# Introduction to Differential Geometry with Tensor Applications

Modern Mathematics in Computer Science

1. Edition May 2022

512 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-119-79562-9**

INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH TENSOR APPLICATIONS

This is the only volume of its kind to explain, in precise and easy-to-understand language, the fundamentals of tensors and their applications in differential geometry and analytical mechanics with examples for practical applications and questions for use in a course setting.

Introduction to Differential Geometry with Tensor Applications discusses the theory of tensors, curves and surfaces and their applications in Newtonian mechanics. Since tensor analysis deals with entities and properties that are independent of the choice of reference frames, it forms an ideal tool for the study of differential geometry and also of classical and celestial mechanics. This book provides a profound introduction to the basic theory of differential geometry: curves and surfaces and analytical mechanics with tensor applications. The author has tried to keep the treatment of the advanced material as lucid and comprehensive as possible, mainly by including utmost detailed calculations, numerous illustrative examples, and a wealth of complementing exercises with complete solutions making the book easily accessible even to beginners in the field.

Groundbreaking and thought-provoking, this volume is an outstanding primer for modern differential geometry and is a basic source for a profound introductory course or as a valuable reference. It can even be used for self-study, by students or by practicing engineers interested in the subject.

Whether for the student or the veteran engineer or scientist, Introduction to Differential Geometry with Tensor Applications is a must-have for any library.

This outstanding new volume:

* Presents a unique perspective on the theories in the field not available anywhere else

* Explains the basic concepts of tensors and matrices and their applications in differential geometry and analytical mechanics

* Is filled with hundreds of examples and unworked problems, useful not just for the student, but also for the engineer in the field

* Is a valuable reference for the professional engineer or a textbook for the engineering student

About the Book xvii

Introduction 1

Part I: Tensor Theory 7

1 Preliminaries 9

1.1 Introduction 9

1.2 Systems of Different Orders 9

1.3 Summation Convention Certain Index 10

1.3.1 Dummy Index 11

1.3.2 Free Index 11

1.4 Kronecker Symbols 11

1.5 Linear Equations 14

1.6 Results on Matrices and Determinants of Systems 15

1.7 Differentiation of a Determinant 18

1.8 Examples 19

1.9 Exercises 23

2 Tensor Algebra 25

2.1 Introduction 25

2.2 Scope of Tensor Analysis 25

2.2.1 n-Dimensional Space 26

2.3 Transformation of Coordinates in Sn 27

2.3.1 Properties of Admissible Transformation of Coordinates 30

2.4 Transformation by Invariance 31

2.5 Transformation by Covariant Tensor and Contravariant Tensor 32

2.6 The Tensor Concept: Contravariant and Covariant Tensors 34

2.6.1 Covariant Tensors 34

2.6.2 Contravariant Vectors 35

2.6.3 Tensor of Higher Order 40

2.6.3.1 Contravariant Tensors of Order Two 40

2.6.3.2 Covariant Tensor of Order Two 41

2.6.3.3 Mixed Tensors of Order Two 42

2.7 Algebra of Tensors 43

2.7.1 Equality of Two Tensors of Same Type 45

2.8 Symmetric and Skew-Symmetric Tensors 45

2.8.1 Symmetric Tensors 45

2.8.2 Skew-Symmetric Tensors 46

2.9 Outer Multiplication and Contraction 51

2.9.1 Outer Multiplication 51

2.9.2 Contraction of a Tensor 53

2.9.3 Inner Product of Two Tensors 54

2.10 Quotient Law of Tensors 56

2.11 Reciprocal Tensor of a Tensor 58

2.12 Relative Tensor, Cartesian Tensor, Affine Tensor, and Isotropic Tensors 60

2.12.1 Relative Tensors 60

2.12.2 Cartesian Tensors 63

2.12.3 Affine Tensors 63

2.12.4 Isotropic Tensor 64

2.12.5 Pseudo-Tensors 64

2.13 Examples 65

2.14 Exercises 71

3 Riemannian Metric 73

3.1 Introduction 73

3.2 The Metric Tensor 74

3.3 Conjugate Tensor 75

3.4 Associated Tensors 77

3.5 Length of a Vector 84

3.5.1 Length of Vector 84

3.5.2 Unit Vector 85

3.5.3 Null Vector 86

3.6 Angle Between Two Vectors 86

3.6.1 Orthogonality of Two Vectors 87

3.7 Hypersurface 88

3.8 Angle Between Two Coordinate Hypersurfaces 89

3.9 Exercises 95

4 Tensor Calculus 97

4.1 Introduction 97

4.2 Christoffel Symbols 97

4.2.1 Properties of Christoffel Symbols 98

4.3 Transformation of Christoffel Symbols 110

4.3.1 Law of Transformation of Christoffel Symbols of 1st Kind 110

4.3.2 Law of Transformation of Christoffel Symbols of 2nd Kind 111

4.4 Covariant Differentiation of Tensor 113

4.4.1 Covariant Derivative of Covariant Tensor 114

4.4.2 Covariant Derivative of Contravariant Tensor 115

4.4.3 Covariant Derivative of Tensors of Type (0,2) 116

4.4.4 Covariant Derivative of Tensors of Type (2,0) 118

4.4.5 Covariant Derivative of Mixed Tensor of Type (s, r) 120

4.4.6 Covariant Derivatives of Fundamental Tensors and the Kronecker Delta 120

4.4.7 Formulas for Covariant Differentiation 122

4.4.8 Covariant Differentiation of Relative Tensors 123

4.5 Gradient, Divergence, and Curl 129

4.5.1 Gradient 130

4.5.2 Divergence 130

4.5.2.1 Divergence of a Mixed Tensor (1,1) 132

4.5.3 Laplacian of an Invariant 136

4.5.4 Curl of a Covariant Vector 137

4.6 Exercises 141

5 Riemannian Geometry 143

5.1 Introduction 143

5.2 Riemannian-Christoffel Tensor 143

5.3 Properties of Riemann-Christoffel Tensors 150

5.3.1 Space of Constant Curvature 158

5.4 Ricci Tensor, Bianchi Identities, Einstein Tensors 159

5.4.1 Ricci Tensor 159

5.4.2 Bianchi Identity 160

5.4.3 Einstein Tensor 166

5.5 Einstein Space 170

5.6 Riemannian and Euclidean Spaces 171

5.6.1 Riemannian Spaces 171

5.6.2 Euclidean Spaces 174

5.7 Exercises 175

6 The e-Systems and the Generalized Kronecker Deltas 177

6.1 Introduction 177

6.2 e-Systems 177

6.3 Generalized Kronecker Delta 181

6.4 Contraction of ijk 183

6.5 Application of e-Systems to Determinants and Tensor Characters of Generalized Kronecker Deltas 185

6.5.1 Curl of Covariant Vector 189

6.5.2 Vector Product of Two Covariant Vectors 190

6.6 Exercises 192

Part II: Differential Geometry 193

7 Curvilinear Coordinates in Space 195

7.1 Introduction 195

7.2 Length of Arc 195

7.3 Curvilinear Coordinates in E3 200

7.3.1 Coordinate Surfaces 201

7.3.2 Coordinate Curves 202

7.3.3 Line Element 205

7.3.4 Length of a Vector 206

7.3.5 Angle Between Two Vectors 207

7.4 Reciprocal Base Systems 210

7.5 Partial Derivative 216

7.6 Exercises 219

8 Curves in Space 221

8.1 Introduction 221

8.2 Intrinsic Differentiation 221

8.3 Parallel Vector Fields 226

8.4 Geometry of Space Curves 228

8.4.1 Plane 231

8.5 Serret-Frenet Formula 233

8.5.1 Bertrand Curves 235

8.6 Equations of a Straight Line 252

8.7 Helix 254

8.7.1 Cylindrical Helix 256

8.7.2 Circular Helix 258

8.8 Exercises 262

9 Intrinsic Geometry of Surfaces 265

9.1 Introduction 265

9.2 Curvilinear Coordinates on a Surface 265

9.3 Intrinsic Geometry: First Fundamental Quadratic Form 267

9.3.1 Contravariant Metric Tensor 270

9.4 Angle Between Two Intersecting Curves on a Surface 272

9.4.1 Pictorial Interpretation 274

9.5 Geodesic in Rn 277

9.6 Geodesic Coordinates 289

9.7 Parallel Vectors on a Surface 291

9.8 Isometric Surface 292

9.8.1 Developable 293

9.9 The Riemannian-Christoffel Tensor and Gaussian Curvature 294

9.9.1 Einstein Curvature 296

9.10 The Geodesic Curvature 308

9.11 Exercises 319

10 Surfaces in Space 321

10.1 Introduction 321

10.2 The Tangent Vector 321

10.3 The Normal Line to the Surface 324

10.4 Tensor Derivatives 329

10.5 Second Fundamental Form of a Surface 332

10.5.1 Equivalence of Definition of Tensor balphaß 333

10.6 The Integrability Condition 334

10.7 Formulas of Weingarten 337

10.7.1 Third Fundamental Form 338

10.8 Equations of Gauss and Codazzi 339

10.9 Mean and Total Curvatures of a Surface 341

10.10 Exercises 347

11 Curves on a Surface 349

11.1 Introduction 349

11.2 Curve on a Surface: Theorem of Meusnier 350

11.2.1 Theorem of Meusnier 353

11.3 The Principal Curvatures of a Surface 358

11.3.1 Umbillic Point 360

11.3.2 Lines of Curvature 361

11.3.3 Asymptotic Lines 362

11.4 Rodrigue's Formula 376

11.5 Exercises 379

12 Curvature of Surface 381

12.1 Introduction 381

12.2 Surface of Positive and Negative Curvatures 381

12.3 Parallel Surfaces 383

12.3.1 Computation of a__ and b__ 383

12.4 The Gauss-Bonnet Theorem 387

12.5 The n-Dimensional Manifolds 391

12.6 Hypersurfaces 394

12.7 Exercises 395

Part III: Analytical Mechanics 397

13 Classical Mechanics 399

13.1 Introduction 399

13.2 Newtonian Laws of Motion 399

13.3 Equations of Motion of Particles 401

13.4 Conservative Force Field 403

13.5 Lagrangean Equations of Motion 405

13.6 Applications of Lagrangean Equations 411

13.7 Himilton's Principle 423

13.8 Principle of Least Action 427

13.9 Generalized Coordinates 430

13.10 Lagrangean Equations in Generalized Coordinates 432

13.11 Divergence Theorem, Green's Theorem, Laplacian Operator, and Stoke's Theorem in Tensor Notation 438

13.12 Hamilton's Canonical Equations 442

13.12.1 Generalized Momenta 443

13.13 Exercises 444

14 Newtonian Law of Gravitations 447

14.1 Introduction 447

14.2 Newtonian Laws of Gravitation 447

14.3 Theorem of Gauss 451

14.4 Poisson's Equation 453

14.5 Solution of Poisson's Equation 454

14.6 The Problem of Two Bodies 456

14.7 The Problem of Three Bodies 462

14.8 Exercises 467

Appendix A: Answers to Even-Numbered Exercises 469

References 473

Index 475