þ In another approach, one can calculate the tensor components of the inner product of two contravariant tensors A and B multiplying by the metric tensor. A ijk B lm! A ijk B lm g lk ð2:103þ Using Eq. (2.89a), one obtains the resulting tensor components B lm g lk ¼ B m k ð2:104þ Substituting Eq. (2.104) into Eq. (2.103) and using the tensor contraction law, one obtains the resulting tensor components

80 62 2 Tensor Analysis C ijm A ijk B m k ¼ B m k Aijk ð2:105þ Equation (2.105) denotes that the inner product of the tensor components is commutative Indices Law Using the metric tensors, the operation of moving indices enables changing indices of the tensor components from the upper into lower positions and vice versa. Multiplying a tensor component by the metric tensor components, the lower index (covariant index) is moved into the upper index (contravariant index) and vice versa. Moving covariant indices i, j to the upper position: Aij k! A ij k gil ¼ Aj kl! Aj kl g jm ¼ A klm ð2:106aþ Moving contravariant indices i, j to the lower position: A ij k! Aij k g jl ¼ A i kl! A i kl g im ¼ A klm ð2:106bþ Quotient Law The quotient law of tensors postulates that if the tensor product of AB and B are tensors, A must be a tensor. ðab ¼ C tensorþ \ ðb tensorþ ) ða tensorþ ð2:107aþ Proof Using the contraction law, the barred components in the transformation coordinates { u i } of the tensor product AB result in Aij k Bil k ¼ C j l ) A k ij Bil k ¼ C l j ð2:107bþ According to the transformation law (2.144), the transformed components in the coordinates { u i } of the tensors B and C can be calculated as

81 2.3 Tensor Algebra 63 B il k ¼ Bmn p C l j ¼ C n q u i u l u p u m u n u k ; u l u q u n u j ð2:107cþ Substituting Eq. (2.107c) into Eq. (2.107b) and using the contraction law, one obtains A k ij B mn p u i u l u p u m u n u k ¼ Cq n ¼ u l u q u n u j A p mq Bmn p Rearranging the terms of Eq. (2.107d), one obtains A k u i u p ij u m u k A mq p u q u l u j )8B mn p ; ul u n 6¼ 0 : A k u i ij u Applying the inner product by r u i components of A at r ¼ m and s ¼ p. u k u s u l u n u q u j u n Bmn p ¼ 0 u m u p u k ¼ A p mq u q u j ð2:107dþ ð2:107eþ to Eq. (2.107e), one obtains the barred A k ij δ m r δ s p ¼ A mq p u q u r u k u j u i u s ) A k ij ¼ A mq p u k u m u q u p u i u j ð2:107fþ Equation (2.107f) proves that A is a mixed third-order tensor of type (1, 2), cf. Eq. (2.107c) Symmetric Tensors Tensor T is called symmetric in the given basis if two covariant or contravariant indices of the tensor component can be interchanged without changing the tensor component value.

82 64 2 Tensor Analysis T ij ijij ¼ T ji jiji : symmetric in i and j T ij ¼ T ji : symmetric in i and j T ijk pq ¼ Tikj pq T ijk pq ¼ Tijk qp : symmetric in j and k : symmetric in p and q ð2:108þ In case of a second-order tensor, the tensor T is symmetric if T equals its transpose. T ¼ T T ð2:109þ Skew-Symmetric Tensors The sign of the tensor component is opposite if a pair of the covariant or contravariant indices are interchanged with each other. In this case, the tensor is skew-symmetric (anti-symmetric). Tensor T is defined as a skew-symmetric tensor (anti-symmetric) if T ij ¼T ji : skew-symmetric in i and j T ij ¼T ji : skew-symmetric in i and j T ijk pq ¼ T ijk pq ¼ Tikj pq : skew- symmetric in j and k Tijk qp : skew-symmetric in p and q ð2:110þ In case of a second-order tensor, the tensor T is skew-symmetric if T is opposite to its transpose. T ¼T T ð2:111þ An arbitrary tensor T can be generally decomposed into the symmetric and skewsymmetric tensors: T ¼ 1 2 T þ TT 1 þ T TT Tsym þ T 2 skew ð2:112þ Proof The first tensor T sym is symmetric: T sym 1 2 T þ TT ¼ T T sym ðqedþ

83 2.4 Coordinate Transformations 65 The second tensor T skew is skew-symmetric: T skew 1 T TT ¼T T 2 skew ðqedþ 2.4 Coordinate Transformations Tensors are tuples of independent coordinates in a finite multifold N-dimensional tensor space (R N... R N ). The tensor describes physical states generally depending on different variables (dimensions). Each physical state can be defined as the point P(u 1,...,u N ) with N coordinates of u i. By changing the variables, such as time, locations, and physical characteristics (e.g. pressure, temperature, density, velocity), the physical state point varies in the multifold N-dimensional space. The tensor does not change itself and is invariant in any coordinate system. However, its components change in the new basis by the coordinate transformation since the basis changes as the coordinate system varies. In this case, applications of tensor analysis have been used to describe the transformation between two general curvilinear coordinate systems in the multifold N-dimensional spaces. Hence, tensors are a very useful tool applied to the coordinate transformations in the multifold N-dimensional tensor spaces. High-order tensors can be generated by a multilinear map between two multifold N-dimensional spaces (cf. Sect. 2.2). Their components change in the relating bases by the coordinate transformations, as displayed in Fig In the following section, the relations between the tensor components in different curvilinear coordinates of the finite N-dimensional spaces will be discussed Transformation in the Orthonormal Coordinates The simple coordinate transformation of rotation between the orthonormal coordinates x i and u j in Euclidean coordinate system is carried out. An arbitrary vector r (first-order tensor) can be written in both coordinate systems: r ¼ x 1 e 1 þ x 2 e 2 x i e i ¼ u 1 g 1 þ u 2 g 2 u j g j ð2:113þ

84 66 2 Tensor Analysis Basis{ g i, g } j Tensor components T ij Tensor T invariant in any basis Tensor components dependent on the basis i j T = Tijg g = i T g g ij j Basis{ g i, g } j Tensor components changed T ij T ij Fig. 2.5 Tensor and tensor components in different bases x 2 u 2 u 2 x 2 e 2 g 2 r P( x j ) 0 e 1 x 1 x 1 g 1 u 1 θ =θ 11 u 1 Fig. 2.6 Two-dimensional coordinate transformation of rotation The vector components in the coordinate u j can be calculated in (Fig. 2.6) u 1 ¼ cos θ 11 x 1 þ cos θ 12 x 2 ð2:114þ u 2 ¼ cos θ 21 x 1 þ cos θ 22 x 2 Thus, u 1 u 2 ¼ cos θ 11 cos θ 12 cos θ 21 cos θ 22 x 1 x 2, u ¼ Tx ð2:115þ

85 2.4 Coordinate Transformations 67 where T is the transformation matrix. Setting θ 11 ¼ θ, one obtains cos θ 11 ¼ cos θ; cos θ 12 ¼ cos θ þ π ¼sin θ 2 cos θ 21 ¼ cos θ π ¼ sin θ; cos θ 2 22 ¼ cos θ Therefore, the transformation matrix T becomes T ¼ cos θ 11 cos θ 12 cos θ sin θ ¼ cos θ 21 cos θ 22 sin θ cos θ The transformed coordinates can be computed by the transformation T: T : x! u ¼ Tx : u 1 ¼ u 2 cos θ sin θ sin θ cos θ x 1 x 2 where θ is the rotation angle of the rotating coordinates u j. Transforming backward Eq. (2.117), one obtains the coordinates x i T 1 : u! x ¼ T 1 u : x 1 ¼ x 2 cos θ sin θ sin θ cos θ u 1 u 2 ð2:116þ ð2:117þ ð2:118þ The vector component on the basis is obtained multiplying Eq. (2.113) by the relating basis e i or g j. x i ¼ r e i ; i ¼ 1, 2 u j ¼ r g j ; j ¼ 1, 2 ð2:119þ Substituting Eq. (2.119) into Eq. (2.117), one obtains the transformation matrix between two coordinate systems. g 1 ¼ g 2 cos θ sin θ sin θ cos θ e 1 e 2, g ¼ T e ð2:120þ Similarly, e 1 ¼ e 2 cos θ sin θ sin θ cos θ g 1 g 2, e ¼ T 1 g ð2:121þ

86 68 2 Tensor Analysis Transformation of Curvilinear Coordinates in E N In the following section, second-order tensors T are used in the transformation of general curvilinear coordinates in Euclidean space E N, as shown in Fig The basis g i of the curvilinear coordinate {u i } can be transformed into the new basis g i of the curvilinear coordinate { u i } using the linear transformation S. The new covariant basis can be rewritten as a linear combination of the old basis. S : g i! g i ¼ S j i g j, G ¼ GS ð2:122þ where S i j are the mixed transformation components of the second-order tensor S. The old covariant basis results can be calculated as g j ¼ r u j ¼ r u i u i u j g i S 1 i ) S 1 i ui j j u j ð2:123þ Inversing the basis matrix in Eq. (2.122), the new contravariant basis can be calculated as G 1 ¼ ðgsþ 1 ¼ S 1 G 1 ) g i ¼ S 1 i j g j ð2:124þ Multiplying Eq. (2.124) by the linear transformation S, the old contravariant basis results in G 1 ¼ SG 1 ) g i ¼ S i j gj ð2:125þ According to Eqs. (2.11) and (2.122), the new covariant basis can be calculated as g j ¼ r u j ¼ r u k u k u j g k S j k ) Sj k uk u j ð2:126aþ Combining Eqs. (2.123), (2.124) and (2.126a) and using the chain rule of differentiation, one obtains the relation of the mixed transformation components between two general curvilinear coordinates in Euclidean space E N. Fig. 2.7 Basis transformation of general curvilinear coordinates in E N N N T ( 2) N E T (2) E N { } g i S { } g i

87 2.4 Coordinate Transformations 69 g j g i ¼ g i g j ¼ S 1, SS 1 ¼ S 1 S ¼ I i l S j kgl g k ¼ S 1 i l S j kδ k l u k u k u j ¼ ui u j ¼ δ j i ¼ S 1 i k S j k ¼ ui ð2:126bþ Therefore, the transformation tensor S can be written as 2 u 1 u 1 u 1 3 u 1 u 2 : u N u 2 u 2 u 2 S ¼ u 1 u 2 : : : : u : N 2 R N R N ð2:127aþ u N u N u N 5 u 1 u 2 : u N The transformation tensor S in Eq. (2.127a) is identical to the Jacobian matrix between two coordinate systems {u i } and { u i }. Inverting the transformation tensor S, the back transformation results in 2 u 1 u 1 u 1 3 u 1 u 2 : u N S 1 u 2 u 2 u 2 ¼ u 1 u 2 : u : : : : N 2 R N R N ð2:127bþ u N u N u N 5 u 1 u 2 : u N The relation between the new and old components of an arbitrary vector v (firstorder tensor) is similarly given in the coordinate transformation S according to Eqs. (2.122) and (2.124). v i ¼ g i v ¼ S j i g j v jg j ¼ S j i v j v i ¼ g i v ¼ S 1 i j g j v j g j ¼ S 1 i j vj ð2:128þ Using Eq. (2.74), the relation of the components of the second-order tensor T can be derived. Covariant metric tensor components: T ij ¼ g i T g j ¼ Si kg k T S l j g l ¼ Si ks j l ðg k T g l Þ ¼ Si ks j lt ð2:129þ kl Contravariant metric tensor components:

88 70 2 Tensor Analysis T ij ¼ g i T g j ¼ S 1 ¼ S 1 i k S1 j l i k gk T S 1 j ¼ S 1 g k T g l l gl i k S1 j l Tkl ð2:130þ Mixed metric tensor components: T i j: ¼ g j T gi ¼ Sj kg k T S 1 i l gl ¼ Sj k S 1 i g l k T g l ¼ S k j S 1 i l T k: l ð2:131þ Note that in Eq. (2.131), the dot after the lower index indicates the position of the basis of the upper index locating after the tensor T. In this case, the tensor T is located between the upper basis g l and lower basis g k. :j ¼ gi T g j ¼ S 1 i k gk T Sj lg l ¼ S 1 i k S j l g k T g l ¼ S 1 i k S j lt :l k T i ð2:132þ Note that in Eq. (2.132), the dot before the lower index indicates the position of the basis of the upper index locating in front of the tensor T. In this case, the tensor T is located between the upper basis g k and lower basis g l Examples of Coordinate Transformations Cylindrical Coordinates The transformation S from Cartesian {u i } to cylindrical coordinates { u i }: 8 < u 1 ¼ r cos θ u 1 cos u 2 S : u 2 ¼ r sin θ u 1 sin u : 2 u 3 ¼ z ¼ u 3 The covariant transformation matrix S can be calculated as 2 u 1 u 1 u 1 3 u 1 u 2 u u S ¼ 2 u 2 u 2 cos θ r sin θ 0 sin θ r cos θ 0 A 6 4 u 1 u 2 u 3 7 u 3 u 3 u u 1 u 2 u 3 The determinant of S is called the Jacobian J.

89 2.4 Coordinate Transformations 71 jsj ¼ u 1 u 1 u 1 u 1 u 2 u 3 u 2 u 2 u 2 cos θ r sin θ 0 J ¼ sin θ r cos θ 0 u 1 u 2 u 3 u 3 u 3 u ¼ r u 1 u 2 u 3 The contravariant transformation matrix S 1 results from the inversion of the covariant matrix S. 2 u 1 u 1 u 1 3 u 1 u 2 u S 1 u ¼ 2 u 2 u 2 ¼ 1 r cos θ r sin θ sin θ cos θ 0 A 6 4 u 1 u 2 u 3 7 r u 3 u 3 u r u 1 u 2 u Spherical Coordinates The transformation S from Cartesian {u i } to spherical coordinates { u i }: 8 < u 1 ¼ ρ sin φ cos θ u 1 sin u 2 cos u 3 S : u 2 ¼ ρ sin φ sin θ u 1 sin u 2 cos u : 3 u 3 ¼ ρ cos φ ¼ u 1 cos u 2 The covariant transformation matrix S can be calculated as 2 u 1 u 1 u 1 3 u 1 u 2 u u S ¼ 2 u 2 u 2 sin φ cos θ ρcos φ cos θ ρ sin φ sin θ sin φ sin θ ρcos φ sin θ ρsin φ cos θ A 6 4 u 1 u 2 u 3 7 u 3 u 3 u 3 5 cos φ ρ sin φ 0 u 1 u 2 u 3 The determinant of S is called the Jacobian J.

90 72 2 Tensor Analysis u 1 u 1 u 1 u 1 u 2 u 3 u jsj ¼ 2 u 2 u 2 J u 1 u 2 u 3 u 3 u 3 u 3 u 1 u 2 u 3 sin φ cos θ ρcos φ cos θ ρ sin φ sin θ ¼ sin φ sin θ ρcos φ sin θ ρsin φ cos θ cos φ ρ sin φ 0 ¼ ρ2 sin φ The contravariant transformation matrix results from the inversion of the matrix S. 2 u 1 u 1 u u 1 u 2 u 3 ρ sin φ cos θ ρsin φ sin θ ρcos φ S 1 u ¼ 2 u 2 u 2 ¼ 1 cos φ cos θ cos φ sin θ sin φ B 6 4 u 1 u 2 u 3 7 ρ u 3 u 3 u 3 5 sin θ cos θ A 0 sin φ sin φ u 1 u 2 u Transformation of Curvilinear Coordinates in R N In the following section, second-order tensors T will be used in the transformation of general curvilinear coordinates in Riemannian manifold R N, as shown in Fig In Riemannian manifold, the bases g i and g i of the curvilinear coordinates u i and u i do not exist any longer. Instead of the metric coefficients, the transformation coefficients that depend on the relating coordinates have been used in Riemannian manifold [3]. The new barred curvilinear coordinate u i is a function of the old curvilinear coordinates u j, j ¼ 1,2,...,N. Therefore, it can be written in a linear function of u j. u i ¼ u i ðu 1 ;...; u N Þ ) u i ¼ aj iuj for j ¼ 1, 2,..., N ) du i ¼ aj iduj for j ¼ 1, 2,..., N ð2:133þ where ā i j is the transformation coefficient in the coordinate transformation S. Using the chain rule of differentiation, one obtains

91 2.4 Coordinate Transformations 73 Fig. 2.8 Basis transformation of general curvilinear coordinates in R N N N T ( 2) N R T (2) R N { } a i S { } a i du i ¼ ui u j duj ¼ a i j duj ) a i j ¼ ui u j ð2:134þ Analogously, the transformation coefficients of the back transformation result in du i ¼ ui u j duj ¼ a i j duj ) aj i ¼ ui u j ð2:135þ Combining Eqs. (2.134) and (2.135) and using the Kronecker delta, one obtains the relation between the transformation coefficients aj ia j k ¼ ui u j u j u k ¼ ui u k ¼ δ k i, aj i a j k ¼ a j k aj i ¼ I ð2:136þ The relation of the second-order tensor components between the new and old curvilinear coordinates can be calculated using Eq. (2.133). T ¼ T kl u k u l ¼ T kl ai ka j l u i u j ð2:137þ T ij u i u j Thus, T ij ¼ ai ka j lt kl ¼ uk u l u i u j T kl; T ij ¼ ai ka j lt kl ¼ uk u l u i u j T kl ð2:138þ In the same way, the covariant, contravariant, and mixed components of the secondorder tensor T between both coordinates in the transformation can be derived as:

92 74 2 Tensor Analysis Covariant tensor components: T ij ¼ a k i a l j T kl, T ij ¼ a k i a l j T kl ð2:139þ Contravariant tensor components: T ij ¼ a i k a j l Tkl, T ij ¼ a i k a j l Tkl ð2:140þ Mixed tensor components: T j i: ¼ a i k l T l k:, T j i: ¼ a k i a j l T k: l ð2:141þ T:j i k a j l :l, T i :j ¼ a k i a l j T :l k ð2:142þ Generally, the transformation coefficients of high-order tensors can be alternatively computed as ij ¼ uk u l u m u s u t u p u q u r u i u j T pqr st ¼ ap ka q la r m ai sa j t T pqr st T klm ð2:143þ Therefore, ij ¼ uk u l u m u s u t u p u q u r u i u j T pqr st ¼ ap ka q la r m ai sa j t T pqr st T klm ð2:144þ 2.5 Tensor Calculus in General Curvilinear Coordinates In the following sections, some necessary symbols, such as the Christoffel symbols, Riemann-Christoffel tensor, and fundamental invariants of the Nabla operator have to be taken into account in the tensor applications to fluid mechanics and other working areas Physical Component of Tensors Various types of the second-order tensors are shown in Eq. (2.74). The physical tensor component is defined as the tensor component on its covariant unitary basis

93 2.5 Tensor Calculus in General Curvilinear Coordinates 75 g i *. Therefore, the basis of the general curvilinear coordinates must be normalized (cf. Appendix B). Dividing the covariant basis by its vector length, the covariant unitary basis (covariant normalized basis) results in g * i ¼ g i jg i j ¼ p g i ffiffiffiffiffiffiffi g ðiiþ ) g * i ¼ 1 The covariant basis norm g i can be considered as a scale factor h i. h i ¼ jg i j ¼ p ffiffiffiffiffiffiffi ðno summation over iþ g ðiiþ ð2:145þ Thus, the covariant basis can be related to its covariant unitary basis by the relation of g i ¼ pffiffiffiffiffiffiffi g * ¼ h i i g * i ð2:146þ g ðiiþ The contravariant basis can be related to its covariant unitary basis using Eqs. (2.47) and (2.146). g i ¼ g ij g j ¼ g ij h j g * j ð2:147þ The contravariant second-order tensor can be written in the covariant unitary bases using Eq. (2.146). T ¼ T ij g i g j ¼ T ij h i h j g * i g * j T *ij g * i g* j ð2:148þ Thus, the physical contravariant tensor components result in T *ij T ij h i h j ð2:149þ The covariant second-order tensor can be written in the contravariant unitary bases using Eq. (2.147). T ¼ T ij g i g j ¼ T ij g ik g jl h k h l g * k g * l T * ij g* k g* l ð2:150þ Similarly, the physical covariant tensor component results in T * ij ¼ T ijg ik g jl h k h l ð2:151þ The mixed tensors can be written in the covariant unitary bases using Eqs. (2.146) and (2.147)

94 76 2 Tensor Analysis T ¼ Tj ig i g j ¼ Tj ig i g jk g k ¼ Tj i h i g * i g jk h k g * k ¼ T i j g jk h i h k g * i g* k *g Tj i * i g * k Thus, the physical mixed tensor component results from Eq. (2.152): * Tj i ¼ T i j g jk h i h k ð2:152þ ð2:153þ Derivatives of Covariant Bases Let g i be a covariant basis in the curvilinear coordinates {u i }. The derivative of the covariant basis with respect to the time variable t can be computed as _g i ¼ g i t ¼ r t u i _r,i ð2:154þ Due to u i is a differentiable function of t, Eq. (2.154) can be rewritten as _g i ¼ g i t ¼ g i u j u j t g i, j _u j ð2:155þ where g i,j is called the derivative of the covariant basis g i of the curvilinear coordinates {u i }. Using the chain rule of differentiation, the covariant basis of the curvilinear coordinates {u i } can be calculated in Cartesian coordinates {x i }. g i ¼ r u i ¼ r x p x p u i ¼ e px p,i ð2:156þ Similarly, one obtains the covariant basis of the coordinates {x i }. e p ¼ r x p ¼ r u k u k x p ¼ g k u k, p ð2:157þ The derivative of the covariant basis of the coordinates {u i } can be obtained from Eqs. (2.156) and (2.157).

95 2.5 Tensor Calculus in General Curvilinear Coordinates 77 g i,j ¼ g i u j ¼ e px p,i x p, i u j ¼ e p u j! x p ¼ u,p k,i u j g k ¼ uk 2 x p x p u i u j Γ k ij g k for k ¼ 1, 2,..., N g k ð2:158þ The symbol Γ k ij in Eq. (2.158) is defined as the second-kind Christoffel symbol, which has 27 (¼3 3 ) components for a three-dimensional space (N ¼ 3). Thus, the second-order Christoffel symbols that only depend on both coordinates of {u i } and {x i } can be written as ij ¼ uk 2 x p x p u i u j ¼ uk 2 x p x p u j u i ¼ Γ ji k Γ k ð2:159þ The result of Eq. (2.159) proves that the second-kind Christoffel symbols are symmetric with respect to i and j. The second-kind Christoffel symbols are given by multiplying both sides of Eq. (2.158) by the contravariant basis g l. Γij k g k g l ¼ Γ k ij δk l ¼ gl g i, j ) Γij l ¼ gl g i,j ð2:160þ Substituting Eq. (2.158) into Eq. (2.155), one obtains the relation between the covariant basis time derivative and the Christoffel symbol. _g i ¼ g i,j _u j ¼ Γ k ij _u j g k ð2:161þ Furthermore, the covariant basis derivative can be calculated in Cartesian coordinate {x i } using Eq. (2.156). g i,j ¼ g i u j ¼ e px p, i x p, i u j ¼ e p u j ¼ e p x p, ij ð2:162þ According to Eq. (2.159), the second-kind Christoffel symbols can be rewritten as Γij k ¼ uk 2 x p x p u i u j ¼ uk 2 x p x p u j u i ¼ u k, p x p,ij ¼ u k,p x p,ji ð2:163þ

96 78 2 Tensor Analysis Christoffel Symbols of First and Second Kind According to Eq. (2.160), the second-kind Christoffel symbol can be defined as Γ k ij k ¼ g i j k 2 r u i u j ¼ 2 r gk u j u i ¼ g k g i,j ¼ g k g j,i ¼ Γji k ð2:164þ Equation (2.164) reconfirms the symmetric property of the Christoffel symbols with respect to the indices i and j. Obviously, the Christoffel symbols are coordinate dependent; therefore, they are not tensors. In order to compute the second-kind Christoffel symbols in the covariant metric coefficients, the derivative of g ij with respect to u k has to be taken into account. g ij ¼ g i g j ) g ij,k g ij u k ¼ g i g j ¼ g i,k g j þ g i g j,k,k ð2:165þ Using Eq. (2.158) at changing the index j into k; then, i into j, one obtains the following relations g i,k ¼ Γ p ik g p ; g j,k ¼ Γ p jk g p ð2:166þ Substituting Eq. (2.166) into Eq. (2.165), one obtains the derivative of g ij with respect to u k. g ij,k ¼ g i,k g j þ g j, k g i ¼ Γ p ik g p g j þ Γ p jk g p g i ¼ Γ p ik g pj þ Γ p jk g pi ð2:167þ Interchanging k with i in Eq. (2.167), one obtains g kj, i ¼ Γ p ki g pj þ Γ p ji g pk ð2:168þ Analogously, one reaches the relation interchanging k with j in Eq. (2.167). g ik, j ¼ Γ p ij g pk þ Γ p kj g pi ð2:169þ Combining Eqs. (2.167) (2.169), the Christoffel symbols can be written in the derivatives of the covariant metric coefficients. g pj Γ p ik ¼ 1 2 g ij,k þ g kj,i g ik,j ð2:170þ

97 2.5 Tensor Calculus in General Curvilinear Coordinates 79 Multiplying Eq. (2.170) by g qj, the Christoffel symbols result according to Eq. (2.50) in Γ p ik g pj gqj ¼ Γ p ik δ p q ) Γ q ik ¼ 1 ð2:171þ 2 gqj g ij, k þ g kj, i g ik, j Changing j into p, k into j, and q into k in Eq. (2.171), one obtains Γij k ¼ 1 2 g ip,j þ g jp,i g ij,p g kp Γ ijp g kp ð2:172þ Changing the index p into k, the first-kind Christoffel symbol Γ ijp in Eq. (2.172) that has 27 (¼3 3 ) components for a three-dimensional space (N ¼ 3) is defined as Γ ijk ½ij, k 1 2 g ik,j þ g jk,i g ij,k ¼ g pk Γ p ij for p ¼ 1, 2,..., N ð2:173aþ Other expressions of the Christoffel symbols can be found in some literature. Γ ijk ½ij, k p ¼ g pk i j g pk Γ p ij for p ¼ 1, 2,..., N ð2:173bþ Prove That the Christoffel Symbols Are Symmetric 1. The first-kind Christoffel symbol is symmetric with respect to i, j According to Eq. (2.173a), the first-kind Christoffel symbol can be written as Γ ijk ¼ 1 2 g ik,j þ g jk,i g ij,k Interchanging i with j in the equation, one obtains Γ jik ¼ 1 2 g jk,i þ g ik,j g ji,k ¼ 1 2 g ik,j þ g jk,i g ij,k ¼ Γ ijk ðq:e:d: Þ 2. The second-kind Christoffel symbol is symmetric with respect to i, j Using Eq. (2.172), the second-kind Christoffel symbol can be expressed as

98 80 2 Tensor Analysis Γ k ij ¼ gkp Γ ijp Due to the symmetry of the first-kind Christoffel symbol, the second-kind Christoffel symbol results in Γij k ¼ gkp Γ ijp ¼ g kp Γ jip ¼ Γji k ðq:e:d: Þ Examples of Computing the Christoffel Symbols Given a curvilinear coordinate{u i } with u 1 ¼ u;u 2 ¼ v;u 3 ¼ w in another coordinate {x i }, the relation between two coordinate systems can be written as 8 < x 1 ¼ uv x 2 ¼ w : x 3 ¼ u 2 v The covariant basis matrix G can be calculated from G ¼ ½g 1 g 2 g 3 ¼ Therefore, the covariant bases can be given in 8 < g 1 ¼ ðv, 0, 2uÞ g 2 ¼ ðu, 0, 1Þ : g 3 ¼ ð0, 1, 0Þ The determinant of G that equals the Jacobian J of jgj ¼ J ¼ 0 x 1 x 1 x 1 1 u 1 u 2 u x 2 x 2 x 2 v u 0 ¼ u 1 u 2 u 3 C x 3 x 3 x 3 A 2u 1 0 u 1 u 2 u 3 v u u 1 0 ¼ 2u2 þ v 6¼ 0 The contravariant basis matrix G 1 is the inverse matrix of the covariant basis matrix G.

99 2.5 Tensor Calculus in General Curvilinear Coordinates 81 Thus, 0 u 1 u 1 u g 1 x 1 x 2 x 3 G 1 ¼ 4 g 2 5 u ¼ 2 u 2 u 2 g 3 B x 1 x 2 x 3 u 3 u 3 u 3 A x 1 x 2 x ¼ u J 1 0 uj 1 4 2u 0 v 5 ¼ 4 2uJ 1 0 vj 1 J 0 ð2u 2 þ vþ G 1 1 ¼ ð2u 2 þ vþ ¼ 1 J 3 5 Testing: G 1 jgj ¼ J 1 J ¼ jj¼ I 1 ðq:e:d: Þ Thus, the contravariant bases result in 8 < g 1 ¼ J 1 ð1, 0, uþ g 2 ¼ J 1 ð2u, 0, vþ : g 3 ¼ J 1 ð0, 2u 2 þ v, 0Þ Some examples of the second-kind Christoffel symbols of 27 components can be computed from Eq. (2.160). Γij k ¼ g i,j gk ) Γ 1 11 ¼ g 1,1 g1 ¼ J 1 ð0 1 þ 0 0 þ 2 uþ ¼ 2uJ 1 Γ 1 12 ¼ g 1,2 g1 ¼ J 1 ð1 1 þ 0 0 þ 0 uþ ¼ J 1 Γ 1 13 ¼ g 1,3 g1 ¼ J 1 ð0 1 þ 0 0 þ 0 uþ ¼ 0... Γ 3 32 ¼ g 3,2 g3 ¼ J 1 ð0 0 þ 0 ð2u 2 þ vþþ0 0Þ ¼ 0 Γ 3 33 ¼ g 3,3 g3 ¼ J 1 ð0 0 þ 0 ð2u 2 þ vþþ0 0Þ ¼ 0 The first-kind Christoffel symbols containing 27 components in R 3 can be computed from Eq. (2.173a). Γ ijk ¼ g pk Γ p ij ¼ g p g k Γ p ij for p ¼ 1, 2, 3

100 82 2 Tensor Analysis Coordinate Transformations of the Christoffel Symbols The second-kind Christoffel symbols like tensor components strongly depend on the coordinates at the coordinate transformations. The curvilinear coordinates {u i } are transformed into the new barred curvilinear coordinates {u i }. Therefore, the old basis is also changed into the new basis. The second-kind Christoffel symbols can be written in the new basis of the barred coordinates {u i }. Γ k ij ¼ gk g i u j ¼ gk g i, j ð2:174þ Using the chain rule of differentiation, the basis of the coordinates {u i } can be calculated as g p ¼ r u p ¼ r u k u k u p ¼ g k u k u p ð2:175þ Multiplying Eq. (2.175) by the new contravariant basis of the coordinates {u i }, one obtains changing the indices m into k, and p into l. g m g p ¼ g m u k g k u p ¼ δ k m u k u p ¼ um u p ) g m g p g p ¼ g m g p g p ¼ g m δp p ¼ um ) g k ¼ uk u l gl u p gp ð2:176þ The new covariant basis of the coordinates {u i } can be calculated as g i ¼ r u i ¼ r u p u p u p u i ¼ g p u i ð2:177þ Thus, the new covariant basis derivative with respect to j of the coordinates {u i } results in g i, j ¼ g i u j ¼ u j u p u i g p ¼ 2 u p u j u i g p þ up g p u i u j ð2:178þ Substituting Eqs. (2.176) and (2.178) into Eq. (2.174), one obtains the second-kind Christoffel symbols in the new basis.

101 2.5 Tensor Calculus in General Curvilinear Coordinates 83 Γ k ij ¼ gk g i,j ¼ uk u l gl! 2 u p u j u i g p þ up g p u i u j ð2:179þ Using Eq. (2.166) and the chain rule of differentiation, the second term in the parentheses of Eq. (2.179) can be computed as g p u j ¼ g p u q u q u j ¼ uq u j g p,q ¼ uq u j ð2:180þ Γpq r g r Inserting Eq. (2.180) into Eq. (2.179), the transformed Christoffel symbols in the new barred coordinates can be calculated as Γ k ij ¼ gk g i, j ¼ uk u l ¼ uk u l ¼ uk u l 2 u p u j u i g l g p þ up 2 u p u j u i δ l p þ up u i u q u q u i u j Γ pq r! gl g r! u j Γ pq r δ r l! 2 u l u i u j þ up u i u q u j Γ l pq ð2:181þ Therefore, the transformed Christoffel symbols in the new barred coordinates {u i } result in Γ k ij ¼ Γ pq l u k u p u q u l u i u j þ uk 2 u l u l u i u j ð2:182þ Rearranging the terms in Eq. (2.181), the second derivatives of u l with respect to the new barred coordinates result in u l u k Γ k ij ¼! 2 u l u i u j þ up u q u i u j Γ pq l ) 2 u l u i u j ¼ ul u k Γ k up u q ij u i u j Γ pq l ð2:183þ Using Eq. (2.160), all Christoffel symbols in Cartesian coordinates {x i } vanish because the basis e i does not change in any coordinate x j.

102 84 2 Tensor Analysis Γ k ij ¼ e k e i,j ¼ e k e i x j ¼ 0 ð2:184þ Derivatives of Contravariant Bases Like Eq. (2.158), the derivative of the contravariant basis of the curvilinear coordinates {u i } with respect to u j can be defined as g i,j ¼ gi u j ^Γ i jk gk ð2:185þ where ^Γ i jk are the second-kind Christoffel symbols in the contravariant bases gk. In order to compute those Christoffel symbols, some calculating steps are carried out in the following section. The derivative of the product between the covariant and contravariant bases with respect to u j can be computed using Eqs. (2.156), (2.164) and (2.185). g i g ¼ g i j, k g j þ,k gi g j,k ¼ δj i,k ¼ ^Γ i kl g l g j þ Γjk l ð g l giþ ¼ ^Γ i kl δ j l þ Γ jk l δ l i ð2:186þ ¼ ^Γ i kj þ Γ jk i ¼ ^Γ i kj þ Γ kj i ¼ 0 Thus, the relation between the Christoffel symbols of two coordinates results in ^Γ i kj ¼Γ i kj ¼Γ i jk ð2:187þ Using Eqs. (2.164) and (2.187), one obtains ^Γ i kj ¼Γ i kj ¼Γ i jk ¼ ^Γ i jk ð2:188þ It proves that the Christoffel symbol ^Γ i jk is symmetric with respect to j and k. Finally, the derivatives of the contravariant basis g i with respect to u j result from Eqs. (2.185) and (2.187).

103 2.5 Tensor Calculus in General Curvilinear Coordinates 85 g i,j ¼ ^Γ i jk gk ¼Γ i jk gk ¼ ^Γ i kj gk ¼Γ i kj gk ð2:189þ Derivatives of Covariant Metric Coefficients The derivatives of the covariant metric coefficient can be derived from the first-kind Christoffel symbols written as Γ ikj ¼ 1 2 g ij, k þ g kj, i g ik, j ; Γ jki ¼ 1 2 g ji,k þ g ki,j g jk,i ð2:190þ The derivative of the covariant metric coefficient results by adding both Christoffel symbols given in Eq. (2.190). Γ ikj þγ jki ¼ 1 2 g ij,k þg kj,i g ik,j þ 1 2 g ji,k þg ki,j g jk,i!! ¼ 1 2 ¼ g ij,k g ij,k þ g kj,i ::::::::: g ik,j þ 1 2 g ij,k þ g ik,j g kj,i ::::::::: ð2:191þ Therefore, the derivatives of the covariant metric coefficient g ij with respect to u k can be expressed in the first-kind Christoffel symbols. g ij,k g ij u k ¼ Γ ikj þ Γ jki ð2:192þ Using Eq. (2.173a), Eq. (2.192) can be rewritten in the second-kind Christoffel symbols. g ij, k ¼ Γ ikj þ Γ jki ¼ g jl Γ l ik þ g il Γ l jk ð2:193þ Similar to Eq. (2.192), one can write g jk,i ¼ Γ jik þ Γ kij ð2:194þ Subtracting Eq. (2.192) from Eq. (2.194), one obtains the relation

104 86 2 Tensor Analysis g ij, k g jk,i ¼ Γ ikj þγ jki Γ jik Γ kij ¼ Γ jki Γ jik ¼ Γ kji Γ ijk ð2:195þ Covariant Derivatives of Tensors Contravariant First-Order Tensors with Components T i The contravariant first-order tensor (vector) T can be written in the covariant basis. T ¼ T i g i ð2:196þ Using Eq. (2.158), the derivative of the contravariant tensor T with respect to u j results in T, j ¼ T i g i, j ¼ T,j i g i þ Ti g i,j ¼ T, i j g i þ Ti Γij kg ð2:197þ k The derivative of the contravariant tensor component T i with respect to u j in Eq. (2.197) can be defined as T,j i Ti u j ð2:198þ Interchanging i with k in the second term on the RHS of Eq. (2.197), one obtains T i Γ k ij g k ¼ Tk Γ i kj g i ¼ T k Γ i jk g i ð2:199þ Substituting Eq. (2.199) into Eq. (2.197), one obtains the derivative of T with respect to u j. T, j ¼ T, i j g i þ Tk Γ jk i g i ¼ T, i j þ Γ jk i Tk g i T i j gi ð2:200þ Therefore, the covariant derivative with respect to u j of the contravariant first-order tensor (vector) can be written as

105 2.5 Tensor Calculus in General Curvilinear Coordinates 87 T i j ¼ T i, j þ Γjk i Tk ¼ T,j g i ð2:201þ The covariant derivative of the contravariant first-order tensor component is transformed in the new barred coordinates {u i }. T i k ¼ T j u i u n n u j u k ) T jn ¼ T i u j u k k u i u n ð2:202aþ Proof The first-order tensor component can be written as T i ¼ T j ui u j ð2:202bþ Differentiating T i with respect to u k and using the chain rule of differentiation, one obtains T i u k ¼ ¼ Using Eq. (2.183), we have ui Tj u j Tj u n u n u k,k! u i u j þ Tj! 2 u i u n u j u n u k ð2:202cþ 2 u i u n u j ¼ Γ k u i nj u k Γ pq i u p u q u n u j ð2:202dþ Substituting Eq. (2.202d) into Eq. (2.202c) and interchanging the indices k with j and j with m, one obtains Thus, T i u k ¼! Tj u n u i u n u k u j þ Tj Γ k nj T i u k þ Tj Γpq i u p u q u n u n u j u k ¼ Tj u i u n u n u j u k þ Tj Γ k nj ¼ Tj u n u i u n u j u k The terms on the RHS of Eq. (2.202e) can be written as u i u k Γ pq i u p u q u n u n u j u k u i u n u k u k þ T m Γ j u i u n ð2:202eþ nm u j u k

106 88 2 Tensor Analysis T j u n u i u n u j u k þ T m Γ j nm u i u n u j u k h i ¼ T j,n þ Tm Γ j u i u n nm u j u k ¼ T j u i u n n u j u k ð2:202fþ Using Eq. (2.202b) and interchanging the indices p with k and q with m, the terms on the LHS of Eq. (2.202e) are rearranged in T i u k þ Tj Γ i pq u p u q u n u n u j u k ¼ T,k i uj þ Tm u m Γpq i u p u q u n u n u j u k ¼ T, i k þ Tm Γkm i u k u m u n u j ð2:202gþ u n u j u k u m ¼ T,k i þ Tm Γ km i T i k Substituting Eqs. (2.202f) and (2.202g) into Eq. (2.202e), one obtains Eq. (2.202a). T i k ¼ T j u i u n n u j u k ) T jn ¼ T i u j u k k u i u n ðq:e:d: Þ Covariant First-Order Tensors with Components T i The covariant first-order tensor (vector) T can be written in the contravariant basis. T ¼ T i g i ð2:203þ Using Eq. (2.189), the partial derivative of the tensor T results in T,j ¼ ðt i g i Þ,j ¼ T i,j g i þ T i g, i j ¼ T i,j g i T i Γjk i gk ð2:204þ The partial derivative of the covariant tensor component T i with respect to u j in Eq. (2.204) can be defined as T i, j T i u j ð2:205þ Interchanging i with k in the second term on the RHS of Eq. (2.204), one obtains

107 2.5 Tensor Calculus in General Curvilinear Coordinates 89 T i Γ i jk gk T k Γ k ji gi ¼ T k Γ k ij gi ð2:206þ Substituting Eq. (2.206) into Eq. (2.204), one obtains the derivative of the firstorder tensor component (vector) T with respect to u j. T,j ¼ T i,j g i T k Γ ij k gi ¼ T i,j Γij kt k g i T i jg i ð2:207þ Therefore, the covariant derivative of the tensor component T i with respect to u j can be defined as T i j¼ Ti,j Γij k T k ¼ T,j g i ð2:208þ The covariant derivative of the first-order tensor component T i is transformed in the new barred coordinates {u i }, similarly to Eq. (2.202a), cf. [4, 7]. T kl j ¼ T i ui u j j u k u l ) T i j¼ Tk j uk u l l u i u j ð2:209þ Second-Order Tensors Second-order tensors can be written in different expressions with covariant and contravariant bases. T ¼ T ij g i g j ¼ T ij g i g j ¼ T j i gi g j ¼ T i j g i g j ð2:210þ Similarly, the covariant derivatives with respect to u k of the second-order tensor components of T can be calculated as, cf. [3, 4, 7] T ij j k ¼ T ij,k Γik mt mj Γjk mt im T ij jk ¼ T ij, k þ Γ km i Tmj þ Γ j km Tim Tj i jk ¼ Tj, i k þ Γ km i T j m Γjk mt m i T j i jk ¼ T j i, k Γ ik mt m j þ Γ j km T i m ð2:211aþ where T ij,k, T ij,k, and T i j,k are the partial derivatives with respect to u k of the covariant, contravariant, and mixed tensor components. Note that they are different to the covariant derivatives of the tensor components, as defined in Eq. (2.211a).

108 90 2 Tensor Analysis In the coordinate transformation from the curvilinear coordinates {u i } to the new barred curvilinear coordinates {u α }, the covariant derivative of the covariant second-order tensor with respect to u γ can be calculated using the chain rule of differentiation, similarly to Eq. (2.202a), cf. [4, 7]. T γ αβ ¼ T k u i u j u k ij u α u β u γ ð2:211bþ where the partial derivatives u α ;i are called the shift tensor between two coordinate systems. This relation in Eq. (2.211b) is the chain rule of the covariant derivatives of the second-order tensors in the coordinate transformation. Analogously, the covariant derivatives of the second-order tensors of different types in the new barred curvilinear coordinates are calculated using the shift tensors. T γ αβ ¼ T k u i u j u k ij u α u β u γ ; T αβγ ¼ T ij u α u β u k k u i u j u γ ; T α β ¼ Tj i u α u j u k γ k u i u β u γ : ð2:211cþ Riemann-Christoffel Tensor The Riemann-Christoffel tensor is closely related to the Gaussian curvature of the surface in differential geometry that will be discussed in Chap. 3. At first, let us look into the second covariant derivative of an arbitrary first-order tensor. The covariant derivative of the tensor with respect to u j has been derived in Eq. (2.208). T i j¼ Ti,j Γij k T k ð2:212þ Obviously, the covariant derivative T i j is a second-order tensor component. Differentiating T i j with respect to u k, the covariant derivative of the secondorder tensor (component) T i j is the second covariant derivative of an arbitrary firstorder tensor (component) T i. This second covariant derivative has been given from Eq. (2.211a) [3].

109 2.5 Tensor Calculus in General Curvilinear Coordinates 91 T i jk T i j j k ¼ T i j, k Γik mt m j Γ m ¼ T i j, k Γ m Γ m j ik T m j j jk T im jk T im Equation (2.212) delivers the relations of T i j, k ¼ T i, jk Γij,k m T m þ Γij m T m, k Γik m T m j ¼ Γ m ik T m, j Γmj n T n Γjk m T im j ¼ Γjk m T i,m Γim n T n ð2:213þ ð2:214aþ ð2:214bþ ð2:214cþ Inserting Eqs. (2.214a) (2.214c) into Eq. (2.213), one obtains the second covariant derivative of T i. T i jk ¼ T i j, k Γik mt m j Γ m jk T im j ¼ T i,jk Γij, m k T m þ Γij mt m, k Γik m T m,j Γmj n T n Γjk m T i,m Γim n T n ¼ T i, jk Γij, m k T m Γij mt m, k Γik mt m,j þ Γik mγ mj n T n Γjk mt i,m þ Γjk mγ im n T n where the second partial derivative of T i is symmetric with respect to j and k: ð2:215þ T i, jk 2 T i u j u k ¼ 2 T i u k u j T i, kj Interchanging the indices j with k in Eq. (2.215), one obtains T i kj ¼ T i,kj Γ m ik,j T m Γ m ik T m, j Γ m ij T m, k þ Γ m ij Γ n mk T n Γ m kj T i, m þ Γ m kj Γ n im T n ð2:216þ ð2:217þ Using the symmetry properties given in Eqs. (2.164) and (2.216), Eq. (2.217) can be rewritten as T i kj ¼ T i,jk Γ m ik,j T m Γ m ik T m, j Γ m ij T m, k þ Γ m ij Γ n mk T n Γ m jk T i, m þ Γ m jk Γ n im T n ð2:218þ In a flat space, the second covariant derivatives in Eqs. (2.215) and (2.218) are identical. However, they are not equal in a curved space because of its surface curvature. The difference of both second covariant derivatives is proportional to the curvature tensor. Subtracting Eq. (2.215) from Eq. (2.218), the curvature tensor results in

110 92 2 Tensor Analysis T i jk T i kj ¼ Γik, n j Γ ij,k n þ Γ ik mγ mj n Γ m Rijk n T n ij Γ n mk T n ð2:219þ The Riemann-Christoffel tensor (Riemann curvature tensor) can be expressed as R n ijk Γ n ik,j Γ n ij, k þ Γ m ik Γ n mj Γ m ij Γ n mk ð2:220þ Straightforwardly, the Riemann-Christoffel tensor is a fourth-order tensor with respect to the indices of i, j, k, and n. They contain 81 (¼3 4 ) components in a three-dimensional space. In Eq. (2.220), the partial derivatives of the Christoffel symbols are defined by Γ n ik, j ¼ Γ n ik u j ; Γ n ij, k ¼ Γ n ij u k ð2:221þ Furthermore, the covariant Riemann curvature tensor of fourth order is defined by the Riemann-Christoffel tensor and covariant metric coefficients. R lijk ¼ g ln R n ijk, R n ijk ¼ gln R lijk ð2:222þ The Riemann curvature tensor has four following properties using the relation given in Eq. (2.222) [4]: First skew-symmetry with respect to l and i: R lijk ¼R iljk ð2:223þ Second skew-symmetry with respect to j and k: R lijk ¼R likj ; Rijk n ¼R ikj n ð2:224þ Block symmetry with respect to two pairs (l, i) and ( j, k): R lijk ¼ R jkli ð2:225þ Cyclic property in i, j, k: R lijk þ R ljki þ R lkij ¼ 0; Rijk n þ R jki n þ R kij n ¼ 0 ð2:226þ Equation (2.226) is called the Bianchi first identity.

111 2.5 Tensor Calculus in General Curvilinear Coordinates 93 Resulting from these properties, there are six components of R lijk in the threedimensional space as follows [3]: R lijk ¼ R 3131, R 3132, R 3232, R 1212, R 3112, R 3212 ð2:227þ In Cartesian coordinates, all Christoffel symbols equal zero according to Eq. (2.184). Therefore, the Riemann-Christoffel tensor, as given in Eq. (2.220) must be equal to zero. R n ijk ¼ Γ n ik,j Γ n ij, k þ Γ m ik Γ n mj Γ m ij Γ n mk ¼ 0 ð2:228þ Thus, the Riemann surface curvature tensor can be written as R lijk ¼ g ln R n ijk ¼ 0 ð2:229þ In this case, Euclidean N-space with orthonormal Cartesian coordinates is considered as a flat space because the Riemann curvature tensor there equals zero. In the following section, the Riemann curvature tensor can be calculated from the Christoffel symbols of first and second kinds. From Eq. (2.222) the Riemann curvature tensor can be rewritten as R hijk ¼ g hl R l ijk Using Eq. (2.220), the Riemann curvature tensor results in R hijk ¼ g hn R ijk n ¼ g hn Γik,j n Γ ij, n k þ Γ ik mγ mj n Γ m ¼ g hn Γ ik n g Γ n u j Γik n g hn, j hn ij u k ¼ g hn Γ ik n g Γ n u j Γik n g hn, j hn ij u k ij Γ n mk þ Γ n ij g hn,k þ g hn Γ m ik Γ n mj g hn Γ m ij Γ n mk þ Γ n ij g hn,k þ Γ m ik Γ mjh Γ m ij Γ mkh ð2:230aþ Changing the index m into n in both last terms on the RHS of Eq. (2.230a), one obtains

112 94 2 Tensor Analysis R hijk ¼ g hn Γ ik n u j ¼ g hn Γ ik n u j g Γ n Γik n g hn, j hn ij u k g hn Γ n ij u k Γ n ik þ Γ n ij g hn,k þ Γ n ik Γ njh Γ n ij Γ nkh g hn,j Γ njh þ Γij n g hn,k Γ nkh ð2:230bþ Using Eq. (2.192) for the first-kind Christoffel symbols in Eq. (2.230b), the Riemann curvature tensor becomes R hijk ¼ g hn Rijk n ¼ Γ ikh u j Γ ijh u k Γ hjn þ Γ njh Γ njh Γ n ik ¼ Γ ikh,j Γ ijh,k Γ n ik Γ hjn þ Γ n ij Γ hkn þ Γij n Γ hkn þ Γ nkh Γ nkh ð2:230cþ Ricci s Lemma The covariant derivative of the metric covariant coefficient g ij with respect to u k results from Eq. (2.211a) changing T ij into g ij. Then, using Eq. (2.193), one obtains g ij jk ¼ g ij u k g mj Γ ik m þ g in Γ jk n ¼ g ij,k g ij,k ¼ 0! ðq:e:d: Þ ð2:231aþ Therefore, g ij,k g ij u k ¼ g mj Γ m ik þ g in Γ n jk ð2:231bþ The Kronecker delta is the product of the covariant and contravariant metric coefficients: δ j l ¼ g li g ij The partial derivative with respect to u k of the Kronecker delta (invariant) equals zero and can be written as

113 2.5 Tensor Calculus in General Curvilinear Coordinates 95 δ j l, k ¼ ð g li gij Þ, k ¼ g li ¼ g li,k g ij þ g li g ij,k ¼ 0 g ij u k gij þ g li u k ð2:232aþ Multiplying Eq. (2.232a) byg lm, one obtains Interchanging the indices, it results in g ij g lm g li,k þ δ m i g ij,k ¼ 0 ) g mj,k ¼gij g lm g li,k ð2:232bþ g mj,k ¼gij g lm g li, k ) g ij, k ¼gmi g nj g mn,k ð2:232cþ Using Eq. (2.211a), the covariant derivative of the metric contravariant coefficient g ij with respect to u k can be written as g ij j k ¼ gij u k þ gmj Γ i km þ gim Γ j km ¼ g ij,k þ gmj Γ i km þ gim Γ j km ð2:233aþ Substituting Eqs. (2.231b), (2.232c) and (2.233a), one obtains after interchanging the indices g ij j k ¼ g ij,k þ gmj Γkm i þ gim Γ j km ¼g mi g nj g mn,k þ g mj Γkm i þ gim Γ j km ¼g mi Γ j mk gnj Γnk i þ gmj Γkm i þ gim Γ j ð2:233bþ km ¼g im Γ j km gmj Γkm i þ gmj Γkm i þ gim Γ j km ¼ 0 ðq:e:d: Þ Note that Eqs. (2.231a) and (2.233b) are known as Ricci s lemma Derivative of the Jacobian In the following section, the derivative of the Jacobian J that is always positive in a right-handed-rule coordinate system can be calculated and its result is very useful in the Nabla operator (cf. [4, 7]). The determinant of the metric coefficient tensor is given from Eq. (2.17):

114 96 2 Tensor Analysis det g ij ¼ g 11 g 12 : g 1N g 21 g 22 : g 2N : : : : ¼ g ¼ J 2 > 0 g N1 g N2 : g NN ð2:234þ The contravariant metric coefficient g ij results from the cofactor G ij of the covariant metric coefficient g ij und the determinant g. g ij ¼ Gij g ) G ij ¼ gg ij ð2:235þ Differentiating both sides of Eq. (2.234) with respect to u k, one obtains Prove Eq. (2.236): g 11 g 12 g g u k ¼ u k u k : 1N u g k 21 g 22 : g 2N : : : : g N1 g N2 : g NN ¼ g 11 u k G11 þ g 12 g u k ¼ g ij u k Gij for i, j ¼ 1, 2,..., N þ...þ g 11 g 12 : g 1N g 21 g 22 : g 2N : : : : g N1 u k u k G12 þ...þ...þ g NN u k ¼ g ij u k Gij for i, j ¼ 1, 2,..., N ðq:e:d: Þ g N2 u k : GNN g NN u k ð2:236þ ð2:237þ Substituting Eq. (2.235) into Eq. (2.236), it gives g u k ¼ g ij u k Gij ¼ g ij u k ggij g ij,k gg ij ð2:238þ Inserting Eq. (2.231b) into Eq. (2.238), one obtains

115 2.5 Tensor Calculus in General Curvilinear Coordinates 97 g u k ¼ g ij, k ggij ¼ g mj Γik m þ g in Γ jk n gg ij ¼ g δm i Γ ik m þ δ n jγ jk n ¼ g Γik i þ Γ j jk ¼ 2gΓik i ð2:239þ Using the chain rule of differentiation, the Christoffel symbol in Eq. (2.239) can be expressed in the Jacobian J > 0. Γik i ¼ 1 g 2g u k ¼ ln pffiffiffi g u k ¼ ðlnjþ u k ¼ 1 J J u k ð2:240þ Prove that R i ijk ¼ 0 Using Eq. (2.220) for n ¼ i, the Riemann-Christoffel tensor can be written as R i ijk ¼ Γ i ik,j Γ i ij, k þ Γ m ik Γ i mj Γ m ij Γ i mk Interchanging j with k in the last term on the RHS of the equation, one obtains R i ijk ¼ Γ i ik, j Γ i ij,k þ Γ m ik Γ i mj Γ m ik Γ i mj ¼ Γ i ik, j Γ i ij,k Using Eq. (2.240), the above Riemann-Christoffel tensor can be rewritten as Rijk i ¼ Γ ik, i j Γ ij,k i p ¼ 2 ln ffiffiffi pffiffi g u j u k 2 ln g u k u j ¼ 0 ðq:e:d: Þ ð2:241þ Ricci Tensor Both Ricci and Einstein tensors are very useful mathematical tools in the relativity theory. Note that tensors using in the relativity fields have been mostly written in the abstract index notation defined by Penrose [8]. This index notation uses the indices to express the tensor types, rather than their covariant components in the basis {g i }. The first-kind Ricci tensor results from the index contraction of k and n for n ¼ k of the Riemann-Christoffel tensor, as given in Eq. (2.220).

116 98 2 Tensor Analysis R ij Rijk k ¼ Γ ik k u j Γ ij k u k Γ ij r Γ rk k þ Γ r ik Γ k rj ð2:242þ The second-kind Ricci tensor can be defined as R i j g ik R kj ¼ g ik Γkm m u j Γ m kj u m Γ m kj Γ n mn þ Γ m kn Γ n mj Using the Christoffel symbol in Eq. (2.240) Γ j ij ¼ p 1 ffiffiffi p ffiffiffi g g u i ¼ ln pffiffi g u i ¼ 1 J J u i ¼ ðlnjþ u i ; the first-kind Ricci tensor can be rewritten as p R ij ¼ 2 ln ffiffi g u i u j Γ ij k p u k Γ ij k ln ffiffi g u k þ Γik r Γ rj k p ¼ 2 ln ffiffi g u i u j Γ ij k p u k þ Γ ij k ln ffiffi! g u k þ Γik r Γ rj k p ¼ 2 ln ffiffi g u i u j 1 pffiffiffi Γ k p ij pffiffi g g u k þ Γ ij k ffiffi! g u k þ Γik r Γ rj k ¼ 2 ln ffiffi p g u i u j 1 pffiffiffi g Γ k ij pffiffi g u k þ Γik r Γ rj k Interchanging i with j in Eq. (2.245), one obtains p R ji ¼ 2 ln ffiffiffi g u j u i p ¼ 2 ln ffiffiffi g u i u j ¼ R ij p 1 ffiffi g p 1 ffiffi g pffiffiffi g Γ k ji u k þ Γjk r Γ ri k pffiffiffi g Γ k ij u k þ Γir k Γ kj r ð2:243þ ð2:244þ ð2:245þ ð2:246þ This result states that the first-kind Ricci tensor is symmetric with respect to i and j. Substituting Eq. (2.245) with k ¼ m, r ¼ n, and i ¼ k into Eq. (2.243), the secondkind Ricci tensor results in

117 2.5 Tensor Calculus in General Curvilinear Coordinates 99 R i j ¼ g ik R kj 0 ¼ g ik 2 p ln ffiffi u k u j p 1 ffiffiffi g pffiffi 1 g Γ m kj u m þ Γkm n Γ nj m A ð2:247þ The Ricci curvature R can be defined as R R i i ¼ g ij R ij Substituting Eq. (2.245) into Eq. (2.248), the Ricci curvature results in 0 R ¼ g ij 2 p ln ffiffiffi u i u j p 1 ffiffi g pffiffiffi 1 g Γ k ij u k þ Γik r Γ rj k A ð2:248þ ð2:249þ Einstein Tensor The Einstein tensor is defined by the second-kind Ricci tensor, Kronecker delta, and the Ricci curvature. G i j R i j 1 2 δ i j R ð2:250aþ The Einstein tensor is a mixed second-order tensor and can be written as G i j ¼ g ik G kj ð2:250bþ Using the tensor contraction rules, the covariant Einstein tensor results in G ij ¼ g ik Gj k ¼ g ik Rj k 1 2 δ j k R ¼ R ij 1 2 g ij R ¼ R ji 1 2 g ji R ð2:251þ ¼ G ji This result proves that the covariant Einstein tensor is symmetric due to the symmetry of the Ricci tensor. The Bianchi first identity in Eq. (2.226) gives

118 100 2 Tensor Analysis R lijk þ R ljki þ R lkij ¼ 0; Rijk n þ R jki n þ R kij n ¼ 0 ð2:252þ Differentiating covariantly Eq. (2.252) with respect to u m, u k, and u l and then multiplying it by the covariant metric coefficients g in, one obtains the Bianchi second identity, cf. [4, 7, 9, 10]. R n jkl þ Rjlm n þ Rjmk n g in ¼ 0 g in m k l ) R m ijkl þ R k ijlm þ R l ijmk ¼ 0 ð2:253þ Due to skew-symmetry of the covariant Riemann curvature tensors, as discussed in Eqs. (2.223) and (2.224), Eq. (2.253) can be rewritten as R ijkl m R ijml k R jimk l ¼ 0 ð2:254þ Multiplying Eq. (2.254) by g il g jk (cf. Sect ), one obtains and using the tensor contraction rules Thus, R m ijkl R k ijml R l jimk ¼ 0, g il g jk R m ijkl g il g jk R k ijml g il g jk R l jimk ¼ g jk R m jk g jk R k jm g il R l im ¼ Rj m Rm k k Rm l l ¼ Rj m 2Rm k k ¼ 0 Rm k k ¼ 1 2 Rj m ðl! kþ ð2:255þ Using Eq. (2.232a) and the symmetry of the Christoffel symbols, the covariant derivative of the Kronecker delta with respect to u k is equal to zero. δ i j ¼ δj,k i þ Γ km i δ j m Γ m k ¼ δj,k i þ Γ kj i Γ jk i ¼ 0 jk δ i m ð2:256þ Differentiating covariantly the Einstein tensor in Eq. (2.250a) with respect to u k and using Eq. (2.256), one obtains the covariant derivative

119 References 101 Gj i ¼ Rj i 1 k 2 δ j ir k ¼ Rj i 1 k 2 δ j i R þ δj i R k k ¼ Rj i 1 k 2 δ j i R k ð2:257þ Changing the index i into k in Eq. (2.257) and using Eq. (2.255), the divergence of the Einstein tensor equals zero. G k j ¼ Rj k 1 k k 2 δ j k R k ¼ Rj k ) Div G G ¼ G k j k 1 2 Rj j ¼ 0 k g j ¼ 0 ðq:e:d: Þ ð2:258þ This result is very important and has been often used in the general relativity theories and other relativity fields. References 1. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 2. Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York, NY (1982) 3. Klingbeil, E.: Tensorrechnung f ur Ingenieure (in German). B.I.-Wissenschafts-Verlag, Mannheim (1966) 4. Nayak, P.K.: Textbook of Tensor Calculus and Differential Geometry. PHI Learning, New Delhi (2012) 5. Itskov, M.: Tensor Algebra and Tensor Analysis for Engineers With Applications to Continuum Mechanics, 2nd edn. Springer, Berlin (2010) 6. Oeijord, N.K.: The Very Basics of Tensors. IUniverse, New York, NY (2005) 7. De, U.C., Shaikh, A., Sengupta, J.: Tensor Calculus, 2nd edn. Alpha Science, Oxford (2012) 8. Penrose, R.: The Road to Reality. Alfred A. Knopf, New York, NY (2005) 9. Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2000) 10. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. AMS, Oxford University Press, Rhode Island (1978) 11. Kay, D.C.: Tensor Calculus. Schaum s Outline Series. McGraw-Hill, New York, NY (2011) 12. McConnell, A.J.: Applications of Tensor Analysis. Dover Publications, New York, NY (1960) 13. Brannon, R.M.: Curvilinear Analysis in a Euclidean Space. University of New Mexico, Albuquerque, NM (2004)

120 Chapter 3 Elementary Differential Geometry 3.1 Introduction We consider an N-dimensional Riemannian manifold M and let g i be a basis at the point P i (u 1,...,u N ) and g j be another basis at the other point P j (u 1,...,u N ). Note that each such basis may only exist in a local neighborhood of the respective points, and not necessarily for the whole space. For each such point we may construct an embedded affine tangential manifold. The N-tuple of coordinates are invariant in any chosen basis; however, its components on the coordinates change as the coordinate system varies. Therefore, the relating components have to be taken into account by the coordinate transformations. 3.2 Arc Length and Surface in Curvilinear Coordinates Consider two points P(u 1,...,u N ) and Q(u 1,...,u N )ofann-tuple of the coordinates (u 1,...,u N ) in the parameterized curve C2 R N. The coordinates (u 1,...,u N ) can be assumed to be a function of the parameter λ that varies from P(λ 1 )toq(λ 2 ), as shown in Fig The arc length ds between the points P and Q results from ds 2 ¼ dr dr dλ dλ dλ where the derivative of the vector r(u,v) can be calculated by ð3:1þ Springer-Verlag Berlin Heidelberg 2017 H. Nguyen-Schäfer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI / _3 103

121 104 3 Elementary Differential Geometry Fig. 3.1 Arc length and surface on the surface (S) da C n dv v P g 2 g 1 ds Q du T (S) r( u( l), v( l)) u O dr dλ ¼ d g ð i ui Þ u i ¼ g dλ i λ g i _u i ðþ; λ 8i ¼ 1, 2 Substituting Eq. (3.2) into Eq. (3.1), one obtains the arc length PQ. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ εðg i _u Þ g j _v dλ ¼ ε g ij _u _v dλ ð3:2þ ð3:3þ where ε(¼1) is the functional indicator, which ensures that the square root always exists. Therefore, the arc length of PQ is given by integrating Eq. (3.3) from the parameter λ 1 to the parameter λ 2. s ¼ ðλ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε g ij _u ðþ_v λ ðþ λ dλ ð3:4þ λ 1 where the covariant metric coefficients g ij are defined by g ij ¼ g ji ¼ g i g j ¼ r r ð3:5þ ; i u, j v u v Thus, the metric coefficient tensor of the parameterized surface S is given in

122 3.2 Arc Length and Surface in Curvilinear Coordinates 105 M ¼ g ij ¼ g 11 g 12 g 21 g 22 ¼ r ur u r u r v E F ð3:6þ r v r u r v r v F G The area differential of the tangent plane T at the point P can be calculated by da ¼ jdu dvj ¼ jg 1 du g 2 dvj ¼ jg 1 g 2 jdudv ð3:7þ Using the Lagrange s identity, Eq. (3.7) becomes da ¼ jg 1 g 2 jdudv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ g 11 g 22 ðg 12 Þ 2 dudv ¼ det g ij dudv p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG F 2 dudv ð3:8þ Integrating Eq. (3.8), the area of the surface S results in A ¼ ðλ 2 ðλ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG F 2 dudv ð3:9þ λ 1 λ 1 In the following section, the circumference at the equator and surface area of a sphere with a radius R are calculated (see Fig. 3.2). The location vector of a given point P(u(λ),v(λ)) in the parameterized surface of the sphere (S) can be written as ðþ: S x 2 þ 0y 2 þ z 2 ¼ R 2 ) 1 R sin φ cos θ rðφ; θþ R sin φ sin θ A; u φ 2 ½0; π½; v θ 2 ½0, 2π½ R cos φ The covariant bases can be calculated in g u ¼ r φ ¼ g v ¼ r θ ¼ 1 R cos φ cos θ R cos φ sin θ A; R sin φ 1 R sin φ sin θ R sin φ cos θ A 0 ð3:10þ ð3:11þ Thus, the metric coefficient tensor results from Eq. (3.11).

123 106 3 Elementary Differential Geometry Fig. 3.2 Arc length and surface of a sphere (S) z R sinϕ (ϕ, θ) (u, v): u ϕ ; v θ A s P n g v C eq e 1 e 3 ϕ R e 0 2 θ g u y R cosϕ x (S) M ¼ g 11 g 12 g 21 g 22 E F ¼ R2 0 F G 0 R 2 sin 2 φ ð3:12þ The circumference at the equator is given at u φ ¼ π/2 and v θ (λ) ¼ λ. C eq ¼ ðλ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ij _u ðþ_v λ ðþ λ dλ ¼ ðλ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 11 _u 2 þ 2g 12 _u _v þ g 22 _v 2 dλ ð3:13þ λ 1 λ 1 where _u ¼ 0 ; _v ¼ 1. Therefore, C eq ¼ ðλ 2 λ 1 pffiffiffiffiffiffiffi 2π ð g 22 dλ ¼ R sin π dλ ¼ 2πR 2 ð3:14þ 0 The surface area of the sphere can be computed according to Eq. (3.9). A S ¼ ¼ ðλ 2 ðλ 2 λ 1 λ 1 ð2π ð π 0 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG F 2 du dv ¼ ð2π R 2 sin φ dφ dθ ¼ ð2π ð π pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG F 2 dφ dθ R 2 cos φ 0 π dθ ¼ 4πR 2 ð3:15þ

124 3.3 Unit Tangent and Normal Vector to Surface Unit Tangent and Normal Vector to Surface The unit tangent vectors to the parameterized curves u and v at the point P have the same direction as the covariant bases g 1 and g 2. Both tangent vectors generate the tangent plane T to the differentiable Riemannian surface (S) at the point P. The unit normal vector n is perpendicular to the tangent plane T at the point P, as shown in Fig The unit tangent vector t is defined, as given in Eq. (B.1a). t i ¼ g * i p g i ffiffiffiffiffiffiffi g ¼ 1 ffiffiffiffiffiffiffi r p ðiiþ g ðiiþ u i 8 t 1 ¼ g * 1 ¼ 1 r >< pffiffiffiffiffiffi g ) 11 u t 2 ¼ g * 2 ¼ 1 r >: pffiffiffiffiffiffi g 22 v ; 8i ¼ 1, 2 ; ð1 uþ ; ð2 vþ ð3:16þ The unit normal vector is perpendicular to the unit tangent vectors at the point P and can be written as n ¼ ðt 1 t 2 Þ ¼ g 1 g r 2 jg 1 g 2 j ¼ u r r v r u v ð3:17þ Using Eq. (3.8), the unit normal vector can be rewritten as n ¼ ðt 1 t 2 Þ ¼ g 1 g 2 jg 1 g 2 j ¼ r r u r v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u r v EG F det g 2 ij ð3:18þ in which the cross product of g 1 and g 2 can be calculated by e 1 e 2 e 3 g 1 g 2 ¼ r r r ¼ ε ijk u1 u2 u3 r v 1 r v 2 r v 3 r u i r v e k j ð3:19þ where ε ijk is the permutation symbol given in Eq. (A.5) in Appendix A. The unit normal vector to the differentiable spherical surface (S) at the point P in Fig. 3.2 can be computed from Eq. (3.11).

125 108 3 Elementary Differential Geometry g 1 ¼ r ¼ φ 0 1 R cos φ cos R cos φ sin θ R sin φ A; g 2 ¼ r θ ¼ 0 1 R sin φ sin R sin φ cos θ A 0 ð3:20þ Therefore, e 1 e 2 e 3 g 1 g 2 ¼ R cos φ cos θ R cos φ sin θ R sin φ R sin φ sin θ R sin φ cos θ R 2 sin 2 φ cos θ B ¼ R 2 sin 2 φ sin θ A R 2 sin φ cos φ ð3:21þ Thus, the unit normal vector results from Eqs. (3.12), (3.18), and (3.21) n ¼ g 1 g 2 1 R 2 sin 2 φ cos θ sin φ cos θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EG F 2 R 2 R 2 sin 2 φ sin θ A sin φ sin θ A sin φ R 2 sin φ cos φ cos φ ð3:22þ Straightforwardly, the unit normal vector depends on each point P(φ, θ) on the spherical surface (S) and has a vector length of The First Fundamental Form The first and second fundamental forms of surfaces are two important characteristics in differential geometry as they are used to measure arc lengths and areas of surfaces, to identify isometric surfaces, and to find the extrema of surfaces. The Gaussian and mean curvatures of surfaces are based on both fundamental forms. Initially, the first fundamental form is examined in the following section. Figure 3.3 displays the unit tangent vector t to the parameterized curve C at the point P in the differentiable surface (S). The unit normal vector n to the surface (S) at the point P is perpendicular to t and (S) at the point P. Both unit tangent and normal vectors generate a Frenet orthonormal frame {t, n, (n t)} in which three unit vectors are orthogonal to each other, as shown in Fig The curvature vector k to the curve C can be rewritten as a linear combination of the normal curvature vector k n and the geodesic curvature vector k g at the point P in the Frenet frame. k ¼ k n þ q k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g, κn C ¼ κ n n þ κ g ðn tþ ) κ ¼ κ 2 n ðþþκ2 λ g ðþ λ ð3:23þ

126 3.4 The First Fundamental Form 109 Fig. 3.3 Normal and geodesic curvatures of the surface S k = κnn n k = κn C C θ n P k g = κg ( n t) n t t 1 dr 2 II Q T r( u( λ), v( λ)) r + dr S O where κ is the curvature of the curve C at P; κ n is the normal curvature of the surface (S) atp in the direction t; κ g is the geodesic curvature of the surface (S) atp. The first fundamental form I of the surface (S) is defined by the arc length on the curve C in the surface (S). I ds 2 ¼ dr dr ¼ r u i du j r u j du j ð3:24þ The first term on the RHS of Eq. (3.24) can be rewritten in the parameterized coordinate u i (λ). r u i dui ¼ g i du i du i ðþ λ ¼ g i dλ ¼ g dλ i _u i ðþdλ λ ð3:25þ in which g i is the covariant basis of the curvilinear coordinate u i, as shown in Fig Inserting Eq. (3.25) into Eq. (3.24), the first fundamental form results in I ¼ g i g j _u i ðþ_u λ j ðþdλ λ 2 ¼ g ij du i du j ¼ g 11 du 2 þ 2g 12 dudv þ g 22 dv 2 ð3:26þ Using Eq. (3.6), the first fundamental form of the surface (S) can be rewritten as

127 110 3 Elementary Differential Geometry I ¼ Edu 2 þ 2Fdudv þ Gdv 2 Therefore, the arc length ds can be rewritten as p ds ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E _u 2 þ 2F _u _v þ G _v 2 dλ ð3:27aþ ð3:27bþ where E, F, and G are the covariant metric coefficients of the metric tensor M, as given in M ¼ g ij E F F G ¼ r u r u r u r v r v r u r v r v ð3:28þ 3.5 The Second Fundamental Form The second fundamental form II is defined as twice of the projection of the arc length vector dr on the unit normal vector n of the parameterized surface (S) at the point P, as demonstrated in Fig II dr n ð3:29þ 2 Using the Taylor s series for a vectorial function with two variables u and v, the differential of the arc length vector dr(u,v) can be written in the second order. dr ¼ r du þ r dv u v þ 1 2 r 2 u 2 du2 þ 2 2 r u v dudv þ 2 r v 2 dv2! þ Odr 3 r u du þ r v dv þ 1 2 r uudu 2 þ 2r uv dudv þ r vv dv 2 þ Odr 3 ð3:30þ Therefore, the second fundamental form can be computed as II 2ðr u ndu þ r v ndvþþ r uu ndu 2 þ 2r uv ndudv þ r vv ndv 2 ð3:31þ Due to the orthogonality of (r u, r v ) and n, one obtains the inner products r u n ¼ r v n ¼ 0 ð3:32þ where the covariant bases of the curvilinear coordinate (u,v) are shown in Fig. 3.1.

128 3.5 The Second Fundamental Form 111 r u ¼ r u ¼ g 1 ; r v ¼ r v ¼ g 2 ð3:33þ Substituting Eqs. (3.31) and (3.32), the second fundamental form results in II ¼ r uu ndu 2 þ 2r uv ndudv þ r vv ndv 2 Ldu 2 þ 2Mdudv þ Ndv 2 in which L, M, and N are the elements of the Hessian tensor [1, 2] L M H ¼ h ij ¼ r uu n r uv n M N r uv n r vv n ð3:34þ ð3:35aþ with n ¼ ðt 1 t 2 Þ ¼ r r u r v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u r v EG F det g 2 ij ð3:35bþ In case of the projection equals zero, the second fundamental form II is also equal to zero. It gives the quadratic equation of du according to Eq. (3.34). Ldu 2 þ 2Mdudv þ Ndv 2 ¼ 0 Resolving Eq. (3.36) for du, one obtains the solution 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 M M 2 LN du A dv L ð3:36þ ð3:37þ There are three cases for Eq. (3.37) with L 6¼ 0[3, 4]: (M 2 LN) > 0: two different solutions of du. The surface (S) cuts the tangent plane T with two lines that intersect each other at the point P (hyperbolic point); (M 2 LN) ¼ 0: two identical solutions of du. The surface (S) cuts the tangent plane T with one line that passes through the point P (parabolic point); (M 2 LN) < 0: no solution of du. The surface (S) does not cut the tangent plane T except at the point P (elliptic point). In another way, the second fundamental form II can be derived by the change rate of the differential of the arc length ds when the surface (S) moves along the unit normal vector n with a parameterized variable α according to [2]. The location vector of the point P can be written in the parameterized variable α.

129 112 3 Elementary Differential Geometry R u ðu; v; αþ ¼ rðu; vþαnðu; vþ ð3:38þ The second fundamental form II can be calculated at α ¼ 0 using Eq. (3.27a): II¼ ds: ðdsþ j ¼ 1 ðds 2 Þ j ¼ 1 I j α α¼0 α¼0 2 α 2 α α¼0 ¼ 1 2 α Edu2 þ 2Fdudv þ Gdv 2 j α¼0 ¼ 1 E j 2 α α¼0 du 2 þ F j α α¼0 dudv þ 1 G j 2 α α¼0 dv 2 ð3:39þ in which the first term on the RHS of Eq. (3.39) can be calculated as E ¼ R u ðu; v; αþr u ðu; v; αþ ¼ ðr u α n u Þðr u α n u Þ ¼ n 2 u α2 2r u n u α þ r 2 u ð3:40þ Thus, 1 E j 2 α α¼0 ¼ n 2 u α r un u j ¼r u n u ð3:41þ Further calculations deliver the second and third terms on the RHS of Eq. (3.39): α¼0 F j α α¼0 ¼r ð u n v þ r v n u Þ; ð3:42þ 1 G 2 α Using the orthogonality of r u and n, one obtains j α¼0 ¼r v n v ð3:43þ ðr u nþ ¼ r uu n þ r u n u ¼ 0 ) r uu n ¼r u n u u ð3:44þ Similarly, one obtains using the orthogonality of r u and n; r v and n ðr u nþ ¼ ð r v nþ ¼ 0 ) 2r uv n ¼r ð u n v þ r v n u Þ; v u ðr v nþ ¼ 0 ) r vv n ¼r v n v v ð3:45þ Substituting Eqs. (3.41) (3.45) into Eq. (3.39), the second fundamental form II can be written as

130 3.6 Gaussian and Mean Curvatures 113 II ¼ 1 E j 2 α α¼0 du 2 þ F j α α¼0 dudv þ 1 G j 2 α α¼0 dv 2 ¼r u n u du 2 ðr u n v þ r v n u Þdudv r v n v dv 2 ¼ r uu ndu 2 þ 2r uv ndudv þ r vv ndv 2 Ldu 2 þ 2Mdudv þ Ndv 2 where L, M, and N are the components of the Hessian tensor H ¼ L M M N ¼ ruu n r uv n r uv n r vv n ð3:46þ ð3:47þ 3.6 Gaussian and Mean Curvatures The Gaussian and mean curvatures are based on the principal normal curvatures κ 1 and κ 2 in the direction t 1 and t 2 of the surface (S) at a given point P, as shown in Fig The unit tangent vectors t 1 and t 2 and the unit normal vector n at the point P generate two principal curvature planes that are perpendicular to each other. The normal curvature κ 1 of the surface (S) in the principal direction t 1 at the point P is defined as the maximum normal curvature in the curvature plane P 1 ; the normal curvature κ 2 in the principal direction t 2 is the minimum normal curvature in the curvature plane P 2. The maximum and minimum normal curvatures κ 1 and κ 2 of the surface (S) at the point P are the eigenvalues of the corresponding eigenvectors t 1 and t 2 [1, 2, 4, 5]. These eigenvalues are given from the characteristic equation that can be derived from the first and second fundamental forms in Eqs. (3.27a) and (3.46). The Gaussian curvature of the surface (S) at the point P is defined by K ¼ κ 1 κ 2 ð3:48þ The mean curvature of the surface (S) at the point P is defined by H ¼ 1 ð 2 κ 1 þ κ 2 Þ ð3:49þ The covariant metric tensor related to the first fundamental form can be written as I ¼ Edu 2 þ 2Fdudv þ Gdv 2 ; M ¼ g ij E F ¼ r u r u r u r v F G r v r u r v r v ð3:50þ The Hessian tensor related to the second fundamental form can be written as

131 114 3 Elementary Differential Geometry Fig. 3.4 Gaussian and mean curvatures of the surface S principal curvature planes P 2 tangent plane unit normal vector P 1 unit tangent vectors r( u( l), v( l)) C 2 S O II ¼ Ldu 2 þ 2Mdudv þ Ndv 2 ; L M H ¼ h ij ¼ r uu n r uv n M N r uv n r vv n The characteristic equation of the principal curvatures results in detðh κmþ ¼ ðl κeþ ð M κf Þ ðm κfþ ðn κgþ ¼ 0 ð3:51þ ð3:52þ Therefore, ðl κeþ: ðn κgþðm κfþ 2 ¼ 0, EG F 2 κ 2 þ ðen 2MF þ LGÞκ þ LN M 2 ð3:53þ ¼ 0 The Gaussian curvature results from Eq. (3.53) LN M2 K ¼ κ 1 κ 2 ¼ EG F 2 ¼ det h ij ð3:54þ det g ij Note that the Gaussian curvature K at a point in the surface is the product of two principal curvatures at this point. According to Gauss s Theorema Egregium (remarkable theorem) in [1, 2, 4, 5], the Gaussian curvature depends only on the first fundamental form I.

132 3.6 Gaussian and Mean Curvatures 115 Similarly, the mean curvature results in H ¼ 1 ð 2 κ 1 þ κ 2 EN 2MF þ LG Þ ¼ ð Þ 2 EG F 2 ð3:55þ The maximum and minimum principal curvatures κ 1 and κ 2 of the surface S at the point P result from Eqs. (3.54) and (3.55). p κ 1 ¼ κ max ¼ H þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ph 2 K κ 2 ¼ κ min ¼ H ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2 ð3:56þ K In the following section, the Gaussian and mean curvatures of the rotational paraboloid surface (S) inr 3 are computed as follows: ðsþ : z ¼ p x 2 þ y 2 ; ht ðþ¼ ffi t; t > 0 The location vector of the curvilinear surface (S) can be written as rðt; hþ ¼ 0 1 ht ðþcos ht ðþsin ϕ A h 2 ðþ t The bases of the curvilinear coordinate (t, ϕ) result from Eq. (3.58) in r t ¼ r t ¼ 0 1 _hðþcos t _hðþsin t ϕ 2h _hðþ t A; r ϕ ¼ r ϕ ¼ 0 1 h sin h cos ϕ A 0 ð3:57þ ð3:58þ ð3:59þ Thus, the covariant metric tensor can be further computed as M ¼ r t r t r t r ϕ ¼ E F r ϕ r t r ϕ r ϕ F G h ¼ _ 2 1 þ 4h 2 ð3:60þ 0 0 h 2 The unit normal vector can be calculated as n ¼ g 1 g e 1 e 2 e 3 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ _h cos ϕ _h sin ϕ 2h _h EG F 2 h sin ϕ h cos ϕ _ hh 2 cos ϕ 2h cos ϕ B ¼ h h _ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 _hh 2 C 1 B sin ϕ A ¼ 2h sin ϕ A 1 þ 4h 2 1 þ 4h _hh 2 1 ð3:61þ

133 116 3 Elementary Differential Geometry The components of the Hessian tensor are calculated by differentiating Eq. (3.59) with respect to t and ϕ. r tt ¼ 2 r t 2 ¼ r ϕϕ ¼ 2 r ϕ 2 ¼ 0 1 h cos h sin ϕ A ; r tϕ ¼ 2 r 2 hh þ 2 _h 2 t ϕ ¼ 0 1 h cos h sin ϕ A _h sin _h cos ϕ A ; 0 ð3:62þ The Hessian tensor results from Eqs. (3.61) and (3.62) in H ¼ r tt n r tϕ n ¼ L M r ϕt n r ϕϕ n M N 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _h 2 ð3:63þ 0 1 þ 4h 2 0 h 2 Therefore, the Gaussian curvature can be calculated from Eqs. (3.60) and (3.63). 2 pffiffiffiffiffiffiffiffiffi 2 h _ 2 h 2 LN M2 K ¼ κ 1 κ 2 ¼ EG F 2 ¼ 1þ4h 2 _h 2 h 2 1 þ 4h ¼ 1 þ 4h 2 2 ¼ ð1 þ 4tÞ 2 > 0 ð3:64þ In case of (M 2 LN) < 0, no solution of du exists. Thus, the surface (S) does not cut the tangent plane T except at the point P that is called the elliptic point. Analogously, the mean curvature results from Eqs. (3.60) and (3.63) in H ¼ 1 2 ðκ 1 þ κ 2 Þ ¼ EN 2MF þ LG 2 EG F 2 2 _! h 2 h 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 4h 2 1 þ 4h 2 ¼ 2 _h 2 h 2 1 þ 4h 2 ¼ 21þ ð 2t Þ ð1 þ 4tÞ 3 2 ð3:65þ in which h 2 ¼ t.

134 3.7 Riemann Curvature Riemann Curvature The Riemann curvature (also Riemann curvature tensor) is closely related to the Gaussian curvature of the surface in differential geometry [1 3]. At first, let us look into the second covariant derivative of an arbitrary first-order tensor. The covariant derivative of the tensor with respect to u j was derived in Eq. (2.208). T i j¼ Ti,j Γij k T k ð3:66þ Obviously, the covariant derivative T i j is a second-order tensor component. Differentiating T i j with respect to u k, the covariant derivative of the secondorder tensor (component) T i j is the second covariant derivative of an arbitrary firstorder tensor (component) T i. This second covariant derivative has been given from Eq. (2.211a) [3]. T i jk T i j j k ¼ T i j, k Γik m T m j Γ m jk T im j ð3:67þ ¼ T i j, k Γ m Γ m j ik T m j jk T im Equation (3.66) delivers the relations of T i j, k ¼ T i, jk Γij,k m T m þ Γij m T m, k Γik m T m j ¼ Γ m ik T m, j Γmj n T n Γjk m T im j ¼ Γjk m T i,m Γim n T n ð3:68aþ ð3:68bþ ð3:68cþ Inserting Eqs. (3.68a) (3.68c) into Eq. (3.67), one obtains the second covariant derivative of T i. T i jk ¼ T i j, k Γik m T m j Γ m jk T im j ¼ T i,jk Γij,k m T m þ Γij m T m,k Γik m T m,j Γmj n T n Γjk m T i, m Γim n T n ¼ T i,jk Γij,k m T m Γij m T m, k Γ m ik T m,j þ Γ m ik Γ n mj T n Γ m jk T i,m þ Γ m jk Γ n im T n where the second partial derivative of T i is symmetric with respect to j and k: ð3:69þ T i, jk 2 T i u j u k ¼ 2 T i u k u j T i,kj ð3:70þ Interchanging the indices j with k in Eq. (3.69), one obtains

135 118 3 Elementary Differential Geometry T i kj ¼ T i,kj Γik,j m T m Γik m T m,j Γ m ij T m,k þ Γ m ij Γ n mk T n Γ m kj T i,m þ Γ m kj Γ n im T n ð3:71þ Using the symmetry properties given in Eq. (3.70), Eq. (3.71) can be rewritten as T i kj ¼ T i,jk Γik,j m T m Γik m T m, j Γij m T m,k þ Γij m Γ mk n T n Γjk m T i,m þ Γjk m Γ im n T ð3:72þ n In a flat space, the second covariant derivatives in Eqs. (3.69) and (3.72) are identical. On the contrary, they are not equal in a curved space because of its surface curvature. The difference of both second covariant derivatives is proportional to the curvature tensor. Subtracting Eq. (3.69) from Eq. (3.72), the curvature tensor results in T i jk T i kj ¼ Γik,j n Γ ij, n k þ Γ ik m Γ mj n Γ m R n ijk T n ij Γ n mk T n ð3:73þ Thus, the Riemann curvature (also Riemann-Christoffel tensor) can be expressed as R n ijk Γ n ik, j Γ n ij,k þ Γ m ik Γ n mj Γ m ij Γ n mk ð3:74þ It is straightforward that the Riemann-Christoffel tensor is a fourth-order tensor with respect to the indices of i, j, k, and n. They contain 81 (¼3 4 ) components in a three-dimensional space. In Eq. (3.74), the partial derivatives of the Christoffel symbols are defined by Γ n ik, j ¼ Γ n ik u j ; Γ n ij, k ¼ Γ n ij u k ð3:75þ According to Eq. (2.172), the second-kind Christoffel symbol is given Γij k ¼ 1 2 g ip,j þ g jp,i g ij,p g kp ð3:76þ Therefore, the Riemann curvature tensor in Eq. (3.74) only depends on the covariant and contravariant metric coefficients of the metric tensor M, as given in Eq. (3.28). Furthermore, the covariant Riemann curvature tensor of fourth order is defined by the Riemann-Christoffel tensor and covariant metric coefficients. R lijk g ln R n ijk, R n ijk ¼ gln R lijk ð3:77þ

136 3.7 Riemann Curvature 119 For a differentiable two-dimensional manifold of the curvilinear coordinates (u,v), the Bianchi first identity gives the relation between the Riemann curvature tensors R and Gaussian curvature K, cf. Eq. (3.117b). R lijk K g lj g ki g lk g ji ð3:78þ Equation (3.78) indicates that the Gaussian curvature K of the two-dimensional surface only depends on the metric coefficients of E, F, and G. Therefore, the Gaussian curvature is only a function of the first fundamental form I. This result was proved by Gauss Theorema Egregium [1, 4 6]. The Riemann curvature tensor has the following properties: First skew-symmetry with respect to l and i: R lijk ¼R iljk ð3:79þ Second skew-symmetry with respect to j and k: R lijk ¼R likj ; R n ijk ¼R n ikj ð3:80þ Block symmetry with respect to two pairs (l, i) and ( j, k): R lijk ¼ R jkli ð3:81þ Cyclic property in i, j, k: R lijk þ R ljki þ R lkij ¼ 0; Rijk n þ R jki n þ R kij n ¼ 0 ð3:82þ Resulting from these properties, there are six components of R lijk in the threedimensional space as follows [2]: R lijk ¼ R 3131, R 3132, R 3232, R 1212, R 3112, R 3212 ð3:83þ In Cartesian coordinates, all second-kind Christoffel symbols equal zero according to Eq. (2.184). Therefore, the Riemann-Christoffel tensor, as given in Eq. (3.74) must be equal to zero. R n ijk ¼ Γ n ik, j Γ n ij,k þ Γ m ik Γ n mj Γ m ij Γ n mk 0 ð3:84þ Therefore, the Riemann curvature tensor in Cartesian coordinates becomes

137 120 3 Elementary Differential Geometry R lijk g ln R n ijk ¼ 0 ð3:85þ 3.8 Gauss-Bonnet Theorem The Gauss-Bonnet theorem in differential geometry connects the Gaussian and geodesic curvatures of the surface to the surface topology by means of the Euler s characteristic. Figure 3.5 displays a differentiable Riemannian surface (S) surrounded by a closed boundary curve Γ. The Gaussian curvature vector K is perpendicular to the manifold surface at the point P lying in the curve C and has the direction of the unit normal vector n. The geodesic curvature vector k g has the amplitude of the geodesic curvature κ g ; its direction of (n t) is perpendicular to the unit normal and tangent vectors in the Frenet orthonormal frame. The Gauss-Bonnet theorem is formulated for a simple closed boundary curve Γ [1, 2, 4]. ðð þ KdA þ κ g dγ ¼ 2π ð3:86þ S Γ The compact curvilinear surface S is triangulated into a finite number of curvilinear triangles. Each triangle contains a point P on the surface. This procedure is called the surface triangulation where two neighboring curvilinear triangles have one common vertex and one common edge (see Fig. 3.6). Therefore, the integral of the geodesic curvature over all triangles on the compact curvilinear surface S equals zero [2]. þ κ g dγ ¼ 0 ð3:87þ Γ In case of a compact triangulated surface, the Gauss-Bonnet theorem can be written in Euler s characteristic χ of the compact triangulated surface S T [1, 2, 4]. ðð KdA ¼ 2πχðS T Þ ð3:88þ S T The Euler s characteristic of the compact triangulated surface can be defined by χðs T Þ V E þ F ð3:89þ

138 3.8 Gauss-Bonnet Theorem 121 Fig. 3.5 Gaussian and geodesic curvatures for a simple closed curve Γ K = Kn g = g k κ ( n t) da t S dγ C P Γ Fig. 3.6 Gaussian and geodesic curvatures for a compact triangulated surface F (face) V (vertex) K = Kn da E (edge) P ( S T ) in which V, E, and F are the numbers of vertices, edges, and faces of the considered compact triangulated surface, respectively. Substituting Eqs. (3.8), (3.54), and (3.89) into Eq. (3.88), the Gauss-Bonnet theorem can be written for a compact triangulated surface. ðð LN M 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dudv ¼ 2πðV E þ FÞ ð3:90þ EG F 2 S T

139 122 3 Elementary Differential Geometry 3.9 Gauss Derivative Equations Gauss derivative equations were derived from the second-kind Christoffel symbols and the basis g 3 that denotes the normal unit vector n (¼g 3 ) of the curvilinear surface with the covariant bases (g i, g j ) for i, j ¼ 1, 2 at any point P(u,v) in the surface. Note that all indices i, j, k in the curvature surface S vary from 1 to 2. The partial derivatives of the covariant basis g i with respect to u j can be written according to Eq. (2.166) as g i, j ¼ Γij k g k þ Γ3 ij g 3 for i, j, k ¼ 1, 2 ð3:91þ The Christoffel symbols in the normal direction n result from Eq. (2.164). Γ 3 ij ¼ n g i, j ¼ g 3 g i, j ð3:92þ Differentiating g i.g 3 and using the orthogonality of g i and g 3, the partial derivative of the covariant basis g i with respect to u j can be calculated as ðg i g 3 Þ, j ¼ g i, j g 3 þ g i g 3,j ¼ 0 ) g i,j ¼g 3 g i g 3, j ¼ g 3 g 3, j g i ð3:93þ Substituting Eq. (3.93) into Eq. (3.92), the Christoffel symbols in the normal direction n can be expressed as Γ 3 ij ¼ g 3 g i,j ¼g ð 3 g 3Þg 3,j g i ¼g 3, j g i ¼n j g i ð3:94aþ h ij ¼ h ji ¼ Γ 3 ji for i, j ¼ 1, 2 Thus, n j ¼h ij g i for i, j ¼ 1, 2 ð3:94bþ where h ij is the symmetric covariant components of the Hessian tensor H, as given in Eq. (3.104b). Inserting Eq. (3.94a) into Eq. (3.91), the covariant derivative of the basis g i with respect to u j results in g i,j ¼ Γ k ij g k þ h ijg 3 ¼ Γ k ij g k þ h ijn for i, j, k ¼ 1, 2 ð3:95þ This equation is called Gauss derivative equations in which the second-kind Christoffel symbol is defined as

140 3.10 Weingarten s Equations 123 Γij k ¼ 1 2 gkp g jp,i þ g pi,j g ij,p ð3:96þ 3.10 Weingarten s Equations The Weingarten s equations deal with the derivatives of the normal unit vector n (¼g 3 ) of the surface at the point P(u,v) in the curvilinear coordinates {u i }. The partial derivative of the normal unit vector n results from Eq. (3.91). n i g 3,i ¼ Γ3i k g k þ Γ3 3i g 3 for i, k ¼ 1, 2 ð3:97þ Differentiating g 3 g 3 ¼ 1 with respect to u i, one obtains ðg 3 g 3 Þ,i ¼ 2g 3 g 3,i ¼ 0 ) g 3 g 3,i ¼ 0 ð3:98þ Using Eqs. (3.92) and (3.98), one obtains Γ 3 ji ¼ g 3 g j,i ) Γ 3 3i ¼ g 3 g 3,i ¼ 0 for i ¼ 1, 2 ð3:99þ Inserting Eq. (3.99) into Eq. (3.97) and using Eq. (3.94b), it gives n i ¼ Γ3i k g k ¼h ikg k ¼ h ik g kj gj ¼h j i g j ) n u i n i ¼h j i g ð3:100þ j for i, j ¼ 1, 2 The mixed components h i j are calculated from the Hessian tensor H and metric tensor M, as shown in Eq. (3.105). h j i ¼ h ik g kj ¼ HM 1 ð3:101þ Using Eq. (3.101) for i, j ¼ u, v, the Weingarten s equations (3.100) can be written as FM LG FL EM n u ¼ EG F 2 r u þ EG F 2 r v, n ð3:102þ u ¼ FM LG r EG F 2 u þ FL EM r EG F 2 v

141 124 3 Elementary Differential Geometry FN GM FM EN n v ¼ EG F 2 r u þ EG F 2 r v, n v ¼ FN GM r EG F 2 u þ FM EN r EG F 2 v ð3:103þ where the covariant metric and Hessian tensors result from the coefficients of the first and second fundamental forms I and II. M ¼ g ij E F ¼ r u r u r u r v F G r v r u r v r v ð3:104aþ ) M 1 ¼ ðg ij 1 G F Þ ¼ EG F 2 F E L M H ¼ h ij ¼ r uu n r uv n ð3:104bþ M N r uv n r vv n Therefore, HM 1 ¼ h j i ¼ 1 ðfm LGÞ ðfl EMÞ EG F 2 ðfn GMÞ ðfm ENÞ ð3:105þ 3.11 Gauss-Codazzi Equations The Gauss-Codazzi equations are based on the Gauss derivative and Weingarten s equations. The Gauss derivative equation (3.95) can be written as g i,j ¼ Γij k g k þ Γ3 ij n ¼ Γij k g k þ K ijn for i, j, k ¼ 1, 2 ð3:106þ in which the symmetric covariant components K ij (¼h ij ) of the surface curvature tensor K are given in Eq. (3.94a). Γ 3 ij ¼ K ij 2 K ð HÞ ¼ K ji ¼ Γ 3 ji The covariant derivative of g i with respect to u j results from Eq. (3.106). g i j ¼ gi,j Γij k g k ¼ K ij n for i, j ¼ 1, 2 ð3:107þ ð3:108þ

142 3.11 Gauss-Codazzi Equations 125 The Weingarten s equation (3.100) can also be written in the mixed components of the surface curvature tensor K. n i g 3, i ¼K j i g j for i, j ¼ 1, 2 ð3:109þ in which the mixed components of the surface curvature tensor K is defined according to Eq. (3.101) as K j i ¼ K ik g kj 2 KM 1 for i, j, k ¼ 1, 2 ð3:110þ Differentiating Eq. (3.108) with respect to u k and using Eq. (3.109), the covariant second derivatives of the basis g i can be calculated as g i jk ¼ K ij, k n þ K ij n k ¼ K ij, k n K ij Kk l g l ð3:111þ Similarly, the covariant second derivatives of the basis g i with respect to u k and u j result from interchanging k with j in Eq. (3.111). g i kj ¼ K ik,j n þ K ik n j ¼ K ik,j n K ik K l j g l ð3:112þ The Riemann curvature tensor R results from the difference of the covariant second derivatives of Eqs. (3.111) and (3.112) according to Eq. (3.73). g i jk g i kj ¼ Rijk l g l ¼ K ij,k K ik,j n þ Kik K l j K ijk l k g l ð3:113þ Multiplying Eq. (3.113) by the normal unit vector n and using the orthogonality between g l and n, one obtains K ij,k K ik,j n n þ Kik Kj l K ijkk l g l n ¼ Rijk l g l n ) K ij,k K ik,j 1 þ Kik Kj l K ijkk l 0 ¼ Rijk l 0 Thus, K ij, k K ik,j ¼ 0 for i, j, k ¼ 1, 2 ð3:114þ Equation (3.114) is called the Codazzi s equation. Multiplying both sides of Eq. (3.113) byg m, using the Codazzi s equation, and employing the tensor contraction rules, one obtains

143 126 3 Elementary Differential Geometry R l ijk g l g m ¼ K ikkj l K ijkk l ) R m ijk g lm ¼ K ikk m j K ij K m k g l g m g lm Thus, the Riemann curvature tensors can be calculated as R lijk ¼ K ik K lj K ij K lk ð3:115þ Equation (3.115) is called the Gauss equation. As a result, both Eqs. (3.114) and (3.115) are defined as the Gauss-Codazzi equations. The Codazzi s equation (3.114) gives only two independent nontrivial terms [3, 6]: K ij, k ¼ K ik,j ) ðk 11,2 ¼ K 12,1 ; K 21,2 ¼ K 22,1 Þ ð3:116þ On the contrary, the Gauss equation delivers only one independent nontrivial term [3, 6]: R 1212 ¼ K 11 K 22 K 2 12 ð3:117aþ Therefore, the Gaussian or total curvature K in Eq. (3.54) can be rewritten as K ¼ det K j i ¼ K 11K 22 K 2 12 g 11 g 22 g 2 12 ¼ K 1 1 K2 2 K1 2 K2 1 ¼ R 1212 g ð3:117bþ in which K i j is the mixed components of the mixed tensor KM 1, as shown in Eq. (3.105); K ij is the covariant components of the surface curvature tensor K; R 1212 is the covariant component of the Riemann curvature tensor; g is the determinant of the covariant metric tensor (g ij ) Lie Derivatives The Lie derivatives (pronouncing/li:/) named after the Norwegian mathematician Sophus Lie ( ) are very useful geometrical tools in Lie algebras and Lie groups in differential geometry of curved manifolds. The Lie derivatives are based on vector fields that are tangent to the set of curves (also called congruence) of the curved manifold.

144 3.12 Lie Derivatives 127 Fig. 3.7 Vector field on the tangent space T p M of the manifold M j u (β) T p M Y d dβ P 11 f ( P 11 ) Δβ Δα f P 21 X P 12 d dα i u (α) f ( P 12 ) (A) (B) M Vector Fields in Riemannian Manifold The vector field tangent to the curve (A) parameterized by the geodesic parameter α can be written in the coordinate u i of the N-dimensional manifold M, as displayed in Fig The vector field lies on the tangent space T p M (tangent surface) that is tangent to the manifold M at the contact point P. As the contact point moves along the curve on the manifold M, a tangent bundle TM of the manifold M is generated. Therefore, the tangent bundle consists of all tangent spaces of the manifold M. Using Einstein summation convention (cf. Sect ), the vector field X on the tangent space T p M can be written with the basis vectors u i of the coordinates u i. X d dα ¼ dui dα u i ð3:118þ X i for i ¼ 1, 2,..., N ui where X i is the vector component in the coordinate u i. Similarly, the vector field tangent to the curve (B) parameterized by another geodesic parameter β can be written in the coordinate u j. Y d dβ ¼ duj dβ u j Y j for j ¼ 1, 2,..., N uj ð3:119þ where Y j is the vector component in the coordinate u j.

145 128 3 Elementary Differential Geometry Lie Bracket Let X and Y be the vector fields of the congruence in the N-dimensional manifold M and f be a mapping function of the coordinate u i in the curve. The commutator of a vector field is called the Lie bracket and can be defined by ½X; Y XYf ð ðþþyxf ð ðþþ ð3:120þ The first mapping operator on the RHS of Eq. (3.120) can be calculated using the chain rule of differentiation. XYf ð ðþþ ¼ X j Y i u j u i ¼ X j Yi u j u i þ Xj Y i 2 u i u j ð3:121þ Analogously, the second mapping operator on the RHS of Eq. (3.120) results in YXf ð ðþþ ¼ Y j X i u j u i ¼ Y j Xi u j u i þ Xi Y j 2 u i u j ð3:122þ Interchanging the indices i with j in the second term on the RHS of Eq. (3.122), one obtains YXf ð ðþþ ¼ Y j Xi u j u i þ Xj Y i 2 u j u i Subtracting Eq. (3.121) from Eq. (3.123), the Lie bracket is given. ½X; Y ¼ X j Yi Xi Yj uj u j u i ½X; Y i u i for i ¼ 1, 2,..., N ð3:123þ ð3:124þ Thus, the component i in the coordinate u i of the Lie bracket is defined by ½X; Y i X j Yi Xi Yj uj u j for i, j ¼ 1, 2,..., N ð3:125þ The vectors X and Y commute if its Lie bracket equals zero. ½X; Y ¼ 0 ð3:126þ According to Eq. (3.125), the Lie bracket is skew-symmetric (anti-symmetric).

146 3.12 Lie Derivatives 129 ½X; Y ¼ ½Y; X ¼ Y j Xi Yi Xj uj u j u i ð3:127þ The Lie bracket (commutator) of the vector field (X, Y) can be expressed in another way as ½X; Y ¼ X j u j Y i u i Y j X i u j u i ¼ X d dβ Y d ð3:128þ dα Therefore, the Lie bracket of the vector field can be written as ½X; Y d dα ; d dβ ¼ d d dαdβ d d dβdα ð3:129þ The Lie bracket of a vector field is generally not equal to zero in a curved manifold due to space torsions and Riemann surface curvatures that will be discussed later in Sect The Lie bracket has some cyclic permutation properties of the vector field (X, Y, and Z) in the N-dimensional manifold M. ½X; ½Y; Z ¼ XYZ XZY YZX þ ZYX ½Y; ½Z; X ¼ YZX YXZ ZXY þ XZY ½Z; ½X; Y ¼ ZXY ZYZ XYZ þ YXZ ð3:130þ The Jacobi identity written in the Lie brackets results from substituting the properties of Eq. (3.130): ½X; ½Y; Zþ½Y; ½Z; Xþ½Z; ½X; Y ¼ 0 ð3:131þ Lie Dragging Lie Dragging of a Function Let f be a function that maps a point P 11 to another point P 12 on the curve (A) bya geodesic parameter distance Δα. The function f(p 12 ) at the point P 12 is called the image of f(p 11 ) at the point P 11 carrying by the mapping function f at Δα on the same curve (A) 2 M (see Fig. 3.7). If the image function f(p 12 ) equals the function f(p 11 ), the function f is invariant under the mapping. Furthermore, the mapping function f can be defined as Lie

147 130 3 Elementary Differential Geometry dragged if f is invariant for any geodesic parameter distance Δα along any congruence on the manifold M. df ¼ 0, f is Lie dragged: dα ð3:132þ Lie Dragging of a Vector Field Let X and Y be vector fields on the tangent space T p M of the tangent bundle TM in an N-dimensional manifold M, as shown in Fig The congruence consists of α- and β-curves with the coordinates u i and u j. The tangent vector X to the curve (A 1 ) at the point P 11 is dragged to the curve (A 2 ) at the point P 21 by a geodesic parameter distance Δβ. The image vector X* of the original vector X is dragged by Δβ from (A 1 )to(a 2 ) and tangent to the curve (A 2 ) at the point P 21. Generally, both tangent vectors X and X* are different to each other under the Lie dragging by Δβ. However, if they are equal for every geodesic parameter distance Δβ, the Lie dragging is invariant. In this case, the vector field is called Lie dragged on the manifold M [7, 8]. Similarly, the tangent vector Y to the curve (B 1 ) at the point P 11 is dragged to the image vector Y* tangent the curve (B 2 ) at the point P 12 by a geodesic parameter distance Δα in the tangent bundle TM. Thus, the vector fields of X and Y along the congruence are generated on the manifold M. The vector field is defined as Lie dragged if its Lie bracket or the commutator given in Eq. (3.129) equals zero. ½X; Y d dα ; d dβ ¼ d d dαdβ d d dβdα ¼ 0 ð3:133þ ) d d dαdβ ¼ d d dβdα In this case, the vector field is commute under the Lie-dragged procedure on the manifold M. In general, the Lie bracket of a vector field is not always equal to zero due to space torsions besides the Riemann surface curvature on the manifold M Lie Derivatives The Lie derivatives of a function with respect to the vector field X are defined by the change rate of the function f between two different points P 11 and P 12 in the same curve under the Lie dragging by a geodesic parameter distance Δα.

148 3.12 Lie Derivatives 131 Fig. 3.8 Lie dragging vector fields on the manifold M β -curves (u j ) T p M TM ( A 1 ) Y ( B 1 ) d dβ P 11 ( A2) Δα Δβ X P 21 P 12 d dα * d X d α Δβ P 22 * d Y d β Δα M ( B ) 2 α -curves (u i ) f ðα 0 þ ΔαÞf ðα 0 Þ X f lim Δα!0 Δα dfðuiþ ¼ ¼ dui f dα dα u i ¼ X i f u i α0 α0 ð3:134þ Thus, the Lie derivative of a function f with respect to the vector field X can be simply expressed in X f ¼ X i u i f ¼ Xf ð3:135þ Analogously, the Lie derivative of a vector Y with respect to the vector field X results from the Lie bracket with the basis vectors u i of the coordinates u i [8, 9]. X Y ¼ ½X; Y ¼ ½X; Y i u i ¼ ð X YÞ i u i for i ¼ 1, 2,..., N ð3:136þ According to Eq. (3.125), the component i in the coordinate u i of the Lie derivative of the vector Y with respect to the vector field X can be calculated as ð X Y Þ i ¼ ½X; Y i ¼ X j u j Y i Y j u j X i ¼ d dα Yi d dβ Xi for i ¼ 1, 2,..., N ð3:137þ

149 132 3 Elementary Differential Geometry Using Eqs. (3.136) and (3.137) the Lie derivative of a vector field is skewsymmetric because of the skew-symmetry of the Lie bracket, as shown in Eq. (3.127). X Y ¼ ð X YÞ i u i ¼ ¼ Y X Yi Xi Xj Yj uj u j u i In the following section, the properties of the Lie derivatives are proved. ð3:138þ Lie Derivative of a Function Product Let f and g be functions. The Lie derivative of product of two functions is given as X ðfg Þ ¼ ð X f Þg þ f ð X gþ ð3:139þ Proof X ðfg Þ ¼ XðfgÞ ¼ X i ðfgþ u i ¼ X i f u i g þ f X i g u i ¼ ð X f Þg þ f ð X gþ ðq:e:d: Þ Lie Derivative of a Tensor Product Let S and T be tensor fields and X a vector field. The Lie derivative of tensor product is given as X ðs T Þ ¼ ð X SÞT þ S ð X TÞ ð3:140þ Proof X ðs TÞ ¼ ½X, S T ¼ ½X; ST þ S ½X; T ¼ ð X SÞT þ S ð X TÞ The Lie derivative of mixed tensors T with respect to a vector field X can be written in the Lie derivative components and their bases of the coordinates. X T ½X; T ¼ ð X T Þ i...k l...n i... k du l... du n

150 3.12 Lie Derivatives 133 where i,..., k and du l,..., du n are the covariant and contravariant bases of the local coordinates {u i }, respectively (cf. Sect. 3.14), is the tensor product, and T i l k n are the components of the mixed tensor T. The Lie derivative components of the tensor field T with respect to a vector field X are computed in [9]: ð X TÞ i...k l...n ¼ Xp T i...k l...n,p þ X p, l Ti...k p...n þ... X, k p Ti...p l...n... ð3:141þ in which the partial derivatives are defined as T i...k l...n, p Ti...k l...n u p ; X p Xp,l u l ; X,p k Xk u p Lie Derivative of a Differential Form Let dω be a differential k-form that is dual to its k-order covariant tensor field on a differentiable manifold M and X be a vector field. The Lie derivative of the differential k-form dω with respect to X is given as X ðdωþ ¼ d ð X ωþ ð3:142þ Proof At first, Eq. (3.142) is derived for a differential zero-form on a differentiable N- dimensional manifold M. The zero-form f is defined as a smooth function on the manifold M. Its differential zero-form df can be written in the dual bases du j using Einstein s summation convention. df ¼ f uj ð Þ u j du j for j ¼ 1, 2,..., N Using the chain rule of differentiation, the LHS of Eq. (3.142) for ω f can be calculated as f X ðdfþ ¼ X u j duj f ¼ X u j du j þ f u j X du j Interchanging the index i with j in the second term in Eq. (3.143a), one obtains

151 134 3 Elementary Differential Geometry X ðdfþ ¼ X i f u i u j du j þ f X j u j u i dui ¼ X i f u i u j du j þ f X i u i u j duj ¼ X i 2 f u i u j þ f u i X i u j! du j ð3:143aþ Using the chain rule of differentiation, the RHS of Eq. (3.142) for ω f is computed as d ð X f Þ ¼ dðxf Þ ¼ d X i f u i ¼ X i d f u i þ f u i dxi ¼ X i 2 f u i u j þ f u i X i u j! du j ð3:143bþ Subtracting Eq. (3.143a) from Eq. (3.143b), Eq. (3.142) is proved for a differential zero-form. X ðdfþd ð X f Þ ¼ 0 ) X ðdfþ ¼ d ð X f Þ This equation is called the Cartan s formula in the special case for a differential zero-form df. In the following step, Eq. (3.142) is derived for a differential one-form on a differentiable N-dimensional manifold M. The one-form ω on the manifold M can be generally written in terms of the dual bases du j using Einstein summation convention, cf. Eq. (3.179). Thus, the differential one-form results in ω ¼ ω j du j dω ¼ ω u j duj The LHS of Eq. (3.142) can be calculated as ω X ðdωþ ¼ X u j duj ω ¼ X u j du j þ ω u j X du j

152 3.12 Lie Derivatives 135 within X ðdu j Þ ¼ d ð X u j Þ ¼ d X i uj u i ¼ d X i δ j i ¼ dx j ¼ Xj u i dui for i ¼ 1, 2,..., N Further calculating the LHS of Eq. (3.142), one obtains interchanging i with j in the second term on the RHS of Eq. (3.143c) X ðdωþ ¼ X i u i ¼ X i u i ω u j du j þ ω X j u j u i dui ω u j du j þ ω X i u i u j duj! du j ¼ X i 2 ω u i u j þ ω u i X i u j ð3:143cþ Next, the RHS of Eq. (3.142) is computed using the chain rule of differentiation. d ð X ωþ ¼ d X i u iω ¼ X i d ω u i þ ω u i dxi ð3:143dþ! ¼ X i 2 ω u i u j þ ω X i u i u j du j Comparing Eqs. (3.143c) and (3.143d), Eq. (3.142) is proved for a differential one-form. X ðdω Þ ¼ d ð X ωþ Finally, Eq. (3.142) is derived for an arbitrary differential k-form (k > 0) on a differentiable N-dimensional manifold M using the Cartan s formula. Let C be a fiber bundle of the manifold M and a point p 2 C.Ak-form at the point p on C 2 M is an element of the cotangent bundle T * M M consisting of k cotangent spaces (cf. Sect. 3.14). Generally, the k-form on the manifold M is defined in local coordinates with the dual bases du i as

153 136 3 Elementary Differential Geometry ω ¼ XN i 1 <i 2 <<i k ω i1 i 2...i k u i du i1 ^ du i2 ^^du i k where ^ is the wedge (exterior) product; ω i are smooth functions of the coordinates u i. The Cartan s formula of a k-form ω (k > 0) is derived in [8]. X ω ¼ i X dω þ di ð X ωþ ð3:144aþ where the notation i X ω is called the interior product of the k-form ω with respect to the vector field X on the differentiable N-dimensional manifold M and defined as i X ω ðω; XÞ ¼ ωðxþ ¼ X j ω u j From the Cartan s formula, one obtains the interior product of the differential k- form. i X dω ¼ X ω di ð X ωþ ð3:144bþ Changing the k-form ω into its exterior derivative dω in the Cartan s formula in Eq. (3.144a), using Eq. (3.144b) and dd ¼ 0, Eq. (3.142) has been derived. X ðdω Þ ¼ i X ddω ð Þþdi ð X dωþ ¼ i X ddω ð Þþd ð X ω di ð X ωþþ ¼ i X ddω þ d ð X ωþddði X ωþ ¼ d ð X ωþ : q:e:d: Lie Derivative of a One-Form and Vector Product Let X and Y be vector fields, and ω a one-form. The Lie derivative of product between the one-form and vector is given as X ðωyþ ð X ωþy ¼ ω½x; Y ð3:145þ Proof X ðωyþ ¼ ½X, ωy ¼ ½X; ωy þ ω½x; Y ¼ ðxωþy þ ω½x; Y ¼ ð X ωþy þ ω ð X YÞ Therefore,

154 3.12 Lie Derivatives 137 X ðωyþ X ω Y ¼ ω½x; Y ðq:e:d: Þ Lie Derivative of a One-Form Let ω be a one-form and X a vector field. The Lie derivative of one-form is given as X ω ¼ ðxω i Þdu i þ ω i dx i ð3:146þ Proof From Eqs. (3.137) and (3.145) one obtains interchanging the index i with j. ð X ω Equation (3.147) gives Þ i Y i ¼ X ω i Y i ωi ð X YÞ i ¼ X j ω iy i u j ω i X j Yi Xi Yj uj u j ¼ X j Y i ω i u j þ Xj Y i ω i u j ω ix j Yi u j þ ω iy j Xi u j ¼ X j Y i ω i u j þ ω iy j Xi u j ði $ jþ ¼ X j ω i u j þ ω X j j u i Y i ð3:147þ ð X ωþ i ¼ X j ω i u j þ ω X j j u i ð3:148þ Multiplying Eq. (3.148) bydu i and interchanging i with j in the second term on the RHS of Eq. (3.149), one obtains ð X ωþ i du i ¼ X j ω i X j u j dui þ ω j u i dui ð3:149þ ¼ ðxω i Þdu i þ ω i dx i Using the chain rule of differentiation, the RHS of Eq. (3.149) can be written as ðxω i Þdu i þ ω i dx i ¼ ð X ω i Þdu i þ ω i X du i ¼ X ω i du i X ω ð3:150þ where the one-form ω can be expressed in the contravariant basis vector du i using Einstein summation convention, cf. Eq. (3.179).

155 138 3 Elementary Differential Geometry ω ¼ ω i du i Thus, one obtains the Lie derivative of the one-form in the direction of the vector field X. X ω ¼ ð X ωþ i du i ¼ ðxω i Þdu i þ ω i dx i ðq:e:d: Þ Torsion and Curvature in a Distorted and Curved Manifold The normal vector field N perpendicular to the surface of the manifold M is dragged in two different paths from the same point P 11 via P 21 to Q in the one path; and via P 12 to S in the other path, as shown in Fig Due to the effect of space torsions and surface curvatures, the vector field N does not close the connection loop at the path ends Q and S of the dragging paths. The gap of the path ends is O(ε 2 )ina distorted and curved manifold and is reduced to the order of O(ε 3 ) in an only curved manifold [7]. The Lie derivative of the vector Y with respect to the vector field X is induced by the space torsion of the distorted manifold. As a consequence, the torsion tensor ε 2 [X,Y] generates the open connection gap QR (see Fig. 3.9). The Riemann surface curvature is to blame for the other open connection gap RS on the order of O(ε 3 )in the curved manifold. Therefore, the connection loop is always closed in a torsion-free and flat space. ½Y; X½X; Y ¼ 0, Y X ¼ X Y ð3:151þ The curvature equation of a distorted and curved manifold can be written by means of the Lie formulations, Riemann curvature tensors, and covariant metric coefficients [7]. ½Y; X½X; Y ¼ ε 2 ½X; Yþε 2 R lijk, Y X X Y ¼ ε 2 X Y þ ε 2 g nl Rijk n ð3:152þ where R lijk is the Riemann curvature tensors of the curved manifold M Killing Vector Fields The Killing vector field K is defined as a vector field in an N-dimensional manifold in which the Lie derivative of the metric tensor g with respect to the vector field K along the congruence equals zero.

156 3.12 Lie Derivatives 139 Fig. 3.9 Connection loop of vector fields in a distorted and curved manifold M * εx 2 2 ε Y = ε X 2 ε R lijk [ X, Y] : torsion tensor : curvature tensor N P 11 εy εx P 21 * εy Q R S M P 12 K g ¼ 0 ð3:153þ Equation (3.153) shows that the metric tensor g is invariant on the manifold with respect to the Killing vector field K. The covariant tensor components of the Lie derivative of the metric tensor with respect to the Killing vector field K can be expressed in [8]. ð K gþ ij ¼ K k g ij u k þ g K k K k ik u j þ g kj u i ¼ 0 ð3:154þ Equation (3.154) can be written in one-dimensional coordinate u k with respect to the Killing vector field K. ð K gþ ij ¼ g ij u k ¼ 0 ð3:155þ Therefore, if the covariant metric coefficient is independent of any coordinate, the basis of the coordinate is a Killing vector. As an example for the Killing vector field, the covariant metric coefficients of the spherical coordinates (r,φ,θ) are given in g rr ¼ g r g r r r ¼ 1 g φφ ¼ g φ g φ φ φ ¼ r2 g θθ ¼ g θ g θ θ θ ¼ r2 sin 2 φ ð3:156þ Equation (3.156) shows that the metric coefficients are independent of the coordinates (r,φ,θ). Hence, the basis vectors g r, g φ, and g θ are the Killing vectors.

157 140 3 Elementary Differential Geometry Fig Invariant fields on a moving surface S(t) i V P S ( t + Δt) α u α V g α Δt N T ( t + Δt, S ) T ( t, S) i x S(t) g i 3.13 Invariant Time Derivatives on Moving Surfaces In the following section, the invariant time derivatives of tensors are applied to a surface S(t) moving with a velocity vector V in the ambient coordinate system. For this case, the invariant time derivative of an invariant field T(t, S) parameterized by the time t and moving surface S can be calculated in the surface coordinate. Generally, two coordinates of the unchanged ambient coordinate x i with the covariant basis g i and the moving surface coordinate u α with the covariant basis g α are used in the moving surface S(t), as shown in Fig The surface S at the time t moves to the new surface position S at the time t + Δt at which the invariant field T(t, S) at the time t is changed into T(t + Δt, S ) in a very short time interval Δt. The time-dependent surface S moves with a coordinate velocity V i in the ambient coordinate x i [10]. The ambient coordinate velocity of the moving surface S(t) in the coordinate x i can be defined by V i xi ðt; SÞ t ð3:157þ The tangential coordinate velocity V α results from projecting the ambient coordinate velocity V i onto the surface along the surface coordinate u α. To calculate the tangential coordinate velocity, the surface velocity vector V can be formulated in both ambient and surface coordinates using the chain rule of differentiation.

158 3.13 Invariant Time Derivatives on Moving Surfaces 141 V ¼ V α g α ¼ V i g i ¼ V i r x i ¼ V i r u α u α x i ¼ V i uα x i g α V i x α gα, i ð3:158þ where the derivative x, i α is called the shift tensor between the ambient and surface coordinates. Thus, the tangential coordinate velocity V α results from the ambient coordinate velocity and shift tensor. Analogously, one obtains V α ¼ V i uα x i ¼ V i x α, i ð3:159aþ V i ¼ V α xi u α ¼ Vα x i,α ð3:159bþ Invariant Time Derivative of an Invariant Field The invariant time derivative of an invariant field T(t, S) can be defined as the time change rate of the invariant field itself and its change rates along the surface coordinates between the old and new surface positions [10]. _ T Tt; ð SÞ V α α T t ð3:160þ At first, the covariant surface derivative of a first-order tensor T can be written as α T ¼ T u α ¼ Ti g i u α ¼ Ti u α g i þ Ti g i u α ¼ Ti u α g i þ g Ti i x j x j u α Using Eq. (2.158), the partial derivative of the basis g i results in ð3:161aþ

159 142 3 Elementary Differential Geometry g i x j ¼ Γ k ij g k ð3:161bþ Inserting Eq. (3.161b) into Eq. (3.161a), one obtains α T ¼ Ti u α g i þ xj u α Γ ij k Ti g k ¼ Tk uα þ xj uα Γij k ð3:161cþ Ti g k α T k gk Thus, α T k ¼ Tk u α þ xj u α Γ k ij Ti, α T i ¼ Ti u α þ xj u α Γ i jm Tm ð3:161dþ The covariant surface derivative of a contravariant first-order tensor results in using Eq. (3.161d) and chain rule of coordinates. α T i T i α ¼ Ti u α þ xj u α Γ i jm Tm ¼ Ti x j x j u α þ xj u α Γ jm i Tm ¼ xj T i u α x j þ Γ jm i Tm ¼ x j,α Ti j x j,α jt i ð3:161eþ Analogously, the covariant surface derivative of a mixed second-order tensor results using Eq. (2.211a) and chain rule of coordinates in α Tj i Tj i ¼ T i j α u α þ xk u α Γkm i T j m Γkj n T n i ¼ T j i x k x k u α þ xk u α Γkm i T j m Γkj n T n i! ¼ xk Tj i u α x k þ Γ km i T j m Γkj n T n i ¼ x, k α T j i x, k α ktj i k ð3:161fþ The ambient coordinate is dependent on time and the surface coordinate of the moving surface S. Thus, the invariant field T can be expressed as

160 3.13 Invariant Time Derivatives on Moving Surfaces 143 Tt; ð SÞ ¼ Tt, ð xt; ð SÞ Þ ð3:162þ Using the Taylor series and the chain rule of differentiation, the partial time derivative of T with two independent variables of t and S can be calculated as because Tðt; SÞ ¼ Tt; ð xþ þ Tt; ð xþ x i ðt; SÞ t t x i t ¼ Tt; ð xþ ð3:163þ þ ð i TÞV i t i T ¼ Tt; ð xþ x i ; V i ¼ xi ðt; SÞ t ð3:164þ The invariant time derivative in Eq. (3.160) can be rewritten using Eqs. (3.161e) and (3.163). _ T Tt; ð SÞ V α α T t ¼ Tt; ð xþ þ V i i T V α x,α k t kt ¼ Tt; ð xþ þ V i i T V α x,α i t it ¼ Tt; ð xþ þ V i x, i α t Vα i T ð3:165aþ The second term on the RHS in Eq. (3.165a) can be further calculated using Eq. (3.159a). V i x, i α Vα ¼ V i xi u α u α x j Vj ¼ V i x, i α x, α j Vj ¼ V j δj i x,α i x, α j ð3:165bþ The useful relation between the contra- and covariant normal vector components and shift tensors of coordinates is derived by [10]: N i N j þ x i,α x α,j ¼ δ i j ð3:165cþ in which δ j i is the Kronecker delta. Substituting Eq. (3.165c) into Eq. (3.165b), the invariant time derivative of T given in Eq. (3.165a) can be rewritten as

161 144 3 Elementary Differential Geometry _ T ¼ Tt; ð xþ þ V j N j N i i T t ¼ Tt; ð xþ þ PN i i T t ð3:165dþ where P is the normal velocity at a given point on the moving surface S, as displayed in Fig In fact, the normal velocity is the projection of the ambient coordinate velocity V i on the surface normal N i. P ¼ V i N i ) P ¼ PN ð3:165eþ ¼ V i N i N ¼ V i N i N j g j Invariant Time Derivative of Tensors Analogously, the invariant time derivative of tensors can be derived from the invariant field. The contravariant tensor can be written in the covariant basis g i. T ¼ T i g i ð3:166þ The invariant time derivative of the tensor T can be expressed on the moving surface S according to Eq. (3.160) [10]. T t; S _ T ¼ ð Þ V α α T t ð3:167þ Substituting Eq. (3.166) into Eq. (3.167), one obtains _ T ¼ Ti g i t V α α T i g i ¼ Ti t g i þ Ti g i t Vα α T i gi ð3:168þ Using Eqs. (2.158) and (3.157), the time derivative of the coordinate basis g i in Eq. (3.168) can be calculated. _g i g i t ¼ g i x j x j t ¼ Γ k ij x j t g k ¼ Γ k ij Vj g k ¼ g i, j x j t ð3:169þ Therefore, the invariant time derivative of the tensor T can be rewritten as

162 3.13 Invariant Time Derivatives on Moving Surfaces 145 _ T ¼ Ti t g i þ Vj Γij k Ti g k V α α T i ¼ Tk þ V j Γij k t Ti V α α T k g k gi ð3:170þ _ T k g k The invariant time derivative of a contravariant tensor T i is given from Eq. (3.170). _ T i ¼ Ti t þ Vj Γ i jk Tk V α α T i ð3:171þ Similarly, one obtains the invariant time derivative of a covariant tensor T i _ T i ¼ T i t Vj Γ k ji T k V α α T i ð3:172þ The invariant time derivative of a mixed tensor T j i can be derived in _ Tj i ¼ T j i t þ Vk Γkm i T j m ¼ T j i t þ Vk Γ i km T m j V k Γkj n T n i Vα α Tj i Γkj n T n i V α α Tj i : ð3:173þ The general invariant time derivative of a mixed fourth-order tensor can be derived in [10]. _ T iα jβ ¼ Tiα jβ t V γ γ T iα jβ þ Vp Γ i pq Tqα jβ Vp Γ q pj Tiα qβ þ _Γ δ αtiδ jβ _Γ β δtiα jδ ð3:174þ where the time derivative of the Christoffel symbols for a moving surface is defined as _Γ α β ¼ βv α PR α β ð3:175þ in which P is the normal velocity in Eq. (3.165e) and R β α is the mean curvature of the moving surface.

163 146 3 Elementary Differential Geometry Fig Transport of a vector field Y in the tangent bundle TM C α 0 congruence C tangent space T p M P 1 M X Δα Y P 2 i u (α) α0 + Δα X tangent bundle TM Y M manifold M T p 3.14 Tangent, Cotangent Bundles and Manifolds Some definitions of spaces and manifolds are discussed before dealing with the Levi-Civita connection. Let C 1 be a set of infinitely differentiable curves (congruence) on the manifold M. The vector X is tangent to the curve C 1 at the point P 1 and moves along it as the parameter α varies in the local coordinate u i (α), as shown in Fig The tangent vector X lies on the tangent space T p M (tangent surface) that is tangent to the curve C 1 at the point P 1 on the manifold M. As the vector X moves along C 1, the tangent space also changes from P 1 to P 2. Thus, the tangent spaces T p M generate a tangent bundle TM that consists of all tangent spaces T p M of the manifold M. The covariant basis of the local coordinate u i (α) on the tangent space T p M is written as i u i ð3:176þ Therefore, the tangent vector X on the tangent space T p M is expressed in the covariant basis. X ¼ X i u i ¼ ui ðαþ α u i ¼ d dα ð3:177þ Furthermore, the dual space of the tangent space T p M is called the cotangent space T p * M in which the contravariant basis vectors du i of the local coordinates u i (α) are used for covariant tensors, covariant vectors, and differential forms. Analogously, the cotangent bundle T * M covers all cotangent spaces T p * M of the manifold M.

164 3.14 Tangent, Cotangent Bundles and Manifolds 147 Due to orthonormality of the covariant and contravariant bases, their inner product (dot or scalar product) vanishes for i 6¼ j. Using the Kronecker delta, the contravariant basis (dual basis) du j on the cotangent space T * p M is defined as i, du j ¼ δ j i ð3:178þ In differential geometry, a differential one-form dω on a differentiable manifold M is defined as a smooth cross section of the cotangent bundle T * M of the manifold M. As a result, the differential one-form dω is naturally dual to its first-order covariant tensor field of type (0,1) in the dual cotangent space. Sometimes, differential one-forms are called covariant vector fields or dual vector fields within physics. Using Einstein summation convention, the one-form ω p at the point P is generally used on the dual space (cotangent space) with the contravariant basis du i. ω p ¼ ω i du i where ω i is a function of u i. The differential one-form results from Eq. (3.179) in dω p ¼ ω u i du i ¼ ω p p u i dui ð3:179þ ð3:180þ Using the Leibniz s rule, the product of two differential one-forms f p and g p is calculated. dfg ð Þ p ¼ f p dg p þ g p df p ð3:181þ Differential 0-forms, 1-forms, and 2-forms are the special cases of the k-form ω that can be generally expressed in terms of the dual bases on a differentiable N- dimensional manifold M as (cf. Chap. 4) ω ¼ XN hu i 1 ;...; u i k du i1 ^ du i2 ^^du i k i 1 <i 2 <<i k where ^ is the wedge (exterior) product; h is a smooth function of the coordinates u i. The differential k-form dω, that results from the k-form ω, is called the exterior derivative of ω. Note that differential forms, wedge products, and exterior derivatives are independent of any coordinates on the manifold M.

165 148 3 Elementary Differential Geometry 3.15 Levi-Civita Connection on Manifolds Levi-Civita connection (LC connection) describes the process of transporting (dragging) a vector field Y with respect to another vector field X on a smooth differentiable N-dimensional manifold M (affine connection). This connection is related to the Riemann connection; therefore, it is sometime called the Riemann Levi-Civita connection (RLC connection). The covariant derivative of the vector field Y with respect to the vector X moving along to a differentiable curve C 1 on the manifold M is generally used to formulate the transport of the vector field Y on the N-dimensional manifold M, as displayed in Fig As a result, the Levi-Civita connection on (M, g) associates the covariant derivative (absolute derivative) with respect to the geodesic parameter α of the coordinates u i (α) on the manifold M. The Levi-Civita connection is defined as the covariant derivative of Y with respect to the vector field X. X Y DYðαÞ Dα ¼ _Y i þ Y j _u k Γjk i i ð3:182þ ð X YÞ i i where X ¼ d dα ; _Y i dyi ðαþ dα ; _u k duk ðαþ dα The component i of the Levi-Civita connection of the vector field Y with respect to the vector field X results from Eq. (3.182) in ð X YÞ i ¼ _Y i þ Y j _u k Γjk i ð3:183þ Proof Using Leibniz rule of differentiation, one obtains X Y ¼ X Y i i ¼ X Y i i þ Y i ð X i Þ ð3:184þ The covariant derivative of a function f ¼ Y i (α) is calculated as X f ¼ Xf ) X Y i ðαþ ¼ XY i ðαþ ¼ d dα Yi ðαþ _Y i ðαþ ð3:185þ The covariant derivative of the coordinate basis with respect to vector X is expressed as

166 3.15 Levi-Civita Connection on Manifolds 149 Fig Levi-Civita connection of vector field Y with respect to X Y C P 1 X Y M P 2 i u (α) X X i ¼ X j j ð ¼ X j Γ k ij k i Þ ð3:186þ in which the covariant derivative of the basis j of the coordinate u j with respect to i is defined as j ð i Þ ¼ Γji k k ¼ Γij k k ð3:187þ Substituting Eqs. (3.185) and (3.186) into Eq. (3.184), one obtains interchanging k with i in the second term on the RHS. X Y ¼ _Y i i þ Y i X j Γij k k ¼ _Y i i þ Y k X j Γkj i i ¼ _Y i þ Y k X j Γkj i i ð3:188þ The vector X is written in the bases of the coordinates u i. X ¼ d dα ¼ dui dα u i ¼ X i u i ¼ Xi i Thus, X j ¼ duj dα _u j ðαþ ð3:189þ Inserting Eq. (3.189) into Eq. (3.188) and interchanging k with j, the Levi-Civita connection in Eq. (3.182) is derived.

167 150 3 Elementary Differential Geometry X Y ¼ _Y i þ Y k _u j Γkj i i ¼ _Y i þ Y j _u k Γjk i i An affine connection is called the Levi-Civita connection on a smooth differentiable geometric manifold (M, g) if It preserves the metric: X g ¼ 0 for all vector fields on M (Riemann connection) and is torsion-free: X Y Y X ¼ ½X; Y where [X,Y] is defined as the Lie bracket. Generally, the covariant derivative of a vector field Y with respect to X at a non-parallel transport does not equal zero (cf. Fig. 3.12). In the case of a parallel transport, the direction of the vector field Y does not change as the vector X moves along the curve C 1 on M. Then, the covariant derivative of Y with respect to X vanishes: X Y ¼ _Y i þ Y j _u k Γ i jk i ¼ 0 ð3:190þ Therefore, the first-order differential equation of Y for a parallel transport results in dy i ðαþ dα þ Γ jk i du k ðαþ dα Yj ðαþ ¼ 0, duk Y i dα u k þ Γ jk i Yj ¼ 0 ð3:191þ Thus, _Y i ðαþ ¼ Γjk i _u k Y j Sj i ðαþy j ðαþ A geodesic is the shortest distance between two any points on the manifold M. The curve C 1 is geodesic (or auto-parallel) if the covariant derivative of vector fields X with respect to itself vanishes: X X ¼ 0 ð3:192þ Substituting X into Y in Eq. (3.191), the second-order differential equation of a geodesic results in

168 3.15 Levi-Civita Connection on Manifolds 151 X i ðαþ ¼ dui dα! Yi ðαþ ) d2 u i dα 2 þ Γ i jk du j dα du k ¼ 0for8i ¼ 1, 2, ::, N dα ð3:193þ Furthermore, let X, Y, Z be vector fields, f and h be linear functions. The Levi- Civita connection on a smooth differentiable geometric manifold (M, g) satisfies the following properties: Linearity: Leibniz rule of differentiation: X ðy þ ZÞ ¼ X Y þ X Z ð3:194aþ X ðf YÞ ¼ f X Y þ ð X f ÞY ¼ f X Y þ ðxf ÞY ð3:194bþ Function linearity: ðf XþhYÞ Z ¼ f X Z þ h Y Z ð3:194cþ Free torsion: TX; ð YÞ X Y Y X ½X; Y ¼ 0 ) X Y Y X ¼ ½X; Y ð3:194dþ where T(X,Y) is the torsion tensor; [X,Y] is the Lie bracket. Metric compatibility: X gðy; Z Þ ¼ gð X Y, ZÞþgðY, X ZÞ ð3:194eþ where g is the bilinear function g(.,.) relating to the metric tensor on the geometric manifold (M, g). The metric tensor g, a covariant second-order tensor of type (0,2) on the manifold M is defined as g ¼ g ij du i du j ; ð3:195þ g ij g i ; j ¼ i j The differential distance between two points can be written in a two-form of the metric tensor g:

169 152 3 Elementary Differential Geometry qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds ¼ g ij du i du j The metric tensor g generates a bilinear function g(.,.) on the tangent space T p M at any point p on the manifold M. In fact, the metric operator g(.,.) is a bilinear mapping of two arbitrary vectors at any point p to a real function. g : T p M T p M! R gðx; YÞ p ¼ g p X p ; Y p ¼ gij X i Y j p ð3:196þ where X p and Y p are tangent vectors at the point p on T p M. The vector fields X and Y are written on the manifold M as X ¼ X i i ; Y ¼ Y j j : Using Eq. (3.196), one obtains for any point p on the manifold M g : T p M T p M! R gðx; YÞ ¼ gx i i, Y j j ¼ g i ; j X i Y j ) gðx; YÞ ¼ g ij X i Y j for 8i, j ¼ 1, 2,..., N ð3:197þ The metric operator g(.,.) has the following properties: gðx; YÞ ¼ gðy; XÞ : symmetric; gðαx þ βy, ZÞ ¼ αgðx; ZÞþβgðY; ZÞ : bilinear ð3:198þ in which α and β are scalars. The Levi-Civita connection satisfies the Koszul formula for vector fields X, Y, and Z on the tangent space T p M. 2gð X Y, ZÞ ¼ X gðy; ZÞþ Y gðx; ZÞ Z gðx; YÞ þ gð½x; Y; ZÞgð½X; Z; YÞgð½Y; Z; XÞ ð3:199þ Proof Using the metric compatibility in Eq. (3.194e), one obtains X gðy; ZÞ ¼ gð X Y, ZÞþgðY, X ZÞ; Y gðx; ZÞ ¼ gð Y X, ZÞþgðX, Y ZÞ; Z gðx; YÞ ¼ gð Z X, YÞþgðX, Z YÞ: ð3:200þ Substituting Eq. (3.200) into the RHS of Eq. (3.199), its new RHS becomes

170 References 153 RHS gð X Y þ Y X þ ½X; Y, ZÞ þ gð X Z Z X ½X; Z, YÞ þ gð Y Z Z Y ½Y; Z, XÞ ð3:201þ Levi-Civita connection is torsion free and therefore satisfies using Eq. (3.194d) X Y Y X ¼ ½X; Y; X Z Z X ¼ ½X; Z; Y Z Z Y ¼ ½Y; Z: ð3:202þ Inserting Eq. (3.202) into Eq. (3.201), the Koszul formula has been proved. RHS gð2 X Y, Z Þ ¼ 2gð X Y, ZÞ LHS In general, the Levi-Civita connection of tensor fields T with respect to a vector field X on the manifold M is calculated in [9] X T ¼ ð X TÞ i...k l...n i... k du l... du n ð3:203þ in which ð X TÞ i...k l...n ¼ Xp Tl...n,p i...k þ Xp Γpq k Ti...q l...n þ... Xp Γ q lp Ti...k q...n... ð3:204þ Let S and T be arbitrary tensor fields, and X be a vector field. The Levi-Civita connection on a smooth differentiable geometric manifold (M, g) fulfills the following properties: Linearity: X ðs þ TÞ ¼ X S þ X T ð3:205aþ Leibniz rule for tensor product: X ðs TÞ ¼ ð X SÞT þ S ð X TÞ ð3:205bþ References 1. Bär, C.: Elementare Differentialgeometrie (in German), Zweiteth edn. De Gruyter, Berlin (2001) 2. Chase, H.S.: Fundamental Forms of Surfaces and the Gauss-Bonnet Theorem. University of Chicago, Chicago, IL (2012)

171 154 3 Elementary Differential Geometry 3. Klingbeil, E.: Tensorrechnung f ur Ingenieure (in German). B.I.-Wissenschafts-verlag, Mannheim (1966) 4. K uhnel, W.: Differentialgeometrie Kurven, Flächen, Mannigfaltigkeit (in German), 6th edn. Springer-Spektrum, Wiesbaden (2013) 5. Lang, S.: Fundamentals of Differential Geometry, 2nd edn. Springer, New York, NY (2001) 6. Danielson, D.A.: Vectors and Tensors in Engineering and Physics, 2nd edn. ABP, Westview Press, CO (2003) 7. Penrose, R.: The Road to Reality. Alfred A. Knopf, New York, NY (2005) 8. Schutz, B.: Geometrical Methods of Mathematical Physics. Cambridge University Press (CUP), Cambridge (1980) 9. Fecko, M.: Differential Geometry and Lie Groups for Physicists. Cambridge University Press, Cambridge (2011) 10. Grinfeld, P.: Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer Science + Business Media, New York, NY (2013)

172 Chapter 4 Differential Forms 4.1 Introduction Alternative to tensors, differential forms are very useful in differential geometry without considering the coordinates compared to tensors. Differential forms are based on exterior algebra in which the coordinates are not taken into account. The exterior algebra was developed by Élie Cartan ( ) and Henri Poincaré ( ). Before going into details in differential forms in exterior algebra, we look into multi-dimensional spaces and shapes of the objects under points of view between topology and differential geometry. Geometers are concerned with the exact shape, size, and curvature of the object in all possible dimensions. On the contrary, topologists generally look into the overall shape of the object as a whole without considering it in detail as the geometers did. In topology, there are usually two basic kinds of spaces with one-dimensional and two-dimensional shapes of any object [1]. The one-dimensional spaces are a straight line or a circle; however, both are fundamentally different from each other. The difference is that the circle can be transformed into any shape of loops, but a circle cannot be made in a line without cutting it. The two-dimensional spaces are classified into two basic types: either a sphere or a donut (torus) with two-dimensional surfaces. The big difference between them is that the sphere has no hole in it and the donut with a hole through it. Whatever what you do, it is definitely that a donut cannot be transformed into a sphere without cutting a hole through the middle. Cubes, hexahedrons, pyramids, and tetrahedrons are topologically homeomorphic (similar shape) to a sphere with genus 0. Note that the genus denotes the number of holes in the object. Obviously, a donut (torus) has genus 1, which denotes one hole in it. Pretzels with two holes have genus 2; pretzels with three holes, genus 3 (see Fig. 4.1). Springer-Verlag Berlin Heidelberg 2017 H. Nguyen-Schäfer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI / _4 155

173 156 4 Differential Forms Fig. 4.1 Topological manifolds with various genera of 0, 1, 2, and 3 If different objects have the same genus, they are homeomorphisms or topological isomorphisms in the mathematical field of topology. Any cube with a hole through it can be somehow reshaped into the shape of a torus (genus 1) by squeezing and stretching. However, an object with any genus other than one (genus 6¼ 1); e.g., bun (genus 0), American pretzel (genus 2), and German pretzel (genus 3) cannot be molded into a donut (genus 1) without punching a hole through it or cutting the holes through the middles. Objects with surfaces at topological dimensions equal to or higher than three are quite difficult to display. They are called hypersurfaces of N-manifolds (N 3). In fact, a sphere has a topological dimension of two because only its 2-D surface is considered. Therefore, it is called 2-sphere S 2 or a sphere. Any surface on which every loop can be shrinked to a point is defined as a sphere (genus 0). On the contrary, loops on the torus (genus 1) surface cannot be tightened to a point, as shown in Fig In differential geometry, the dimension of a space (manifold) is the necessary numbers of coordinates to determine the characteristic of a given point (location and physical states) on the manifold. Therefore, the manifold of the object in differential geometry is studied in more detail in higher-dimensional Riemannian spaces, different to topology in which the object is considered as a whole. Hence, the sphere S 2 is considered as a three-dimensional space in differential geometry. The computations in N-dimensional spaces are carried out using tensors and differential forms without drawing them on the paper. In order to maintain that the maximum velocity of any object must be less than or equal to the light speed (v c), the four-dimensional Minkowski spacetime (t, x, y, z) is generally used in the Maxwell s equations, special relativity theory, and cosmology (cf. Chap. 5).

174 4.3 Differential k-forms Definitions of Spaces on the Manifold Differential forms deal with calculations in differential geometry without considering any coordinate contrary to vector and tensor calculus. However, they use many mapping operators in various spaces, vector and differential form bundles using exterior algebra that is quite different to linear algebra. Unfortunately, they are very confused in the applications of differential forms. In order to comprehend the differential forms, some necessary spaces and bundles on the manifold are defined in the following section [2]. Let M be a smooth and differentiable N-dimensional manifold, M* be a dual manifold of M, and p be a point on the manifold M. The tangent space T p M at the point p on M is defined as the space that contains the tangent vector to the manifold M at the point p. The basis vectors in the tangent space T p M have been discussed in Sect The cotangent space T * p M at the point p on M is defined as the dual space of the manifold M. The dual bases dx i for i ¼ i 1,...,i k are used in the cotangent space T * p M. The tangent bundle TM is a set of all tangent spaces T p M as the point p moves on the manifold M. The cotangent (dual) bundle T * M is a set of all dual spaces T * p M as the point p moves on the manifold M. ^k(t p M) is defined as the subset of the k-tangent-vector space at the point p on the tangent space T p M. ^k(t * p M) is defined as the subset of the k-form space at the point p on the cotangent (dual) space T * p M. ^k(m) is defined as the exterior k-vector-bundle space on M that consists of all subsets of the tangent vector spaces ^k(t p M). ^k(m * ) is defined as the exterior k-form-bundle space on M that consists of all subsets of the k-form spaces ^k (T * p M). A k (M) denotes the space of smooth sections of the exterior k-form bundle of ^k (M * ). 4.3 Differential k-forms The differential k-form ω p at the point p on M is an alternating (skew-symmetric) multilinear map from p into the subset of the k-form space ^k(t p * M). ω p : T * p M... T* p fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} M k times, ω p : dx i1 ^...^ dx i k ð! Λ k T * p M Þ! ω p ¼ X I X I Λ k ðmþ f I xp1,..., x pn dx i1 ^...^ dx i k f I xp1,..., x pn dx I 2 Λ k T * p M ð4:1þ

175 158 4 Differential Forms cotangent space ω p k form space exterior k form bundle k * Λ ( ) M R N T * p M p I dx manifold ω p = I f dx I I k Λ ( T * p M ) N M R Fig. 4.2 A differential k-form ω p of a point p on M The elementary k-form of ω p is defined as dx I dx i1 ^ dx i2 ^...^ dx i k ð4:2þ for I ¼ ði 1 ;...; i k Þ; i 1 <...< i k in which the dual bases dx i for i ¼ i 1,...,i k are used in the cotangent (dual) space T p * M on the manifold M. The k-form ω p is a subset of the exterior form bundle on M (see Fig. 4.2). ω p 2 Λ k T * p M Λ k M * [ Λ k p2m T * p M ð4:3þ Some examples of k-forms are given as Zero-form (k ¼ 0): ω ¼ 1 One-form (k ¼ 1): ω ¼ yzdx þ xydy x 2 zdz Two-form (k ¼ 2): ω ¼ x 2 y 3 ðdx ^ dyþþxy 3 ðdy ^ dzþxyz 2 ðdx ^ dzþ Three-form (k ¼ 3): ω ¼ xy 2 z 3 ðdx ^ dy ^ dzþ The wedge product (exterior product) of two any differential one-forms ω and η is defined as ω ^ ηω η η ω ð4:4þ where is the tensor product (cf. Chaps. 2 and 6). Let ω and η be one-forms for k ¼ 1. If the order of two dual bases is interchanged, one permutation is carried out. As a result, the sign of the k-form is changed. In the case of an even number of permutations, the sign of the k-form is +1 and the sign is 1 for an odd number of permutations (see Table 4.1).

176 4.3 Differential k-forms 159 Table 4.1 Permutations of the dual bases in 3-forms i 1 i 2 i 3 3-forms dx I σ x y z dx ^ dy ^ dz ¼þdx ^ dy ^ dz 0 x z y dx ^ dz ^ dy ¼dx ^ dy ^ dz 1 y x z dy ^ dx ^ dz ¼dx ^ dy ^ dz 1 y z x dy ^ dz ^ dx ¼þdx ^ dy ^ dz 2 z x y dz ^ dx ^ dy ¼þdx ^ dy ^ dz 2 z y x dz ^ dy ^ dx ¼dx ^ dy ^ dz 3 Generally, the elementary k-forms result after σ permutations as dx σ ð i Þ^ 1 dx σ ð i Þ^ 2...^ dx σ ð i k Þ ¼ ð1þ m σ dx i1 ^ dx i2 ^...^ dx i k ð4:5þ where m σ is the modulo 2 of σ that is the residual of the permutation number σ divided by 2. m σ ¼ σmod2 ð4:6þ Some examples of the permutations of k-forms are given in the following section. If no permutation (σ ¼ 0) is carried out, one obtains Thus, m 0 ¼ 0mod2 ¼ 0 þ 0 ) m 0 ¼ 0 dx i1 ^ dx i2 ^...^ dx i k If σ equals one permutation (σ ¼ 1), one obtains Thus, ¼ ðþ1þdx i1 ^ dx i2 ^...^ dx i k m 1 ¼ 1mod2 ¼ 0 þ 1 ) m 1 ¼ 1 dx i2 ^ dx i1 ^...^ dx i k If σ permutations are carried out, one obtains Therefore, dx ik ^ dx i1 ^ dx i2 ^...^ dx i k1 ¼ ð1þdx i1 ^ dx i2 ^...^ dx i k m σ ¼ σmod2; σ > 1 1 if σ is an odd number ¼ 0 if σ is an even number ¼ ð1þ m σ dx i1 ^ dx i2 ^...^ dx i k

177 160 4 Differential Forms Table 4.1 displays the permutations of the dual bases in six 3-forms (k! ¼ 3! ¼ 6) with the number of permutations σ. The first 3-form of the dual bases is called the elementary form without any permutation (σ ¼ 0). Moving the dual bases in the first 3-form, there are fives possibilities of the 3-forms that are transformed into the elementary form using Eq. (4.5). Interchanging dz with dy in the second 3-form, one obtains the elementary form at one permutation (σ ¼ 1); thus, its sign is changed. The fourth 3-form needs two permutations (σ ¼ 2) of dz with dx and dy with dx to become the elementary form; therefore, its sign is unchanged. Similarly, other remaining 3-forms are easily transformed into the elementary form using Table 4.1. In general, let ω be a k-form, η be an l-form, and v be a m-form. The wedge product of two differential forms has the following properties: ω ^ ðη þ vþ ¼ ðω ^ ηþþðω ^ vþ : left distributive ðη þ vþ^ω ¼ ðη ^ ωþþðv ^ ωþ : right distributive ω ^ ðη ^ vþ ¼ ðω ^ ηþ^v : associative ω ^ η ¼ ð1þ kl η ^ ω : graded anticommutative ð4:7þ Using the graded-anticommutative law in Eq. (4.7), one obtains for ω ¼ η ω ^ ω ¼ 0 ) dω ^ dω ¼ 0 ð4:8þ Moving k dual bases of the k-form behind the l dual bases of the l-form, the wedge product of two differential forms in Eq. (4.7) is proved. ω ^ η ¼ X ω i1...i k η j1...j l dx i1 ^...^ dx i k ^ dx j 1 ^...^ dx j l i, j ¼ X ð1þ l ω i1...i k η j1...j l dx i1 ^...^ dx i k1 ^ i, j dx j 1 ^ dx j 2 ^...^ dx j ð l Þ^dx i k ¼ X ð4:9þ ð1þ lk η j1...j l ω i1...i k dx j 1 ^...^ dx j l ^ dx i1 ^...^ dx i k i, j ¼ ð1þ kl η ^ ω ðq:e:d: Þ According to Eq. (4.9), the wedge product of two differential k- and l-forms has the order of (k + l) and belongs to the exterior (k + l)-form space in exterior algebra. ω 2 Λ k T * p M Λ k M * ; η 2 Λ l T * p M Λ l M * ) ω ^ η ¼ ð1þ kl η ^ ω 2 Λ kþl T * p M Λ kþl M *

178 4.4 The Notation ω X 161 The differential forms of ω and η are written in their dual bases as ω ¼ X X ω i1...i k dx i1 ^ dx i2 ^...^ dx i k ω I dx I ; i 1 <...<i k I η ¼ X X η j1...j l dx j 1 ^ dx j 2 ^...^ dx j l η J dx J : j 1 <...<j l J where ω I and η J are smooth differentiable functions of x I and x J, respectively. The dimension of the k-form space ^k(t * p M)inanN-dimensional space R N is defined as the number of choosing k-element subsets of an N-element set that consists of N distinct elements. The dimension of the k-form space is defined as the possible number of k permutations from N dimensions disregarding the permutation order. As a result, the k-form dimension is calculated as the binomial coefficient in combinatorics. N N! ¼ k k! ðn kþ! ¼ dim Λk T * p M ; 0 k N ð4:10þ The factorial of N is defined by N! ¼ YN k¼1 k ¼ 1 2 ðn 1ÞNfor8N > 0; 0! 1 Thus, the factorial of N can be written in the recurrence relation as N! ¼ 1 if N ¼ 0; ðn 1Þ!N if N > 0: Using Eq. (4.10), the dimensions of k-form spaces in a three-dimensional space R 3 are calculated and shown in Table The Notation ω X Let ω be a one-form in the 1-form space ^1 (T p * M); X be a vector field in the tangent space T p M 2 R N. A linear mapping of y 2 M to ωx( y) by the mapping function ωx is written as ω X : y 2 M! ω Xy ðþ2r

179 162 4 Differential Forms Table 4.2 Dimensions of the k-form spaces in R 3 ω (0 k 3) Form space Dual bases of k-forms Dimensions of form space 0 form ^0 R 3 ¼ R form ^1 R 3 dx 1, dx 2, dx form ^2 R 3 dx 1 ^ dx 2, dx 2 ^ dx 3, dx 1 ^ dx form ^3 R 3 dx 1 ^ dx 2 ^ dx 3 1 The notation ωx linking the differential forms to vector fields is defined as y 2 M! ω Xy ðþωðþxy y ðþ2r Using Einstein s summation convention, the one-form ω in the 1-form space ^1 (T * p M) and the vector field X in the tangent space T p M are written as ω ¼ h i dx i 2 Λ 1 T * p M ; i ¼ 1,..., N X ¼ ξ j x j 2 T pm; j ¼ 1,..., N Due to orthonormality of the bases of the tangent and cotangent spaces according to Eq. (3.178), the notation ωx( y) is written using Kronecker delta as ω Xy ðþ¼ωðþxy y ðþ¼h i ðþξ y j ðþδ y j i ¼ h i ðþξ y i ðþ2r; y i ¼ 1,..., N Analogously, the notation is extended to p-forms ω in the p-form space ^p(t * p M) and vector bundles (X 1,..., X p ) in the p-vector-bundle space ^p(m) as y 2 M! ω X 1 ^...^ X p ðþ¼ω y ðþ y X1 ðþ^...^ y X p ðþ y 2 R Due to linearity of the p-forms, an element (monomial) of the p-form is used in this case. The monomial ω of the p-form for y 2 M is written as ωðþ¼hy y ðþdx i1 ^...^ dx i p 2 Λ p T * p M Using orthonormality of the bases, the extended notation ωx results as [3] X ω X 1 ^...^ X p ¼ ð1þ m σ h X σ ð i1 Þx i 1 Xσ ip σ2σ p where σ is the number of permutations; Σ p is a set of permutations {1,2,...,p}; ðþ xi p

180 4.5 Exterior Derivatives 163 m σ ¼ σ mod 2, cf. Eq. (4.6); x i are the local coordinates in the cotangent space T p * M. 4.5 Exterior Derivatives Let ω be any k-form; η be any l-form in any N-dimensional space R N. The exterior derivative dω of the k-form ω is a (k + 1)-form [3 5]. ω 2 Λ k R N Λ k M * ) dω 2 Λ kþ1 R N The k-form ω is written in the dual bases as ω ¼ X X f i1...i k dx i1 ^ dx i2 ^...^ dx i k f I dx I i 1 <...<i k I where f I is a smooth function 2 R. The exterior derivative dω of the k-form ω is defined as dω ¼ d ¼ X I ¼ X I! X f I dx I ¼ X df I ^ dx I 2 Λ kþ1 R N I I df I ^ dx i1 ^ dx i2 ^...^ dx i k! X k f I x j dxj ^ dx i1 ^ dx i2 ^...^ dx i k j¼1 ð4:11þ Generally, the exterior derivative upgrades the differential form by one order. The exterior derivative has the following properties: dðω þ ηþ ¼ dω þ dη ð4:12aþ df I ¼ X i f I x idxi ð4:12bþ dðω ^ ηþ ¼ dω ^ η þð1þ ω ^ dη ð4:12cþ ddω ð Þ ¼ 0 : The Poincare Lemma ð4:12dþ Proof of Eq. (4.12c) Using Eq. (4.11) and the chain rule of differentiation, one obtains the exterior derivative of the wedge product of two differential forms.

181 164 4 Differential Forms dðω ^ ηþ ¼ d X ðω I η J Þdx I ^ dx J ¼ X d ω I η J dx I ^ dx J I, J I, J ¼ X X ðω I η J Þ I, x J i i dx i ^ dx I ^ dx J ¼ X X ω I I, x J i i η J þ η J x i ω I ¼ X X ω I x I i i dxi ^ dx I ^ X η J dx J þ J ð1þ kx ω I dx I ^ X X η J x I J i i dxi ¼ ðdω ^ ηþþð1þ k ðω ^ dηþ ðq:e:d: Þ dx i ^ ðdx I ^ dx J Þ ^ dx J Proof of the Poincaré Lemma Eq. (4.12d) Using Eqs. (4.11) and (4.12a c), the exterior derivative of any differential k-form ω is written as dω ¼ d ¼ X I X I! ω I dx I ¼ X! I X ω I x i dxi i d ω I dx I X ¼ dω I ^ dx I ^ dx i1 ^...^ dx i k ð Þ I ð4:13þ Using Eq. (4.13) and the chain rule of differentiation, the exterior derivative of dω is calculated as ddω ð Þ ¼ d X! X ω I x I i i dxi ^ dx I ¼ X! X 2 ω I I i, x j j x idxj ^ dx i ^ dx I ¼ X! X 2 ω I x I i<j j x i 2 ω I x i x j dx j ^ dx i ^ dx I ¼ 0 ðq:e:d: Þ fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼ 0 This equation is called the Poincaré Lemma. If f is any zero-form (a smooth function with k ¼ 0) and ω is any one-form, one obtains using Eqs. (4.12c) and (4.12d) dðf ^ dωþdðfdωþ ¼ df ^ dω þð1þ 0 f ^ ddω ð Þ ¼ df ^ dω fflffl{zfflffl} ¼0 ð4:14þ

182 4.5 Exterior Derivatives 165 Using Eq. (4.12b), Eq. (4.14) can be expressed as dðf ^ dωþ ¼ df ^ dω ¼ X i f x i dxi ^ dω ð4:15þ Some examples of differential forms and their exterior derivatives are given in the following section. 1. One-form (k ¼ 1) A one-form ω is given in R 3 as ω ¼ xydx þ xzdy þ xyzdz 2 Λ 1 R 3 Using Eq. (4.13), the exterior derivative of ω is calculated as dω ¼ dxy ð Þ^dx þ dxz ð Þ^dy þ dðxyzþ^dz ¼ xdy ^ dx þ ðzdx ^ dy þ xdz ^ dyþþðyzdx ^ dz þ xzdy ^ dzþ ¼xdx ^ dy þ zdx ^ dy xdy ^ dz þ yzdx ^ dz þ xzdy ^ dz ¼ ðx þ zþdx ^ dy þ xz ð 1Þdy ^ dz þ yzdx ^ dz 2 Λ 2 R 3 The exterior derivative of dω is calculated further using Eq. (4.5) as ddω ð Þ ¼ ddωd 2 ω ¼ dz ^ dx ^ dy þ ðz 1Þdx ^ dy ^ dz þ zdy ^ dx ^ dz ¼ ð1 þ z 1 zþdx ^ dy ^ dz 2 Λ 3 R 3 ¼ 0 ðq:e:d: Þ 2. Two-form (k ¼2) A two-form ω is given in R 3 as ω ¼ x 2 ðy þ zþdx ^ dy þ yx 2 þ 2z dy ^ dz þ 2xy 2 dx ^ dz 2 Λ 2 R 3 Using Eq. (4.13), the exterior derivative of ω is calculated as dω ¼ dx 2 ðy þ zþ dx ^ dy þ dyx 2 þ 2z dy ^ dz þ d 2xy 2 dx ^ dz ¼ x 2 dz ^ dx ^ dy þ 2xydx ^ dy ^ dz þ 4xydy ^ dx ^ dz ¼ x 2 þ 2xy 4xy dx ^ dy ^ dz ¼ xx ð 2yÞdx ^ dy ^ dz 2 Λ 3 R 3 Due to (dx i^ dx i ) ¼ 0 and using Eq. (4.13), one obtains Eq. (4.12d).

183 166 4 Differential Forms ddω ð Þ ¼ dxx ½ ð 2yÞdx ^ dy ^ dz ¼ dxx ½ ð 2yÞ^dx ^ dy ^ dz ¼ X ½xx ð 2yÞ x i i dx i ^ dx ^ dy ^ dz ¼ 0 ðq:e:d: Þ 3. Three-form (k ¼3) A three-form ω is given in R 3 as ω ¼ fðx; y; zþdx ^ dy ^ dz ¼ x 2 yz 3 dx ^ dy ^ dz 2 Λ 3 R 3 Using (dx i^ dx i ) ¼ 0 and Eq. (4.13), the exterior derivative of ω is calculated as dω ¼ dx 2 yz 3 dx ^ dy ^ dz ¼ dx 2 yz 3 ^ dx ^ dy ^ dz ¼ X ðx 2 yz 3 Þ x i i dx i ^ dx ^ dy ^ dz ¼ 0 for dx i ¼ dx, dy, dz Thus, ddω ¼ dð0þ ¼ 0 ðq:e:d: Þ 4.6 Interior Product The interior product or interior derivative degrades the differential form by one order. It is usually applied to exterior algebra of k-differential forms on a smooth N- dimensional manifold M. In fact, the interior product is a linear map of a k-form ω to the (k 1)-form i X ω in the vector fields (X 1,...,X k1 ) on the manifold M. In exterior algebra, the interior product is an operator for contracting vectors and differential forms to one-order lower forms. The interior product of any differential k-form ω on the manifold M is defined as i X : ω 2 Λ k T * p M! i X ω 2 Λ k1 T * p M ði X ωþðx 1 ;...; X k1 Þ ¼ ωðx; 1 ;...; X k1 Þi Xk1... i X1 i X ω ¼ X X ð4:16þ X i ω ii1 i k1 dx i1 ^...^ dx i k1 i 1 <::::<i k1 i

184 4.6 Interior Product 167 Using Einstein summation convention, the vector field X is written as X ¼ X i x i Xi i Let f be a smooth function (a zero-form), X be a vector field on M, ω be a one-form, η and ξ be k- and l-forms, respectively. The interior product has the following properties: i X f ¼ 0 i X ω ¼ ωðxþhω; Xi : inner product i X ðη ^ ξþ ¼ ði X ηþ^ξ þð1þ k η ^ ði X ξþ i X i Y η ¼i Y i X η ð4:17þ Furthermore, if α and β are one-forms; X and Y are the vector fields on the manifold M, one obtains from Eqs. (4.16) and (4.17) αðx; YÞ ¼ i Y i X α ¼ i Y ðαðxþþ ¼ 0 fflffl{zfflffl} 0form ð4:18aþ and ðα ^ β h i ÞðX; YÞ ¼ i Y ½i X ðα ^ βþ ¼ i Y i X α ^ β þð1þ 1 α ^ i X β ¼ i Y ½αðXÞ^β α ^ βðxþ ¼ i Y ½αðXÞβ βðxþα ¼ αðxþði Y βþβðxþði Y αþ ¼ αðxþβðyþβðxþαðyþ ð4:18bþ Therefore, ðα ^ β ÞðX; YÞ ¼ αðxþβðyþβðxþαðyþ ¼ hα; Xi hβ; Yi hβ; Xi hα; Yi ð4:19þ The Cartan s formula for any differential form ω is given in [4, 6] X ω ¼ i X dω þ di ð X ωþ ð4:20þ where ω is the k-form on M; X ω is the Lie derivative of ω, cf. Eq. (3.144a). Changing ω into a zero-form f and using the property of i X f ¼ 0, one obtains the relation between the Lie derivative and interior product of any function f X f Xf ¼ i X df þ di ð X f Þ ¼ i X df ð4:21þ

185 168 4 Differential Forms In general, the Cartan s formula (known as Weil s formula) in differential calculus is written as X ¼ di X þ i X d ¼ ½d; i X ð4:22þ 4.7 Pullback Operator of Differential Forms The pullback operator of differential forms is used in the transformation of coordinate variables from one manifold to another manifold. It is a smooth linear map of two mapping functions f and g. f : Xx ðþ2mr N! fðxþ ¼ yx ðþ2nr N g : fðxþ 2 N R N! gðfðxþþ ¼ gyx ð ðþþ 2 P R ) f * g : Xx ðþ2mr N! f * gx ð Þ ¼ gðfðxþþ 2 P R Let X (x) be a vector field on the manifold M. The pullback operator f*g of g by f in the vector field X(x) 2 M is defined as f * g : Xx ðþ2m! f * gx ð Þ ¼ gðfðxþþ ¼ gyx ð ðþþ 2 P ð4:23þ The operator f*gis called the pullback g by f since the mapping function g is pulled from N backwards to M (see Fig. 4.3). The function f maps the vector field X to f(x) ¼ y(x) that is further mapped by the function g to g( f(x)). In fact, the pullback operator f*g maps the vector field X 2 M to g o f(x) 2 P directly. An example of the pullback is given in the following section. Let ω be a one-form of y that is written as ω ¼ y 1 y 2 dy 1 þ y 2 2 y 3dy 2 þ y 1 y 2 y 3 dy 3 2 Λ 1 R 3 * f g go f ( X ) = g( f ( X )) f g X ( x) M f ( X ) = y( x) N g( f ( X )) = g( y( x)) P Fig. 4.3 Pullback operator f*g of g by f over M

186 4.7 Pullback Operator of Differential Forms 169 Let X(t) be a vector field (t, 2t, 3t) 2 R 3. The linear function f maps X to f(x) as f : Xt ðþ!fðxþ ¼ yt ðþ¼ ) dfðxþ ¼ dyðþ¼ t 8 < y 1 ¼ t y 2 ¼ 2t : 8 y 3 ¼ 3t < dy 1 ¼ dt dy 2 ¼ 2dt : dy 3 ¼ 3dt Using the above equation, the pullback of the one-form ω by f results as a one-form f * ωðxþ ¼ ωðfðxþþ ¼ ωðyt ðþþ ¼ tð2tþðdtþþð2tþ 2 ð3tþð2dtþþtð2tþð3tþð3dtþ ¼ 2t 2 ð21t þ 1Þdt 2 Λ 1 R 3 The pullback of differential forms has the following properties: f * ðω þ ηþ ¼ f * ω þ f * η; f * ðω ^η Þ ¼ f * ω ^ f * η; d f * ω ¼ f * ðdωþ: ð4:24þ where ω and η are the arbitrary k-forms; and f is the linear mapping function. The function g in the pullback mapping is replaced by a k-form ω that is written in R N as ω ¼ X J a J dy J ¼ X J a J dy j 1 ^...^ dy j k 2 Λ k R N The pullback of the k-form ω by the mapping function f results as f * ω ¼ f * ¼ X J! X a J dy J ¼ X f * a J f * dy j 1 ^...^ dy j k J J ajo f f * dy j 1 ^...^ f * dy j k ¼ X J ¼ X J ¼ X J ajo f df j 1 ^...^ df j k i 1,..., i k x i ajo f! X j f 1 i 1 x i dx i 1 ^...^ X! j f k 1 i k x i dx i k k ajo f X j f 1 x i f j k 1 x i dx i1 ^...^ dx i k k x i

187 170 4 Differential Forms The second term on the RHS of the above equation is in fact the Jacobian J. Thus, J X i 1,..., i k j f 1 x i 1 f j k x i k ¼ yj 1 ;...; y j ð k Þ ðx i 1 ;...; x i k Þ f * ω ¼ ω o f ¼ J X J J X I a I dx I 2 Λ k R N ð4:25þ ajo f dx i1 ^...^ dx i k 4.8 Pushforward Operator of Differential Forms Figure 4.4 shows the mapping scheme of the pushforward operator in an N- dimensional space R N. The linear smooth mapping function f creates the vector field y 2 V from any vector field x 2 U: f : x 2 U R N! y ¼ fðþ2v x R N ) x ¼ f 1 ðþ y ð4:26þ The linear function f * maps the dual space T X * R N of X to the dual space T y * R N of the vector field y: dfðþf x * : Xx ðþ2t * X RN! f * X 2 T * y RN ð4:27þ x U N R f y = f ( x) V N R X f * X X * ( x ) T R X N df ( x) f * f X T * R y * N Fig. 4.4 Pushforward operator f * X of X by f

188 4.9 The Hodge Star Operator 171 The operator f * X is called the pushforward operator of X by the mapping function f that is defined as f * Xx ðþdfðþx x f 1 ðþ y 2 T * y R N ð4:28þ The pullback of a one-form ω by f for any vector field X 2 T * X R N is defined as [3] f * ω Xx ðþ¼ωf * Xy ðþ ð4:29aþ Therefore, the pullback of a k-form ω (k > 1) by f for the bundle of vector fields (X i1,..., X ik ) 2 T * M results from Eq. (4.29a). f * ω ðx i1 ^...^ X ik Þ ¼ ω ðf * X i1 ^...^ f * X ik Þ ð4:29bþ 4.9 The Hodge Star Operator The Hodge star operator (star operator) is used to carry out various operations, such as gradient, div, and curl in exterior algebra, in which the coordinates are not taken into account. The Hodge * operator is a counterpart of Nabla operator in vector calculus in linear algebra that considers the coordinates in calculations. The star operator is a linear map from an exterior k-form bundle to another exterior (N k)-form bundle on the N-dimensional manifold M [5, 6] * : Λ k R N! Λ Nk R N ; 0 k N ð4:30þ where ^k(r N ) is the differential k-form bundle that consists of all differential k- form spaces. Let ω be any k-form that is written for I ¼ (i 1,...,i k ) with 1 i 1 <...< i k N as ω ¼ X I f I dx I ¼ X I f I dx i1 ^...^ dx i k 2 Λ k R N ð4:31þ The Hodge * operator maps the k-form ω to the Hodge dual *ω in the exterior (N k)-form bundle. The Hodge dual *ω is a pseudo (N k)-form that is written as * : dx I 2 Λ k R N! *dx I 2 Λ Nk R N ) *ω ¼ X f I *dx I 2 Λ Nk R N ð4:32þ I Similarly, the Hodge dual elementary (N k)-form is defined as the Hodge * operator of the elementary form dx I as

189 172 4 Differential Forms *dx I ¼ dx i kþ1 ^ dx i kþ2 ^...^ dx i N fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ðnkþ times ð4:33þ Substituting Eq. (4.33) into Eq. (4.32), the Hodge dual of the k-form ω is written as *ω ¼ X I X I, J f I *dx I ¼ X f I dx i kþ1 ^ dx i kþ2 ^...^ dx i N I f I dx J 2 Λ Nk R N ð4:34þ where I ¼ (i 1,...,i k ) with 1 i 1 <...< i k N and J ¼ (i k+1,...,i N ) with 1 i k+1 <...< i N N. Obviously, the wedge product of the elementary k-form and its Hodge dual is the elementary N-form. dx I ^ *dx I ¼ dx i1 ^...^ dx i k ð Þ^ dx i kþ1 ^...^ dx i N ð Þ ¼ dx i1 ^ dx i2 ^...^ dx i N dx N ð4:35þ Some examples of the Hodge * operators of the dual bases (dx,dy,dz) inr 3 are given in the following section. *dx ¼ dy ^ dz ) *dy^ ð dzþ ¼ dx *dy ¼ dz ^ dx ) *dz^ ð dxþ ¼ dy *dz ¼ dx ^ dy ) *dx^ ð dyþ ¼ dz *1 ¼ dx ^ dy ^ dz ) *dx^ ð dy ^ dzþ ¼ 1 ð4:36aþ Proof Using Eqs. (4.5) and (4.35), one writes dx ^ ð*dxþ ¼ dx ^ ðdy ^ dzþ 2 Λ 3 R 3 ) *dx ¼ dy ^ dz; dy ^ ð*dyþ ¼ dx ^ dy ^ dz ¼dy ^ ðdx ^ dzþ 2 Λ 3 R 3 ) *dy ¼dx ^ dz ¼ dz ^ dx; dz ^ ð*dzþ ¼ dx ^ dy ^ dz ¼ dz ^ ðdx ^ dyþ 2 Λ 3 R 3 ) *dz ¼ dx ^ dy; ðdx ^ dy ^ dzþ^*dx^ ð dy ^ dzþ ¼ dx ^ dy ^ dz ) *dx^ ð dy ^ dzþ ¼ 1: In general, Eq. (4.36a) can be expressed in an N-dimensional space R N. *dx i 1 ¼ dx i2 ^...^ dx i N ; *dx i1 ^ dx i2 ^...^ dx i N ð Þ ¼ 1; *1 ¼ dx i1 ^ dx i2 ^...^ dx i N ð4:36bþ

190 4.9 The Hodge Star Operator Star Operator in Vector Calculus and Differential Forms Let f be a function of x, y, and z in R 3. The gradient of a function f(x,y,z) is written in a one-form as df ¼ f x i dxi ¼ f f f dx þ dy þ dz ¼ f dr x y z ð4:37aþ Using the Hodge duals, the Laplacian of f(x,y,z) is calculated as Δf ¼ f ¼ *d ð*df Þ ð4:37bþ Similarly, let v be any vector (one-form) in R 3. The vector v (v 1,v 2,v 3 ) corresponds to a one-form ω v that can be expressed as v $ ω v v 1 dx þ v 2 dy þ v 3 dz ¼ v dr ð4:38þ Using Eq. (4.14), the exterior derivative of the one-form ω v is written as dω v ¼ dv ð 1 dx þ v 2 dy þ v 3 dzþ ¼ ðdv 1 ^ dxþþðdv 2 ^ dyþþðdv 3 ^ dzþ The first term on the RHS of dω v is calculated as dv 1 ^ dx ¼ v 1 x dx þ v 1 y dy þ v 1 z dz ^ dx ¼ v 1 y dy ^ dx þ v 1 dz ^ dx z ¼ v 1 dx ^ dy þ v 1 dz ^ dx y z Analogously, the second and third terms on the RHS of dω v result as dv 2 ^ dy ¼ v 2 dx ^ dy v 2 dy ^ dz x z dv 3 ^ dz ¼ v 3 dz ^ dx þ v 3 dy ^ dz x y Hence, the exterior derivative of the one-form ω v is calculated as dω v ¼ dv ð 1 dx þ v 2 dy þ v 3 dzþ ¼ v 3 y v 2 dy ^ dz þ v 1 z z v 3 x dz ^ dx þ v 2 x v 1 y dx ^ dy

191 174 4 Differential Forms Thus, the Hodge dual of the exterior derivative dω v results as *dω v ¼ v 3 y v 2 dx þ v 1 z z v 3 x dy þ v 2 x v 1 dz y ð4:39aþ The curl of v is calculated in R 3 as v ¼ v 3 y v 2 z h i T ð4:39bþ v 1 z v 3 x v 2 x v 1 y Using Eqs. (4.39a) and (4.39b), one obtains the relation between the Hodge dual *dω v and curl v *dω v ¼ ð vþdr ð4:40þ To calculate the divergence of a vector, a two-form η v is defined using the vector components as η v v 1 dy ^ dz þ v 2 dz ^ dx þ v 3 dx ^ dy The Hodge dual of the one-form ω v in Eq. (4.38) results as *ω v v 1 ð*dx Þþv 2 ð*dyþþv 3 ð*dzþ Using Eq. (4.36a), the two-form η v equals the Hodge dual *ω v. η v ¼ *ω v ð4:41þ The exterior derivative dη v is calculated as dη v ¼ dv ð 1 dy ^ dz þ v 2 dz ^ dx þ v 3 dx ^ dyþ ð4:42þ Using Eq. (4.14), the RHS of Eq. (4.42) is rewritten as RHSdðv 1 dy ^ dz þ v 2 dz ^ dx þ v 3 dx ^ dyþ ¼ dv 1 ^ ðdy ^ dzþþdv 2 ^ ðdz ^ dxþþdv 3 ^ ðdx ^ dyþ ð4:43þ Using the property of the wedge product (dx ^ dx) ¼ 0, the first term on the RHS in Eq. (4.43) is calculated as dv 1 ^ ðdy ^ dzþ ¼ v 1 x dx þ v 1 y dy þ v 1 z dz ¼ v 1 dx ^ dy ^ dz x ^ ðdy ^ dzþ

192 4.9 The Hodge Star Operator 175 Analogously, the second and third terms on the RHS in Eq. (4.43) result as dv 2 ^ ðdz ^ dxþ ¼ v 2 dx ^ dy ^ dz; y dv 3 ^ ðdx ^ dyþ ¼ v 3 dx ^ dy ^ dz: z Therefore, the exterior derivative dη v results as dη v ¼ v 1 x þ v 2 y þ v 3 dx ^ dy ^ dz z Using Eqs. (4.36a) and (4.41), the Hodge dual dη v is written as *dη v ¼ v 1 x þ v 2 y þ v 3 *dx^ ð dy ^ dzþ z ¼ v 1 x þ v 2 y þ v 3 z ¼ *d ð *ω vþ Furthermore, the divergence of v is calculated as v ¼ v 1 x þ v 2 y þ v 3 z Therefore, the relation between *dω v and div v results as *d ð*ω v Þ ¼ v ð4:44þ Table 4.3 shows the overview of gradient, curl, and divergence operators that are displayed in vector calculus and differential forms Star Operator and Inner Product Let ω be any k-form 2^k(R N )inann-dimensional space R N. The Hodge * operator *ω is the (N k)-form 2^Nk (R N ). Table 4.3 Gradient, curl, and divergence in differential forms Operators Variable Grad Curl Divergence Vector calculus f grad! f Laplacian! Δf v curl! v! div v Differential forms f! d df! *d Δf ¼ *d ð*df Þ ω v! *d *dω v! *d *d ð*ω v Þ

193 176 4 Differential Forms If η is any k-form 2^k(R N ), the inner product of two k-forms ω and η is defined so that [6, 7] η ^ *ω ¼ hω; ηidx N ¼ hω; ηidx i1 ^...^ dx i N 2 Λ N R N ð4:45þ The inner product of two orthonormal dual bases is written in Kronecker delta as dx i, dx j ¼ δ j i ¼ 0 for i 6¼ j 1 for i ¼ j ð4:46þ Let dx I and dx J be two k-forms of the dual bases. The inner product of dx I and dx J is defined as hdx I, dx J i ¼ X D ED E ð1þ m σ dx i 1, dx σ ð j 1Þ dx i 2, dx σ ð j 2Þ σ ð4:47þ dx i k, dx σ ð j kþ 2 R where m σ ¼ σ mod2; σ is the number of permutations on k elements. For any i 6¼ j in an N-dimensional space R N, one obtains using Eqs. (4.45) and (4.46) dx i ^ *dx j ¼ hdx j, dx i idx N ¼ δ j i dx i1 ^...^ dx i N ð Þ ¼ 0 Some examples of elementary two- and three-forms are given here to demonstrate the inner product of the elementary k-forms. For elementary two-forms (k ¼ 2), there are two terms of the inner product; they result from k!. Using Eq. (4.47), the inner product is written as dx i1 ^ dx i 2, dx j 1 ^ dx j 2 ¼ ð1þ 0 dx i 1, dx j 1 dx i 2, dx j 2 þ ð1þ 1 dx i 1, dx j 2 dx i 2, dx j 1 ¼ dx i 1, dx j 1 dx i 2, dx j 2 dx i 1, dx j 2 dx i 2, dx j 1 ð4:48þ The terms of the inner product of two elementary 2-forms result from Table 4.4. For elementary three-forms (k ¼ 3), there are six terms of the inner product; they result from k!. These terms with their signs are given in Table 4.5. The contravariant indices of any term of the elementary forms are displayed in each line, in which the same indices of i and j (¼1, 2, 3) could occur only two times. The number of permutations σ of the following terms results from the order of the first term of the

194 4.9 The Hodge Star Operator 177 Table 4.4 Inner product of the elementary 2-forms i j i j 2! ¼ 2 terms Permutation σ þ < i 1, j 1 >< i 2, j 2 > σ ¼ 0! m σ ¼ < i 1, j 2 >< i 2, j 1 > σ ¼ 1! m σ ¼ 1 Table 4.5 Inner product of the elementary 3-forms i j i j i j 3! ¼ 6 terms Permutation σ þ < i 1, j 1 >< i 2, j 2 >< i 3, j 3 > σ ¼ 0! m σ ¼ < i 1, j 2 >< i 2, j 1 >< i 3, j 3 > σ ¼ 1! m σ ¼ þ < i 1, j 3 >< i 2, j 1 >< i 3, j 2 > σ ¼ 2! m σ ¼ < i 1, j 3 >< i 2, j 2 >< i 3, j 1 > σ ¼ 1! m σ ¼ þ < i 1, j 2 >< i 2, j 3 >< i 3, j 1 > σ ¼ 2! m σ ¼ < i 1, j 1 >< i 2, j 3 >< i 3, j 2 > σ ¼ 1! m σ ¼ 1 Table 4.6 Summary of the operators using in differential forms Operators in R N Symbols Forms Results Orders Wedge product (exterior product) ^ k-form ω; p-form η ω^ η k + p Interior product (Interior derivative) i x k-form ω i x ω k 1 Exterior derivative d k-form ω dω k + 1 Pullback operator f* function f; k-form ω f * ω; ω 0 f k Hodge star operator * k-form ω * ω N k elementary forms. Namely, the second term of the elementary forms is given by one interchange of the indices 1 and 2; therefore, the number of permutation σ ¼ 1. Using Eq. (4.47), the inner product of two elementary 3-forms is written from Table 4.5 as dx i1 ^ dx i2 ^ dx i 3, dx j 1 ^ dx j 2 ^ dx j h 3 i ¼ þhdx i 1, dx j 1ihdx i 2, dx j 2ihdx i 3, dx j 3i hdx i 1, dx j 2ihdx i 2, dx j 1ihdx i 3, dx j 3iþ hdx i 1, dx j 3ihdx i 2, dx j 1ihdx i 3, dx j 2i hdx i 1, dx j 3ihdx i 2, dx j 2ihdx i 3, dx j 1iþ hdx i 1, dx j 2ihdx i 2, dx j 3ihdx i 3, dx j 1i hdx i 1, dx j 1ihdx i 2, dx j 3ihdx i 3, dx j 2i ð4:49þ Table 4.6 gives the overview of the discussed operators in differential forms of exterior algebra Star Operator in the Minkowski Spacetime The Minkowski spacetime is applied to the special relativity, in which time t is an extra dimension besides the space coordinates of x, y, and z. As a result, the Minkowski is called the four-dimensional spacetime (t, x, y, z) 2 R 4 (cf. Chap. 5).

195 178 4 Differential Forms In the following section, the Hodge * operator will be used in the Minkowski spacetime [6, 7]. In gravitation, relativity theory, and cosmology, the light speed c is defined as c 2 1, in which c is considered as a constant. The distance ds between two arbitrary points in the Minkowski spacetime for special relativity is written as ds 2 ¼ g μv dx μ dx v ¼ dt 2 dx 2 dy 2 dz 2 ð4:50þ The Minkowski metric with four spacetime coordinates (t, x, y, z) is expressed as g ¼ g μv ¼ B C A ð4:51þ The elementary form of the Minkowski spacetime is defined as dx I ¼ dt ^ dx ^ dy ^ dz 2 Λ 4 R 4 ð4:52þ Note that the inner products of the orthonormal dual bases of the four-dimensional Minkowski spacetime result from Eqs. (4.50) and (4.51) hdt; dti ¼þ1; hdx; dxi ¼ hdy; dyi ¼ hdz; dzi ¼1; hdt, dx i i ¼ hdx i, dx j i ¼ 0 if i 6¼ j: ð4:53aþ Using Eqs. (4.48) and (4.53a), the inner products of 2-forms of the dual bases result as hdt ^ dx, dt ^ dxi ¼ hdt ^ dtihdx ^ dxihdt ^ dxihdx ^ dxi ¼ ðþ1þð1þð0þð0þ ¼ 1 ) hdt ^ dy, dt ^ dyi ¼ hdt ^ dz, dt ^ dzi ¼1 hdx ^ dy, dx ^ dyi ¼ hdx ^ dxihdy ^ dyi hdx ^ dyihdy ^ dxi ¼ ð1þð1þð0þð0þ ¼ þ1 ) hdx ^ dz, dx ^ dzi ¼ hdy ^ dz, dy ^ dzi ¼þ1 ð4:53bþ Using Eqs. (4.49), (4.53a) and (4.53b), the inner products of 3-forms of the dual bases result as hdt ^ dx ^ dy, dt ^ dx ^ dyi ¼ þ1; hdt ^ dy ^ dz, dt ^ dy ^ dzi ¼þ1; hdt ^ dx ^ dz, dt ^ dx ^ dzi ¼þ1; hdx ^ dy ^ dz, dx ^ dy ^ dzi ¼1: ð4:53cþ

196 4.9 The Hodge Star Operator 179 Using the inner product of two k-forms in Eq. (4.45), the Hodge dual *dt is calculated as dt ^ *dt ¼ hdt; dtidx N ¼ hdt; dtidt ^ dx ^ dy ^ dz ) dt ^ *dt ð Þ ¼ dt ^ dx ^ dy ^ dz hdt; dti ð4:54þ ) dt ^ *dt ð Þ ¼ dt ^ dx ^ dy ^ dz ðþ1þ ð Þ Therefore, the Hodge dual of the 1-form dt results as *dt ¼ dx ^ dy ^ dz ð4:55þ Using Eq. (4.54), one obtains three Hodge duals of 1-forms of the dual bases. *dx ¼ dt ^ dy ^ dz *dy ¼ dt ^ dz ^ dx *dz ¼ dt ^ dx ^ dy ð4:56þ Six Hodge duals of 2-forms of the dual bases are computed as [7, 8] *dt^ ð dxþ ¼ dy ^ dz; *dt^ ð dyþ ¼ dz ^ dx; *dt^ ð dzþ ¼ dx ^ dy; *dx^ ð dyþ ¼ dz ^ dt; *dx^ ð dzþ ¼ dt ^ dy; *dy^ ð dzþ ¼ dx ^ dt: ð4:57þ Four Hodge duals of 3-forms of the dual bases are computed as [7, 8] *dx^ ð dy ^ dzþ ¼ dt; *dt^ ð dx ^ dyþ ¼ dz; *dt^ ð dz ^ dxþ ¼ dy; *dt^ ð dy ^ dzþ ¼ dx: ð4:58þ According to Eq. (4.45), the Hodge dual of the 4-form of the dual bases is computed as *dt^ ð dx ^ dy ^ dzþ ¼ 1 ) *1 ¼ dt ^ dx ^ dy ^ dz ð4:59þ

197 180 4 Differential Forms References 1. Yau, S.T., Nadis, S.: The Shape of Inner Space. Basic Books, London (2012) 2. Chern, S.S., Chen, W.H., Lam, K.S.: Lectures on Differential Geometry. World Scientific Publishing, Singapore (2000) 3. Darling, R.W.R.: Differential Forms and Connections. Cambridge University Press, Cambridge (2011) 4. Cartan, H.: Differential Forms. Dover Publications, London (2006) 5. Dray, T.: Differential Forms and the Geometry of General Relativity. CRC Press, Taylor & Francis Group, Boca Raton, FL (2015) 6. Renteln, P.: Manifolds, Tensors, and Forms. Cambridge University Press, Cambridge (2014) 7. Garrity, T.A.: Electricity and Magnetism for Mathematicians. Cambridge University Press, Cambridge (2015) 8. Bachman, D.: A Geometric Approach Differential Forms, 2nd edn. Birkhäuser, Basel (2012)

198 Chapter 5 Applications of Tensors and Differential Geometry 5.1 Nabla Operator in Curvilinear Coordinates Nabla operator is a linear map of an arbitrary tensor into an image tensor in N- dimensional curvilinear coordinates. The Nabla operator can be usually defined in N-dimensional Cartesian coordinates {x i } using Einstein summation convention as e i x i for i ¼ 1, 2,..., N ð5:1þ According to Eq. (2.12), the relation between the bases of Cartesian and general curvilinear coordinates can be written as g i ¼ r u i ¼ r x j x j u i ¼ e j x j u i ð5:2þ Multiplying Eq. (5.2) by g i e j, one obtains the basis of Cartesian coordinates expressed in the curvilinear coordinate basis. e j ¼ g i xj u i ð5:3þ Using chain rule of coordinate transformation, the Nabla operator in the general curvilinear coordinates {u i } results from Eq. (5.3) [1, 2]. Springer-Verlag Berlin Heidelberg 2017 H. Nguyen-Schäfer, J.-P. Schmidt, Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers, Mathematical Engineering, DOI / _5 181

199 182 5 Applications of Tensors and Differential Geometry e i u j u j x i ¼ g k xi u j u k u j x i ¼ g k x i u j u j u k x i ¼ g k u j u j u k ¼ g k u j δ j k ¼ g k u k ð5:4þ Thus, the Nabla operator can be written in the curvilinear coordinates {u i }using Einstein summation convention. g i u i ¼ gi i for i ¼ 1, 2,..., N ð5:5þ 5.2 Gradient, Divergence, and Curl Let φ be a velocity potential that exists only in a vortex-free flow. The velocity potential can be defined as φ ¼ ð vdx ð5:6þ Differentiating Eq. (5.6) with respect to x, the velocity component results in v ¼ φ x ð5:7þ The velocity vector v can be written in the general curvilinear coordinates {u i }with the contravariant basis. v ¼ v i g i ¼ φ u i gi ð5:8þ Gradient of an Invariant The gradient of an invariant φ (function, zero-order tensor) can be defined by

200 5.2 Gradient, Divergence, and Curl 183 Table 5.1 Essential Nabla operators Operand () Function f 2 R (0th order tensor) Vector v 2 R N (1st order tensor) Operator Grad ðþ Div ðþ Curl ðþ Laplacian Δ () Vector 2 R N Scalar 2 R 2nd order tensor 2 R N R N Scalar 2 R Vector 2 R N Vector 2 R N 3rd order tensor Vector 2 R N R N R N 2 R N Tensor T 2 R N R N (2nd order tensor) Order of results One order higher One order lower Order unchanged Order unchanged Grad φ ¼ φ ¼ g i φ u i φ,i g i ¼ v i g i ¼ v ) φ ¼ v ¼ φ u 1 g1 þ φ u 2 g2 þ φ ð5:9þ u 3 g3 Obviously, gradient of a function is a vector (cf. Table 5.1) Gradient of a Vector The gradient of a contravariant vector v can be calculated using the derivative of the covariant basis g j, as given in Eq. (2.158). Grad v ¼ v ¼ g i v j g j u i v j g j ¼ g i u i ¼ g i v j,i g j þ v j g j,i ¼ v,i k g k þ v j Γij k g k g i ¼ v, k i þ v j Γij k g k g i v k i g k g i ð5:10þ Analogously, the gradient of a covariant vector v can be written using the derivative of the contravariant basis g j in Eq. (2.189).

201 184 5 Applications of Tensors and Differential Geometry Grad v ¼ v ¼ g i v jg j u i ¼ g i ¼ v j, i g j þ v j g j,i v k,i v j Γ j ik g k g i ¼ v k,i g k v j Γ j ik gk g i v k j i g k g i ð5:11þ Obviously, grad of a vector is a second-order tensor; grad of a second-order tensor is a third-order tensor (cf. Table 5.1) Divergence of a Vector Let v be a vector in the curvilinear coordinates {u i }; it can be written in the covariant basis g j. The divergence of v can be defined by v ¼ v j g j Div v ¼ v ¼ g i v j g j u i v ¼ g i u i ¼ g i v j,i g j þ v j g j, i g i i v ð5:12þ ð5:13þ Using Eq. (2.158), the derivative of the covariant basis g j results in g j, i ¼ Γ k ji g k ¼ Γ k ij g k Substituting Eq. (5.14) into Eq. (5.13), one obtains the divergence of v. v ¼ g i v j, i g j þ v j Γij k g k ¼ g i v, k i g k þ v j Γij k g k ¼ g i v,i k þ v j Γij k g k ¼ v,i k þ v j Γij k g i g k v k jiδk i ¼ v,i i þ v j Γij i ¼ v i ji ð5:14þ ð5:15aþ According to Eq. (2.240), the second-kind Christoffel symbol can be rewritten as

202 5.2 Gradient, Divergence, and Curl 185 ij ¼ p 1 ffiffi p ffiffi g g u j ¼ ln pffiffiffi g u j Γ i ð5:15bþ Substituting Eq. (5.15b) into Eq. (5.15a), the divergence of v can be expressed in v ¼ 1 pffiffiffi g ¼ p 1 ffiffiffi g pffiffi v i J g þ vi ui u i p ffiffiffi g v i u i ¼ 1 pffiffiffi pffiffi g v i ð5:15cþ g,i Analogously, the covariant vector v can be written in the contravariant basis g j. v ¼ v j g j ð5:16þ The divergence of v can be derived using the contraction law and derivative of the basis g j in Eq. (2.189). v ¼ g i v jg j u i ¼ g i v j,i g j þ v j g j, i ¼ g i v k,i g k v j Γ j ik gk ¼ v k,i v j Γ j ik g i g k v ki j g i g k ¼ v ki j g ik ¼ v m jig mk g ik ¼ v m jiδm i ¼ vi ji ð5:17þ Obviously, divergence of a vector is a scalar (cf. Table 5.1). Some useful abbreviations are listed as follows: Divergence of the contravariant vector v: v g i i v ¼ g i i v j g j ¼ v, i i þ v j Γij i vi ji ¼ p 1 ffiffiffi p ffiffi g v i g u i ¼ 1 pffiffiffi pffiffiffi g v i g, i ð5:18aþ Divergence of the covariant vector v: v ¼ g i i v ¼ g i i v j g j ¼ v k, i v j Γ j ik g ki ¼ v ki j g ki ð5:18bþ

203 186 5 Applications of Tensors and Differential Geometry Covariant derivative of the contravariant vector component: v k ji v, k i þ v j Γij k ¼ vk u i þ v j Γij k ð5:19aþ Covariant derivative of the covariant vector component: v ki j v k, i v j Γ j ki ¼ v k u i v jγ j ki Covariant derivative of the contravariant vector v with respect to u i : i v ¼ v,i k þ v j Γij k g k ¼ v k jig k ð5:19bþ ð5:20aþ Covariant derivative of the covariant vector v with respect to u i : i v ¼ v k,i v j Γ j ik g k ¼ v ki j g k ¼ v m jig mk g k ð5:20bþ Divergence of a Second-Order Tensor Let T be a contravariant tensor in the curvilinear coordinates {u i }; it can be written in the covariant bases g i and g j. T ¼ T ij g i g j ð5:21þ The divergence of T can be calculated from T ¼ g k T ij g u k T ¼ i g j gk u k ð5:22þ ¼ g k T ij,k g i g j þ Tij g i, k g j þ T ij g i g j,k Using Eq. (2.158), the derivative of the covariant basis g i results in g i,k ¼ Γ m ik g m ¼ Γ m ki g m ; g j,k ¼ Γ n jk g n ¼ Γ n kj g n Interchanging the indices, the divergence of a contravariant second-order tensor T becomes

204 5.2 Gradient, Divergence, and Curl 187 T ¼ T ij,k g i g j þ Γ km i Tmj g i g j þ Γ j km Tim g i g j g k ¼ T ij, k δ ð5:23þ i k þ Γkm i Tmj δi k þ Γ j km Tim δi g k j Equation (5.23) can be written in the covariant basis g j at k ¼ i. T ¼ T ij, k þ Γ km i Tmj þ Γ j km Tim δi k g j ¼ T ij, i þ Γ im i Tmj þ Γ j im Tim g j ð5:24aþ T ij jig j Using Eq. (2.240), the covariant derivative of the tensor component T ij with respect u i on the RHS of Eq. (5.24a) can be expressed in T ij ji ¼ T ij, i þ Γ im i Tmj þ Γ j im Tim ¼ Tij u i þ T mj ¼ T im Γ j im þ p 1 ffiffiffi g ¼ T ik Γ j ik þ p 1 ffiffi g 1 p p ffiffiffi g ffiffiffi g u m þ T im Γ j im pffiffi T ij g u i þ T ij pffiffi g u i p ffiffiffi g T ij u i ¼ T ik Γ j ik þ 1 pffiffiffi g pffiffi g T ij, i ð5:24bþ Therefore, T ¼ T ij jig j ¼ T ik Γ j ik þ 1 pffiffi pffiffiffi g T ij g, i g j ð5:24cþ Interchanging the indices, the divergence of a covariant second-order tensor T can be written as T ¼ g k T ijg i g j u k ¼ T ij,k g i g j þ T ij g,k i gj þ T ij g i g j, k g k ¼ T ij,k g i g j T ij Γkm i gm g j T ij Γ j km gi g m g k ð5:25þ ¼ T ij,k T mj Γki m T imγkj m g i g j g k ¼ T ij j k g jk g i Furthermore, the divergence of a mixed second-order tensor T results as the same way at k ¼ i.

205 188 5 Applications of Tensors and Differential Geometry T T ¼ g k j igj g i u k ¼ Tj, i k þ Γ km i T m j Γjk m T m i ¼ Tj, i i þ Γ im i T j m Γji m T m i Tj i jig j ¼ Tj i jig kj g k δ k i gj g j ð5:26aþ Using Eq. (2.240), the covariant derivative of the mixed tensor component with respect to u i on the RHS of Eq. (5.26a) can be written in Tj i ji ¼ Tj,i i þ Γ im i T j m Γij m T m i ¼ 1 pffiffiffi T i p j pffiffi g g u i þ T j i ffiffiffi g u i ¼ p 1 ffiffi g pffiffiffi g T i j,i T i k Γ k ij! Γ m ij T i m ð5:26bþ Therefore, T ¼ Tj i jig j ¼ Tj i jig kj g k ¼ 1 pffiffi pffiffi g T i g T i j k Γ ij k,i g kj g k ð5:26cþ These results prove that the divergence of a second-order tensor T, such as the stress tensor Π or deformation tensor D results in a first-order tensor which is a vector in the curvilinear coordinates {u i }. Obviously, divergence of a second-order tensor is a vector (cf. Table 5.1) Curl of a Covariant Vector Let v be a covariant vector in the curvilinear coordinates {u i }; it can be written in the contravariant basis g j. v ¼ v j g j ð5:27þ The curl (rotation) of v can be defined by

206 5.2 Gradient, Divergence, and Curl 189 Curl v Rot v v ¼ g i u i v jg j ¼ g i v jg j u i ¼ g i v j,i g j þ v j g j,i ¼ v j, i ðg i g j Þþv j g i g j,i ð5:28þ Using Eq. (2.189), the derivative of the contravariant basis g j results in g j,i ¼Γ j ik gk Substituting Eq. (5.29) into Eq. (5.28), one obtains the curl of v. v ¼ v j,i g i g j þ vj g i g j,i ¼ ^ε ijk v j,i g k v j Γ j ik gi g k ð5:29þ ð5:30þ ¼ ^ε ijk v j,i g k ^ε ikm v j Γ j ik g m where the contravariant permutation symbols can be defined as (cf. Appendix A). 8 þp 1 ffiffiffi if ði, j, kþ is an even permutation >< g ^ε ijk ¼ p 1 ffiffiffi if ði, j, kþ is an odd permutation >: g 0 if i ¼ j, ori ¼ k; or j ¼ k ð5:31þ However, the second term in RHS of Eq. (5.30) vanishes due to the symmetric Christoffel symbols with respect to the indices of i and k, and the anticyclic permutation property with respect to i, k, and m. ^ε ikm v j Γ j ik g m ¼ v j p ffiffi Γ j ik Γ j ki g m g p ¼ v j ffiffi g Therefore, the curl of v in Eq. (5.30) becomes Γ j ik Γ j ik g m ¼ 0 ð5:32þ v ¼ ^ε ijk v j, i g k ¼ ^ε ijk v j u i g k ð5:33þ Obviously, curl of a vector is a vector (cf. Table 5.1).

207 190 5 Applications of Tensors and Differential Geometry 5.3 Laplacian Operator Laplacian operator is a linear map of an arbitrary tensor into an image tensor in N- dimensional curvilinear coordinates Laplacian of an Invariant Laplacian of an invariant φ (function, zeroth-order tensor) is the divergence of grad φ. Using Eq. (5.9), this expression can be written in Div ðgrad φþ φ ¼ 2 φ Δφ ð5:34þ Substituting the gradient φ of Eq. (5.9) into Eq. (5.34), one obtains the Laplacian Δφ. Δφ φ ¼ φ, k g k ¼ vk g k ¼ g l ð5:35þ u l φ,k g k ¼ g l φ,kl g k þ φ, k g,l k Using Eq. (2.189), the derivative of the contravariant basis g j results in g k, l ¼Γ k lm gm ð5:36þ Inserting Eq. (5.36) into Eq. (5.35) and using Eq. (5.19b), the Laplacian of φ can be computed as Δφ ¼ 2 φ ¼ g l φ,kl g k þ φ, k g,l k ¼ φ, kl g k φ,k Γ k ¼ φ, kl g k φ,m Γlk m gk g l ¼ φ, kl φ,m Γlk m g k g l ¼ φ, kl φ,m Γkl m g kl lm gm g l ð5:37þ The covariant vector components and their derivatives with respect to u k and u l are defined as φ, k ¼ φ u k ¼ v k; φ, m ¼ φ u m ¼ v m; φ,kl ¼ 2 φ u k u l ¼ v k,l ) Δφ ¼ v k,l v m Γkl m g kl v kl j g kl ð5:38þ Obviously, Laplacian of a function is a scalar (cf. Table 5.1).

208 5.3 Laplacian Operator Laplacian of a Contravariant Vector Laplacian of a contravariant vector (first-order tensor) is the divergence of grad v that can be computed as [3] DivðGrad v Þ ¼ Δv v ¼ 2 v ¼ v k jl, m v k p Γ p lm þ vp jlγpm k g lm g k ð5:39þ v k jlmg lm g k According to Eq. (5.39), Laplacian of a vector is a vector (cf. Table 5.1). The second covariant derivative of the contravariant vector component v k in Eq. (5.39) can be defined as v k jlm v k jl,m v k p Γ p lm þ vp jlγpm k ð5:40þ where v k jl, m ¼ v k jl, m v,lm k þ v,m n Γ nl k þ vn Γnl,m k ð5:41þ v,p k þ vn Γnp k ð5:42þ v k p v p jl v p,l þ vn Γ p nl ð5:43þ The vector triple product gives the relation of a ðb c Þ ¼ ba ð cþða bþc ð5:44þ Thus, Eq. (5.44) can be rewritten in the curl identity of the vector v is set into the position of the vector c, cf. Appendix C. ð v Þ ¼ ð vþð Þv ¼ ð vþ 2 v ð5:45þ The Laplacian of a vector v results from Eq. (5.44) as Δv 2 v ¼ ð Þv ¼ ð vþ, ð vþ ¼ ð vþ ð vþ, Laplacianv DivðGradvÞ ¼ GradðDivvÞCurlðCurlvÞ ð5:46þ

209 192 5 Applications of Tensors and Differential Geometry 5.4 Applying Nabla Operators in Spherical Coordinates Spherical coordinates (ρ,φ,θ) are orthogonal curvilinear coordinates in which the bases are mutually perpendicular but not unitary. Figure 5.1 shows a point P in the spherical coordinates (ρ,φ,θ) embedded in orthonormal Cartesian coordinates (x 1,x 2,x 3 ). However, the vector component changes as the spherical coordinates vary. The vector OP can be written in Cartesian coordinates (x 1,x 2,x 3 ): R ¼ ðρ sin φ cos θþ e 1 þ ðρ sin φ sin θþ e 2 þ ρ cos φ e 3 x 1 e 1 þ x 2 e 2 þ x 3 e 3 ð5:47þ where e 1, e 2, and e 3 are the orthonormal bases of Cartesian coordinates; φ is the equatorial angle; θ is the polar angle. To simplify the formulation with Einstein symbol, the coordinates of u 1, u 2, and u 3 can be used for ρ, φ, and θ, respectively. Therefore, the coordinates of the point P (u 1,u 2,u 3 ) can be written in Cartesian coordinates: 8 9 Pu 1 ; u 2 ; u 3 < x 1 ¼ ρ sin φ cos θ u 1 sin u 2 cos u 3 = ¼ x 2 ¼ ρ sin φ sin θ u 1 sin u 2 cos u : 3 x 3 ¼ ρ cos φ u 1 cos u 2 ; The covariant bases result from Chap. 2. ð5:48þ x 3 ρ sinϕ g 1 (ρ,ϕ,θ) (u 1,u 2,u 3 ): u 1 ρ ; u 2 ϕ ; u 3 θ P g 3 e 1 e 3 ϕ ρ e 0 2 θ g 2 ρ cosϕ x 2 x 1 Fig. 5.1 Orthogonal spherical coordinates

210 5.4 Applying Nabla Operators in Spherical Coordinates 193 g 1 ¼ ðsin φ cos θþ e 1 þ ðsin φ sin θþ e 2 þ cos φ e 3 ) jg 1 j ¼ g ρ ¼ 1 g 2 ¼ ðρ cos φ cos θþ e 1 þ ðρ cos φ sin θþ e 2 ðρ sin φþ e 3 ) jg 2 j ¼ g φ ¼ ρ g 3 ¼ ðρ sin φ sin θþ e 1 þ ðρ sin φ cos θþ e 2 þ 0:e 3 ) jg 3 j ¼ jg θ j ¼ ρ sin φ ð5:49aþ The covariant metric tensor M in the spherical coordinates can be computed from Eq. (5.49a) g 11 g 12 g M ¼ 4 g 21 g 22 g 23 5 ¼ 4 0 ρ ð5:49bþ g 31 g 32 g ðρ sin φþ 2 Similarly, the contravariant bases result from Chap. 2. g 1 ¼ ðsin φ cos θþe 1 þ ðsin φ sin θþ e 2 þ cos φ e 3 ) g 1 ¼ 1 g 2 ¼ 1 cos φ cos θ e 1 þ 1 cos φ sin θ e 2 1 ρ ρ ρ sin φ e 3 ) g 2 1 ¼ ρ g 3 ¼ 1 sin θ e 1 þ 1 cos θ e 2 þ 0:e 3 ) g 3 1 ¼ ρ sin φ ρ sin φ ρ sin φ ð5:50aþ The contravariant metric coefficients in the contravariant metric tensor M 1 can be calculated from Eq. (5.50a) g 11 g 12 g M 1 ¼ 4 g 21 g 22 g 23 5 ¼ 4 0 ρ 2 0 g 31 g 32 g ðρ sin φ Þ ð5:50bþ Gradient of an Invariant The gradient of an invariant Α 2 R can be written according to Eq. (5.9) in A ¼ g i A u i A, i g i ð5:51þ Dividing the covariant basis by its vector length, the normalized covariant basis (covariant unitary basis) results in

211 194 5 Applications of Tensors and Differential Geometry g * i ¼ g i jg i j ¼ g i pffiffiffiffiffiffiffi g ðiiþ ¼ g i h i ð5:52þ The covariant basis in Eq. (5.52) in given in g i ¼ h i g * i where h i are the vector lengths, as given in Eq. (5.49a). p h 1 ¼ ffiffiffiffiffiffi g p 11 h 2 ¼ ffiffiffiffiffiffi g p 22 h 3 ¼ ffiffiffiffiffiffi ¼ j g1 ¼ j g2 g 33 ¼ j g3 jg ρ ¼ 1 jg φ ¼ ρ j jg θ j ¼ ρ sin φ ð5:53þ ð5:54þ The contravariant bases can be transformed into the covariant bases in the orthogonal spherical contravariant basis, as given in Eq. (5.50b). 8 g 1 ¼ g 11 g 1 ¼ g ρ >< g i ¼ g ij g g j ) 2 ¼ g 22 g 2 ¼ 1 ρ 2g φ ð5:55aþ g 3 ¼ g 33 1 >: g 3 ¼ ðρ sin φþ 2g θ Substituting Eqs. (5.53) and (5.54) into Eq. (5.55a), one obtains 8 g 1 ¼ g ρ ¼ h 1 g * ρ ¼ g * ρ >< g 2 ¼ 1 ρ 2g φ ¼ 1 ρ 2 h 2g * φ ¼ 1 ρ g* φ g 3 1 ¼ ðρ sin φþ 2g θ ¼ 1 ðρ sin φþ 2 h 3g * 1 >: θ ¼ ρ sin φ g* θ ð5:55bþ Using Eqs. (5.51) and (5.55b), the gradient of A can be expressed in the physical vector components in the covariant unitary basis. A ¼ A u i gi ¼ A,i g * i h i ¼ A ρ g* ρ þ 1 A ρ φ g* φ þ 1 ð5:56þ A ρ sin φ θ g* θ

212 5.4 Applying Nabla Operators in Spherical Coordinates Divergence of a Vector The divergence of v can be computed using the Christoffel symbols described in Eq. (5.15a). v ¼ g i v v j g u i ¼ j gi u i ¼ v,i i þ v j Γij i v i ji ð5:57þ At first, the covariant derivatives of the contravariant vector components in Eq. (5.57) have to be computed. v 1 1 v 2 2 v 3 3 j ¼ v 1,1 þ Γ1 11 v1 þ Γ 1 12 v2 þ Γ 1 13 v3 for i ¼ 1; j ¼ 1, 2, 3 j ¼ v 2,2 þ Γ2 21 v1 þ Γ 2 22 v2 þ Γ 2 23 v3 for i ¼ 2; j ¼ 1, 2, 3 j ¼ v 3,3 þ Γ3 31 v1 þ Γ 3 32 v2 þ Γ 3 33 v3 for i ¼ 3; j ¼ 1, 2, 3 ð5:58þ The second-kind Christoffel symbols in spherical coordinates can be calculated as [1] ρ Γ 1 ij 0 ρ 0 A; Γ ρ sin 2 ij ¼ 1 B 0 0 A ; φ ρ sin φ cos φ Γ 3 ij ¼ ρ B 0 0 cot φ 1 A cot φ 0 ρ ð5:59þ The physical vector components v *i result from the contravariant vector components in the covariant unitary basis g i * according to Eq. (B.11) in Appendix B. v i ¼ v*i h i ) 8 v 1 ¼ 1 v *1 v ρ >< h 1 v 2 ¼ 1 v *2 1 h 2 ρ v φ >: v 3 ¼ 1 h 3 v *3 1 ρ sin φ v θ ð5:60þ Using Eqs. (5.59) and (5.60), the covariant derivatives of the contravariant vector components can be computed as

213 196 5 Applications of Tensors and Differential Geometry v 1 j1 ¼ v 1, 1 ¼ v ρ ρ v 2 j2 ¼ v 2, 2 þ Γ2 21 v1 ¼ v 2,2 þ 1 ρ v1 ¼ 1 ρ v 3 j3 ¼ v 3, 3 þ Γ3 31 v1 þ Γ 3 32 v2 ¼ 1 ρ sin φ v φ φ þ 1 ρ v ρ v θ θ þ 1 ρ v ρ þ cot φ v φ ρ ð5:61þ Thus, the divergence of v results from Eq. (5.61) in v ¼ v i ji v 1 j1 þ v 2 j2 þ v 3 j3 ¼ v ρ ρ þ 1 v φ ρ φ þ 1 v θ ρ sin φ θ þ 2v ρ ρ þ cot φv φ ρ ð5:62þ Curl of a Vector The curl of a vector results from Eq. (5.33). v ¼ ^ε ijk v j, i g k ¼ 1 ½ð J v 3,2 v 2,3 Þg 1 þ ðv 1,3 v 3,1 Þg 2 þ ðv 2, 1 v 1, 2 Þg 3 ð5:63þ The Jacobian of the spherical coordinates were calculated in Eq. (2.37) as J ¼ ρ 2 sin φ ð5:64þ Using Eqs. (B.19) and (5.49b), the covariant vector components can be computed in their physical vector components. 8 v *1 v 1 ¼ g 11 ¼ v ρ v *j h 1 >< v *2 v i ¼ g ij ) v 2 ¼ g h 22 ¼ ρ v φ j h 2 v *3 >: v 3 ¼ g 33 ¼ ρ sin φ v θ h 3 ð5:65þ According to Eq. (5.53), the covariant bases can be written in the covariant unitary basis.

214 5.5 The Divergence Theorem < g 1 ¼ h 1 g * g i ¼ h i g * 1 ¼ 1 g* 1 g* ρ i ) g 2 ¼ h 2 g : * 2 ¼ ρ g* 2 ρ g* φ g 3 ¼ h 3 g * 3 ¼ ðρ sin φþ g* 3 ð ρ sin φ Þ g* θ ð5:66þ Substituting Eqs. (5.64) (5.66) into Eq. (5.63), the curl of v can be expressed in the unitary covariant basis g i *. v ¼ 1 ð J v 3, 2 v 2, 3 Þ g 1 þ 1 ð J v 1,3 v 3,1 Þg 2 þ 1 ð J v 2,1 v 1,2 Þg 3 ¼ ð ρ sin φ v θþ ρ v φ 1 φ θ ρ 2 sin φ g* 1 þ v ρ θ ð ρ sin φ v θþ 1 ρ ρ sin φ g* 2 þ ρ v φ v ρ 1 ρ φ ρ g* 3 ð5:67þ Computing the partial derivatives in Eq. (5.67), one obtains the curl of v in the unitary spherical coordinate bases. v ¼ 1 v θ ρ φ 1 ρ sin φ 1 þ ρ sin φ þ v φ ρ 1 ρ v φ θ þ cot φv θ ρ v ρ θ v θ ρ v θ ρ v ρ φ þ v φ ρ g * θ g * φ g * ρ ð5:68þ 5.5 The Divergence Theorem Gauss and Stokes Theorems The divergence theorem, known as Gauss theorem deals with the relation between the flow of a vector or tensor field through the closed surface and the characteristics of the vector (tensor) in the volume closed by the surface. Gauss law states that the flux of a vector through any closed surface is proportional to the charge in the volume closed by the surface. This divergence theorem is a very useful tool that can be mostly applied to engineering and physics, such fluid dynamics and electrodynamics to derive the Navier-Stokes equations and Maxwell s equations, respectively.

215 198 5 Applications of Tensors and Differential Geometry Fig. 5.2 Fluid flux through a closed surface S V R 3 v n ds S The Gauss theorem can be generally written in a three-dimensional space (see Fig. 5.2). þ ð v nds ¼ v dv ð5:69þ S where v is the fluid vector through the surface S; n is the normal vector on the surface; and v is the divergence of the vector v. The outward fluid flux from the volume V causes the negative change rate of the volume mass with time. þ S V ð ρv nds ¼ V ρ t dv ð5:70þ Using Gauss divergence theorem, the balance of mass (also continuity equation) in the control volume V can be derived in þ S ð ρv nds ¼ V ρ t dv ¼ ð V ðρvþdv ð5:71þ where ρ is the fluid density. By rearranging the second and third terms in Eq. (5.71), the continuity equation can be written in the integral form for a control volume V: ð V ρ t þ ð ρv Þ dv ¼ 0, ρ þ ρv t ð Þ ¼ 0 ð5:72þ

216 5.5 The Divergence Theorem 199 Fig. 5.3 Fluid flux through an open surface S o v v ds n S o dl (C) Stokes theorem can be used for an open surface S o, as shown in Fig The Stokes theorem indicates that the flow velocity along the closed curve (C) is equal to the flux of curl v going through the open surface S o. þ ð v dl ¼ ð vþnds ð5:73þ ðcþ S o where dl is the length differential on the curve closed (C) of the surface S o ; v is the curl of the vector v Green s Identities The Green s identities can be derived from the Gauss divergence theorem. Sometimes, they can be usefully applied to the boundary element method (BEM) using the Green s function [4 6]. Two Green s identities are discussed in the following section First Green s Identity The vector v can be chosen as the product of two arbitrary scalars ψ and ϕ. v ¼ ψ ϕ ð5:74þ The divergence of v can be computed as follows: