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