| Foreword | v |
| Preface | vii |
| 1. Preliminaries | |
| 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 | |
| 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 |
| 2.7 Closing Remarks | 22 |
| 3. Tensors | |
| 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 | |
| 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 | 93 |
| 5.2 The Torsion of a Curve | 94 |
| 5.3 Serret-Frenet Equations | 101 |
| 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 |
| 5.13 Conclusion | 145 |
| Appendix A Formulary | 147 |
| Appendix B Hints and Answers | 165 |
| Bibliography | 187 |
| Index | 189 |