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Tensor Analysis

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Product Details

  • ISBN-13: 9789812383600
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 7/1/2003
  • Pages: 204
  • Product dimensions: 6.10 (w) x 9.10 (h) x 0.70 (d)

Table of Contents

1 Preliminaries 1
1.1 The Vector Concept Revisited 1
1.2 A First Look at Tensors 2
1.3 Assumed Background 3
1.4 More on the Notion of a Vector 5
2 Transformations and Vectors 9
2.1 Change of Basis 9
2.2 Dual Bases 10
2.3 Transformation to the Reciprocal Frame 14
2.4 Transformation Between General Frames 16
2.5 Covariant and Contravariant Components 18
2.6 The Cross Product in Index Notation 19
3 Tensors 23
3.1 Dyadic Quantities and Tensors 23
3.2 Tensors from an Operator Viewpoint 24
3.3 Dyadic Components Under Transformation 28
3.4 More Dyadic Operations 30
3.5 Properties of Second Rank Tensors 34
3.6 Extending the Dyad Idea 48
3.7 Tensors of the Fourth and Higher Ranks 50
4 Tensor Fields 53
4.1 Vector Fields 53
4.2 Differentials and the Nabla Operator 62
4.3 Differentiation of a Vector Function 67
4.4 Derivatives of the Frame Vectors 67
4.5 Christoffel Coefficients and their Properties 68
4.6 Covariant Differentiation 73
4.7 Covariant Derivative of a Second Rank Tensor 74
4.8 Differential Operations 76
4.9 Orthogonal Coordinate Systems 81
4.10 Some Formulas of Integration 85
4.11 Norms on Spaces of Vectors and Tensors 88
5 Elements of Differential Geometry 93
5.1 Elementary Facts from the Theory of Curves 94
5.2 The Torsion of a Curve 101
5.3 Serret-Frenet Equations 104
5.4 Elements of the Theory of Surfaces 106
5.5 The Second Fundamental Form of a Surface 117
5.6 Derivation Formulas 122
5.7 Implicit Representation of a Curve; Contact of Curves 125
5.8 Osculating Paraboloid 131
5.9 The Principal Curvatures of a Surface 133
5.10 Surfaces of Revolution 138
5.11 Natural Equations of a Curve 140
5.12 A Word About Rigor 142
App. A: Formulary 147
App. B: Hints and Answers 165
Bibliography 187
Index 189
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