Tensor Analysis

by Lebedev, L. P.; Cloud, Michael J.; Lebedev, L. P.; Cloud, Michael J.

• ISBN13: 9789812383600
• ISBN10: 9812383603
• Format: Hardcover
• Copyright: 2003-07-01
• Publisher: World Scientific Pub Co Inc

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.