VIA VECTOR TO TENSORPDF电子书下载

PART Ⅰ-VECTORS 1

CHAPTER 2 VECTOR ALGEBRA 3

What is a vector? 3

Scalars and vectors 3

Some other vector quantities 4

Components and resolved parts 6

Vector addition and subtraction 7

Unit vectors 8

The scalar product 9

The vector product 12

Triple products 14

Instructive Exercises 16

CHAPTER 2 SOME APPLICATIONS TO GEOMETRY AND DYNAMICS 18

Points,lines and planes 18

Differentiation of vectors 19

Some geometry 20

Higher derivatives 21

Instructive Exercises 23

CHAPTER 3 VECTOR FIELDS 25

Work and potential energy 25

The gradient vector 26

Volume and surface integrals 29

Divergence 30

Circuital relations 33

The curl of a vector 34

Instructive Exercises 37

CHAPTER 4 COMBINED OPERATORS 39

Null and meaningless operations 39

The scalar Laplacian 40

The vector Laplacian 41

Representation of an arbitrary vector 41

Another vector operator 42

Curvilinear co-ordinates．Differential elements 42

Vector formulae 44

Conformal transformation 44

Instructive Exercises 45

PART Ⅱ-TENSORS 47

CHAPTER 5 TENSORS AND THEIR TRANSFORMATION 49

Introduction 49

Two types of vector components 50

Some notation 53

Tensors of higher ranks 56

Invariants 57

Densities and tensor densities 59

Contraction 60

Instructive Exercises 61

Appendix 62

CHAPTER 6 COVARIANT DIFFERENTIATION 63

Introduction 63

Differentiation of a tensor 63

Curl of a vector 64

The covariant derivatives 65

Covariant differentiation 68

Some examples 69

A further illustration 72

Parallel transfer 75

An example 77

Instructive Exercises 79

CHAPTER 7 THE METRIC TENSOR 80

Introduction 80

Metric tensor 81

Contravariant metric tensor 82

Raising and lowering indices 84

Covariant vector components 84

Three-index symbols 85

Covariant derivatives of the metric tensor 86

Instructive Exercises 87

CHAPTER 8 SOME GEOMETRY 89

Introduction 89

Curves 89

Tangent vector 90

Intrinsic derivative 91

Principal normal and curvature 92

Binormal and the torsion 93

Geodesics 96

Minimal distance 98

Surfaces 100

Surface metric 101

Curves on a surface 103

Curvature of a surface 106

Curvature．Curved space 106

Integrability 107

The Riemann-Christoffel tensor 108

Gaussian or total curvature of a surface 110

Normal vector of the surface 110

The third fundamental differential form 112

Equations of Gauss and Codazzi 113

Curves on a surface 116

Principal curvatures of a surface 117

Instructive Exercises 119

CHAPTER 9 TENSORS IN MATHEMATICAL PHYSICS 121

Introduction 121

Vectors 121

Maxwell＇s equations 125

Elasticity and fluid mechanics 126

Stream function and stress function 131

Instructive Exercises 132

Appendix 1 Suggestions for further reading 134

Appendix 2 Tensor and physical components 136

Hints and Answers to the Exercises 139

Index 141