Tensor Analysis
by Lebedev, L. P.; Cloud, Michael J.; Lebedev, L. P.; Cloud, Michael J.Format: Hardcover
Pub. Date: 2003-07-01
Publisher(s): World Scientific Pub Co Inc
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Summary
Tensor analysis is an essential tool in any science (e.g. engineering, physics, mathematical biology) that employs a continuum description. This concise text offers a straightforward treatment of the subject suitable for the student or practicing engineer. The final chapter introduces the reader to differential geometry, including the elementary theory of curves and surfaces. A well-organized formula list, provided in an appendix, makes the book a very useful reference. A second appendix contains full hints and solutions for the exercises. Undergraduates in engineering or physics, and engineers.
Table of Contents
| Foreword | |
| Preface | |
| Preliminaries | p. 1 |
| The Vector Concept Revisited | p. 1 |
| A First Look at Tensors | p. 2 |
| Assumed Background | p. 3 |
| More on the Notion of a Vector | p. 5 |
| Transformations and Vectors | p. 9 |
| Change of Basis | p. 9 |
| Dual Bases | p. 10 |
| Transformation to the Reciprocal Frame | p. 14 |
| Transformation Between General Frames | p. 16 |
| Covariant and Contravariant Components | p. 18 |
| The Cross Product in Index Notation | p. 19 |
| Tensors | p. 23 |
| Dyadic Quantities and Tensors | p. 23 |
| Tensors from an Operator Viewpoint | p. 24 |
| Dyadic Components Under Transformation | p. 28 |
| More Dyadic Operations | p. 30 |
| Properties of Second Rank Tensors | p. 34 |
| Extending the Dyad Idea | p. 48 |
| Tensors of the Fourth and Higher Ranks | p. 50 |
| Tensor Fields | p. 53 |
| Vector Fields | p. 53 |
| Differentials and the Nabla Operator | p. 62 |
| Differentiation of a Vector Function | p. 67 |
| Derivatives of the Frame Vectors | p. 67 |
| Christoffel Coefficients and their Properties | p. 68 |
| Covariant Differentiation | p. 73 |
| Covariant Derivative of a Second Rank Tensor | p. 74 |
| Differential Operations | p. 76 |
| Orthogonal Coordinate Systems | p. 81 |
| Some Formulas of Integration | p. 85 |
| Norms on Spaces of Vectors and Tensors | p. 88 |
| Elements of Differential Geometry | p. 93 |
| Elementary Facts from the Theory of Curves | p. 94 |
| The Torsion of a Curve | p. 101 |
| Serret-Frenet Equations | p. 104 |
| Elements of the Theory of Surfaces | p. 106 |
| The Second Fundamental Form of a Surface | p. 117 |
| Derivation Formulas | p. 122 |
| Implicit Representation of a Curve; Contact of Curves | p. 125 |
| Osculating Paraboloid | p. 131 |
| The Principal Curvatures of a Surface | p. 133 |
| Surfaces of Revolution | p. 138 |
| Natural Equations of a Curve | p. 140 |
| A Word About Rigor | p. 142 |
| App. A: Formulary | p. 147 |
| App. B: Hints and Answers | p. 165 |
| Bibliography | p. 187 |
| Index | p. 189 |
| Table of Contents provided by Blackwell. All Rights Reserved. |
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