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‘Tensors’ were introduced by Professor Gregorio Ricci of University of Padua (Italy) in 1887

primarily as extension of vectors. A quantity having magnitude only is called Scalar and a quantity with

magnitude and direction both, called Vector. But certain quantities are associated with two or more

directions, such a quantity is called Tensor. The stress at a point of an elastic solid is an example of a

Tensor which depends on two directions one normal to the area and other that of the force on it.

Tensors have their applications to Riemannian Geometry, Mechanics, Elasticity, Theory of Relativity,

Electromagnetic Theory and many other disciplines of Science and Engineering.

This book has been presented in such a clear and easy way that the students will have no difficulty

in understanding it. The definitions, proofs of theorems, notes have been given in details.

The subject is taught at graduate/postgraduate level in almost all universities.

In the end, I wish to thank the publisher and the printer for their full co-operation in bringing out

the book in the present nice form.

Suggestions for further improvement of the book will be gratefully acknowledged.

CONTENTS :-

Foreword .............................................................................................................vii

Preface ................................................................................................................ ix

Chapter–1 Preliminaries........................................................................................... 1-5

1.1. n-dimensional Space ...............................................................................................1

1.2. Superscript and Subscript .......................................................................................1

1.3. The Einstein's Summation Convention ......................................................................1

1.4. Dummy Index .......................................................................................................1

1.5. Free Index .............................................................................................................2

1.6. Krönecker Delta .....................................................................................................2

Exercises ...............................................................................................................5

Chapter–2 Tensor Algebra ..................................................................................... 6-30

2.1. Introduction ..........................................................................................................6

2.2. Transformation of Coordinates ................................................................................6

2.3. Covariant and Contravariant Vectors .........................................................................7

2.4. Contravariant Tensor of Rank Two ..........................................................................9

2.5. Covariant Tensor of Rank Two ................................................................................9

2.6. Mixed Tensor of Rank Two.....................................................................................9

2.7. Tensor of Higher Order ......................................................................................... 14

2.8. Scalar or Invariant ................................................................................................ 15

2.9. Addition and Subtraction of Tensors....................................................................... 15

2.10. Multiplication of Tensors (Outer Product of Tensors)............................................... 16

2.11. Contraction of a Tensor ........................................................................................ 18

2.12. Inner Product of Two Tensors .............................................................................. 18

2.13. Symmetric Tensors .............................................................................................. 20

2.14. Skew-symmetric Tensor ....................................................................................... 20

2.15. Quotient Law ....................................................................................................... 24

xii Tensors and Their Applications

2.16. Conjugate (or Reciprocal) Symmetric Tensor .......................................................... 25

2.17. Relative Tensor .................................................................................................... 26

Examples ............................................................................................................ 26

Exercises ............................................................................................................. 29

Chapter–3 Metric Tensor and Riemannian Metric ............................................ 31-54

3.1. The Metric Tensor ............................................................................................... 31

3.2. Conjugate Metric Tensor (Contravariant Tensor)...................................................... 34

3.3. Length of a Curve ................................................................................................ 42

3.4. Associated Tensor ................................................................................................ 43

3.5. Magnitude of Vector ............................................................................................. 43

3.6. Scalar Product of Two Vectors .............................................................................. 44

3.7. Angle Between Two Vectors .................................................................................. 45

3.8. Angle Between Two Coordinate Curves .................................................................. 47

3.9. Hypersurface ....................................................................................................... 48

3.10. Angle Between Two Coordinate Hyper surface ........................................................ 48

3.11. n-Ply Orthogonal System of Hypersurfaces ............................................................. 49

3.12. Congruence of Curves .......................................................................................... 49

3.13. Orthogonal Ennuple .............................................................................................. 49

Examples ............................................................................................................ 52

Exercises ............................................................................................................. 54

Chapter–4 Christoffel’s Symbols and Covariant Differentiation ................................ 55-84

4.1. Christoffel’s Symbol............................................................................................. 55

4.2. Transformtion of Christoffel’s Symbols .................................................................. 64

4.3. Covariant Differentiation of a Covariant Vector ........................................................ 67

4.4. Covariant Differentiation of a Contravariant Vector................................................... 68

4.5. Covariant Differentiation of Tensors ....................................................................... 69

4.6. Ricci’s Theorem .................................................................................................. 71

4.7. Gradient, Divergence and Curl ............................................................................... 75

4.8. The Laplacian Operator......................................................................................... 80

Exercises ............................................................................................................. 83

Chapter–5 Riemann-Christoffel Tensor ............................................................ 85-110

5.1. Riemann-Christoffel Tensor................................................................................... 85

5.2. Ricci Tensor ........................................................................................................ 88

5.3. Covariant Riemann-Christoffel Tensor .................................................................... 89

5.4. Properties of Riemann-Christoffel Tensors of First Kind Ri j k l

................................... 91

5.5. Bianchi Identity .................................................................................................... 94

5.6. Einstein Tensor .................................................................................................... 95

5.7. Riemannian Curvature of Vn

.................................................................................. 96

Contents xiii

5.8. Formula For Riemannian Curvature in Terms of Covariant

Curvature Tensor of Vn

......................................................................................... 98

5.9. Schur’s Theorem ............................................................................................... 100

5.10. Mean Curvature ................................................................................................. 101

5.11. Ricci Principal Directions .................................................................................... 102

5.12. Einstein Space ................................................................................................... 103

5.13. Weyl Tensor or Projective Curvature Tensor ......................................................... 104

Examples .......................................................................................................... 106

Exercises ........................................................................................................... 109

Chapter–6 The e-systems and the Generalized Krönecker Deltas ................111-115

6.1. Completely Symmetric .........................................................................................111

6.2. Completely Skew-symmetric ................................................................................111

6.3. e-system ........................................................................................................... 112

6.4. Generalized Krönecker Delta ................................................................................ 112

6.5. Contraction of i jk

áâã ä ............................................................................................ 114

Exercises ........................................................................................................... 115

Chapter–7 Geometry ........................................................................................ 116-141

7.1. Length of Arc .................................................................................................... 116

7.2. Curvilinear Coordinates in E3

.............................................................................. 120

7.3. Reciprocal Base System Covariant and Contravariant Vectors .................................. 122

7.4. On The Meaning of Covariant Derivatives ............................................................. 127

7.5. Intrinsic Differentiation ....................................................................................... 131

7.6. Parallel Vector Fields........................................................................................... 134

7.7. Geometry of Space Curves ................................................................................. 134

7.8. Serret-Frenet Formulae ....................................................................................... 138

7.9. Equations of A Straight Line ................................................................................ 140

Exercises ........................................................................................................... 141

Chapter–8 Analytical Mechanics..................................................................... 142-169

8.1. Introduction ...................................................................................................... 142

8.2. Newtonian Laws ................................................................................................ 142

8.3. Equations of Motion of Particle ............................................................................ 143

8.4. Conservative Force Field ..................................................................................... 144

8.5. Lagrangean Equation of Motion ........................................................................... 146

8.6. Applications of Lagrangean Equations ................................................................... 152

8.7. Hamilton’s Principle ............................................................................................ 153

8.8. Integral Energy .................................................................................................. 155

8.9. Principle of Least Action ..................................................................................... 156

xiv Tensors and Their Applications

8.10. Generalized Coordinates ...................................................................................... 157

8.11. Lagrangean Equation of Generalized Coordinates ................................................... 158

8.12. Divergence Theorem, Green’s Theorem, Laplacian Operator and Stoke’s

Theorem in Tensor Notation ................................................................................ 161

8.13. Gauss’s Theorem ............................................................................................... 164

8.14. Poisson’s Equation ............................................................................................. 166

8.15. Solution of Poisson’s Equation ............................................................................. 167

Exercises ........................................................................................................... 169

Chapter–9 Curvature of a Curve, Geodesic .................................................... 170-187

9.1. Curvature of Curve, Principal Normal................................................................... 170

9.2. Geodesics ......................................................................................................... 171

9.3. Euler’s Condition ............................................................................................... 171

9.4. Differential Equations of Geodesics ...................................................................... 173

9.5. Geodesic Coordinates ......................................................................................... 175

9.6. Riemannian Coordinates ...................................................................................... 177

9.7. Geodesic Form of a Line Element ........................................................................ 178

9.8. Geodesics in Euclidean Space .............................................................................. 181

Examples .......................................................................................................... 182

Exercises ........................................................................................................... 186

Chapter–10 Parallelism of Vectors ................................................................. 188-204

10.1. Parallelism of a Vector of Constant Magnitude (Levi-Civita’s Concept) ..................... 188

10.2. Parallelism of a Vector of Variable Magnitude ............................ 191

10.3. Subspace of Riemannian Manifold ......................... 193

10.4. Parallelism in a Subspace ......... 196

10.5. Fundamental Theorem of Riemannian Geometry Statement ................. 199

Examples .............................. 200

Exercises .................................. 203

Chapter–11 Ricci’s Coefficients of Rotation and Congruence ....................... 205-217

11.1. Ricci’s Coefficient of Rotation ............... 205

11.2. Reason for the Name “Coefficients of Rotation” ................. 206

11.3. Curvature of Congruence ...... 207

11.4. Geodesic Congruence ............ 208

11.5. Normal Congruence .............. 209

11.6. Curl of Congruence .................. 211

11.7. Canonical Congruence ........... 213

Examples ............................... 215

Exercises ..................... 217

Contents xv

Chapter–12 Hypersurfaces ............ 218-242

12.1. Introduction .................. 218

12.2. Generalized Covariant Differentiation ............ 219

12.3. Laws of Tensor Differentiation .......... 220

12.4. Gauss’s Formula ...................... 222

12.5. Curvature of a Curve in a Hypersurface and Normal Curvature, Meunier’s Theorem,

Dupin’s Theorem ....................... 224

12.6. Definitions ....................... 227

12.7. Euler’s Theorem .................... 228

12.8. Conjugate Directions and Asymptotic Directions in a Hypersurface........... 229

12.9. Tensor Derivative of Unit Normal........................ 230

12.10. The Equation of Gauss and Codazzi .................. 233

12.11. Hypersurfaces with Indeterminate Lines of Curvature .................. 234

12.12. Central Quadratic Hypersurfaces ................ 235

12.13. Polar Hyperplane .............. 236

12.14. Evolute of a Hypersurface in an Euclidean Space ......................... 237

12.15. Hypersphere ................238

Exercises ..................... 241

Index .......... 243-245

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