Tensor Calculus for Engineers and Physicists

Presents concepts in a straightforward way, while maintaining a great level of rigor Provides 56 solved exercises and a select set of unsolved problems with answers Presents a didactic and concise text suited to undergraduate and graduate students Enriches understanding of tensor calculus applied to all technical sciences and engineering disciplines, providing the reader with complete illustrations that supplement the presented This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of n-dimensional spaces. The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step. Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without needing to resort to other bibliographical sources on tensors. Chapter 1 deals with Fundamental Concepts about tensors and chapter 2 is devoted to the study of covariant, absolute and contravariant derivatives. The chapters 3 and 4 are dedicated to the Integral Theorems and Differential Operators, respectively. Chapter 5 deals with Riemann Spaces, and finally the chapter 6 presents a concise study of the Parallelism of Vectors. It also shows how to solve various problems of several particular manifolds. Topics Theoretical and Applied Mechanics Mathematical Methods in Physics Mathematical Applications in the Physical Sciences Cover 1 S Title 2 Tensor Calculus for Engineers and Physicists 4 © Springer International Publishing Switzerland 2016 5 ISBN 978-3-319-31519-5 ISBN 978-3-319-31520-1 (eBook 5 DOI 10.1007/978-3-319-31520-1 5 Library of Congress Control Number: 2016938417 5 Dedication 6 Preface 8 Historical Introduction 10 Contents 22 Notations 28 Chapter 1: Review of Fundamental Topics About Tensors 31 1.1 Preview 31 1.1.1 Index Notation and Transformation of Coordinates 31 1.2 Space of N Dimensions 32 1.3 Tensors 32 1.3.1 Vectors 32 1.3.2 and Permutation Symbol 33 1.3.3 Reciprocal) Basis 33 1.3.3.1 Orthonormal Basis 36 1.3.3.2 Transformation Law of Vectors 37 1.3.3.3 Covariant and Contravariant Vectors 37 1.3.3.4 Transformation Covariant Vectors 39 1.3.3.5 Transformation Contravariant Vectors 39 1.3.4 Multilinear Forms 40 1.3.4.1 Transformation Law of the Second-Order Tensors 40 1.3.4.2 Transformation Law of the Third-Order Tensors 41 1.3.4.3 Inverse Transformation 42 1.3.4.4 Transitive Property 42 1.3.4.5 Multiplication of a Tensor by a Scalar 43 1.3.4.6 Addition and Subtraction of Tensors 44 1.3.4.7 Contraction of Tensors 44 1.3.4.8 Product of Tensors 44 1.3.4.9 Inner Product of Tensors 45 1.3.4.10 Quotient Law 45 1.4 Spaces and Isotropic Spaces 46 1.5 Metric Tensor 46 1.5.1 Conjugated Tensor 52 1.5.2 Dot Product in Metric Spaces 60 1.5.2.1 Vector Norm 61 1.5.2.2 Lowering of a Tensor ́s Indexes 62 1.5.2.3 Raising of a Tensor ́s Indexes 62 1.5.2.4 Tensorial Equation 63 1.5.2.5 Associated Tensors 65 1.6 Curves 69 1.6.1 Symmetrical and Antisymmetrical Tensors 73 1.6.1.1 Generalization of the Kronecker Delta 76 1.6.1.2 Fundamental Expressions with the Generalized Kronecker Delta 77 1.6.1.3 Product of the Ricci by the Generalized Kronecker Delta 79 1.6.1.4 Norm of the Antisymmetric Pseudotensor of the Second Order 80 1.6.1.5 Generation of Tensors from the Ricci Pseudotensor 81 1.7 Tensors 82 1.7.1 Multiplication by a Scalar 84 1.7.1.1 Addition and Subtraction 85 1.7.1.2 Product 85 1.7.1.3 Contraction 85 1.7.1.4 Inner Product 86 1.7.1.5 Pseudotensor 86 1.7.1.6 Scalar Capacity 88 1.7.1.7 Scalar Density 90 1.7.1.8 Tensorial Capacity 90 1.7.1.9 Tensorial Density 91 1.8 Physical Components of a Tensor 92 1.8.1 Physical Components of a Vector 92 1.8.1.1 Physical Components of the Second-Order Tensor 96 1.9 Tests of the Tensorial Characteristics of a Variety 96 Chapter 2: Covariant, Absolute, and Contravariant Derivatives 102 2.1 Initial Notes 102 2.2 Cartesian Tensor Derivative 103 2.2.1 Vectors 104 2.2.2 Cartesian Tensor of the Second Order 106 2.3 Derivatives of the Basis Vectors 107 2.3.1 Christoffel Symbols 110 2.3.2 Relation Between the Christoffel Symbols 112 2.3.3 Symmetry 113 2.3.4 Cartesian Coordinate System 113 2.3.5 Notation 114 2.3.6 Number of Different Terms 114 2.3.7 Transformation of the Christoffel Symbol of First Kind 115 2.3.8 Transformation of the Christoffel Symbol of Second Kind 116 2.3.9 Linear Transformations 117 2.3.10 Orthogonal Coordinate Systems 117 2.3.11 Contraction 118 2.3.12 Christoffel Relations 120 2.3.13 Ricci Identity 121 2.3.14 Fundamental Relations 122 2.4 Covariant Derivative 129 2.4.1 Contravariant Tensor 130 2.4.1.1 Contravariant Vector 130 2.4.2 Contravariant Tensor of the Second-Order 133 2.4.2.1 Contravariant Tensor of Order Above Two 135 2.4.3 Covariant Tensor 138 2.4.3.1 Covariant Vector 138 2.4.3.2 Covariant Tensor of the Second Order 140 2.4.3.3 Covariant Tensor of Order Above Two 141 2.4.4 Mixed Tensor 142 2.4.5 Covariant Derivative of the Addition, Subtraction, and Product of Tensors 145 2.4.6 Covariant Derivative of Tensors gij,gij,deltaji 146 Ricci ́s Lemma 146 2.4.7 Particularities of the Covariant Derivative 150 2.5 Covariant Derivative of Relative Tensors 152 2.5.1 Covariant Derivative of the Ricci Pseudotensor 154 2.6 Intrinsic or Absolute Derivative 157 2.6.1 Uniqueness of the Absolute Derivative 160 2.7 Contravariant Derivative 162 Chapter 3: Integral Theorems 165 3.1 Basic Concepts 165 3.1.1 Smooth Surface 165 3.1.2 Simply Connected Domain 165 3.1.3 Multiply Connected Domain 166 3.1.4 Oriented Curve 166 3.1.5 Surface Integral 166 3.1.6 Flow 167 3.2 Oriented Surface 168 3.2.1 Volume Integral 169 3.3 Green ́s Theorem 170 3.4 Stokes ́ Theorem 175 HeadingsFPar20002702246 175 3.5 Gauß-Ostrogradsky Theorem 178 HeadingsFPar30002702246 178 Chapter 4: Differential Operators 182 4.1 Scalar, Vectorial, and Tensorial Fields 182 4.1.1 Initial Notes 182 4.1.2 Scalar Field 183 4.1.3 Pseudoscalar Field 183 4.1.4 Vectorial Field 183 4.1.5 Tensorial Field 185 4.1.6 Circulation 186 4.2 Gradient 187 4.2.1 Norm of the Gradient 191 4.2.2 Orthogonal Coordinate Systems 192 4.2.3 Directional Derivative of the Gradient 193 4.2.4 Dyadic Product 194 4.2.5 Gradient of a Second-Order Tensor 196 4.2.6 Gradient Properties 197 4.3 Divergence 201 4.3.1 Divergence Theorem 204 4.3.2 Contravariant and Covariant Components 206 4.3.3 Orthogonal Coordinate Systems 208 4.3.4 Physical Components 210 4.3.5 Properties 210 4.3.6 Divergence of a Second-Order Tensor 210 4.4 Curl 221 4.4.1 Stokes Theorem 223 4.4.2 Orthogonal Curvilinear Coordinate Systems 228 4.4.3 Properties 229 4.4.4 Curl of a Tensor 229 4.5 Successive Applications of the Nabla Operator 234 4.5.1 Basic Relations 234 4.5.2 Laplace Operator 241 4.5.2.1 Laplacian of a Scalar Function 241 4.5.3 Properties 243 4.5.4 Orthogonal Coordinate Systems 245 4.5.5 Laplacian of a Vector 245 4.5.6 Curl of the Laplacian of a Vector 246 4.5.7 Laplacian of a Second-Order Tensor 247 4.6 Other Differential Operators 251 4.6.1 Hesse Operator 251 4.6.2 D ́Alembert Operator 252 Chapter 5: Riemann Spaces 254 5.1 Preview 254 5.2 The Curvature Tensor 254 5.2.1 Formulation 255 5.2.2 Differentiation Commutativity 258 5.2.3 Antisymmetry of Tensor Rjki 260 5.2.4 Notations for Tensor Rjki 260 5.2.5 Uniqueness of Tensor Rijk 261 5.2.6 First Bianchi Identity 261 5.2.7 Second Bianchi Identity 262 5.2.8 Curvature Tensor of Variance (0, 4) 265 5.2.9 Properties of Tensor Rpijk 267 5.2.10 Distinct Algebraic Components of Tensor Rpijk 268 5.2.11 Classification of Spaces 272 5.3 Riemann Curvature 273 5.3.1 Definition 273 5.3.2 Invariance 274 5.3.3 Normalized Form 275 5.4 Ricci Tensor and Scalar Curvature 277 5.4.1 Ricci Tensor with Variance (0, 2) 278 5.4.2 Divergence of the Ricci Tensor with Variance Ricci (0, 2) 280 5.4.3 Bianchi Identity for the Ricci Tensor with Variance (0, 2) 280 5.4.4 Scalar Curvature 281 5.4.5 Geometric Interpretation of the Ricci Tensor with Variance (0, 2) 281 5.4.6 Eigenvectors of the Ricci Tensor with Variance (0, 2) 283 5.4.7 Ricci Tensor with Variance (1, 1) 284 5.4.8 Notations 286 5.5 Einstein Tensor 289 5.6 Particular Cases of Riemann Spaces 291 5.6.1 Riemann Space E2 292 5.6.2 Gauß Curvature 294 5.6.3 Component R1212 in Orthogonal Coordinate Systems 296 5.6.4 Einstein Tensor 298 5.6.5 Riemann Space with Constant Curvature 300 Schur Theorem 300 5.6.6 Isotropy 301 5.6.7 Minkowski Space 307 5.6.8 Conformal Spaces 308 5.6.8.1 Initial Concept 308 5.6.8.2 Christoffel Symbols 309 5.6.8.3 Riemann-Christoffel tensor 310 5.6.8.4 Ricci Tensor 312 5.6.8.5 Scalar Curvature 312 5.6.8.6 Weyl Tensor 313 Formulation 313 Properties of the Weyl Tensor 315 Uniqueness of the Weyl tensor 316 Contraction of the Weyl Tensor 317 Weyl Tensor in the Riemann Space E4 317 5.7 Dimensional Analysis 318 Chapter 6: Geodesics and Parallelism of Vectors 321 6.1 Introduction 321 6.2 Geodesics 321 6.2.1 Representation by Means of Curves in the Surfaces 325 6.2.2 Constant Direction 325 6.2.3 Representation by Means of the Unit Tangent Vector 327 6.2.4 Representation by Means of an Arbitrary Parameter 328 6.3 Geodesics with Null Length 333 6.4 Coordinate Systems 335 6.4.1 Geodesic Coordinates 335 6.4.2 Riemann Coordinates 337 6.5 Geodesic Deviation 339 6.6 Parallelism of Vectors 345 6.6.1 Initial Notes 345 6.6.2 Parallel Transport of Vectors 347 6.6.2.1 Independence of Path 350 6.6.2.2 Invariance of the Modulus and the Angle Between Vectors 351 6.6.2.3 Space with Affine Connections 352 6.6.2.4 Integrability 352 6.6.3 Torsion 358 Bibliography 362 Index 366 "This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of n-dimensional spaces. The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step. Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without needing to resort to other bibliographical sources on tensors. Chapter 1 deals with Fundamental Concepts about tensors and chapter 2 is devoted to the study of covariant, absolute and contravariant derivatives. The chapters 3 and 4 are dedicated to the Integral Theorems and Differential Operators, respectively. Chapter 5 deals with Riemann Spaces, and finally the chapter 6 presents a concise study of the Parallelism of Vectors. It also shows how to solve various problems of several particular manifolds"--Publisher's website This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of N dimensional spaces. The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step. Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without needing to resort to other bibliographical sources on tensors. Chapter 1 deals with Fundamental Concepts about tensors and chapter 2 is devoted to the study of covariant, absolute and contravariant derivatives. The chapters 3 and 4 are dedicated to the Integral Theorems and Differential Operators, respectively. Chapter 5 deals with Riemann Spaces, and finally the chapter 6 presents a concise study of the Parallelism of Vectors. It also shows how to solve various problems of several particular manifolds Front Matter....Pages i-xxix Review of Fundamental Topics About Tensors....Pages 1-71 Covariant, Absolute, and Contravariant Derivatives....Pages 73-135 Integral Theorems....Pages 137-153 Differential Operators....Pages 155-226 Riemann Spaces....Pages 227-293 Geodesics and Parallelism of Vectors....Pages 295-335 Back Matter....Pages 337-345