《VIA VECTOR TO TENSOR》

PART Ⅰ-VECTORS1

CHAPTER 2VECTOR ALGEBRA3

What is a vector?3

Scalars and vectors3

Some other vector quantities4

Components and resolved parts6

Vector addition and subtraction7

Unit vectors8

The scalar product9

The vector product12

Triple products14

Instructive Exercises16

CHAPTER 2SOME APPLICATIONS TO GEOMETRY AND DYNAMICS18

Points,lines and planes18

Differentiation of vectors19

Some geometry20

Higher derivatives21

Instructive Exercises23

CHAPTER 3VECTOR FIELDS25

Work and potential energy25

The gradient vector26

Volume and surface integrals29

Divergence30

Circuital relations33

The curl of a vector34

Instructive Exercises37

CHAPTER 4COMBINED OPERATORS39

Null and meaningless operations39

The scalar Laplacian40

The vector Laplacian41

Representation of an arbitrary vector41

Another vector operator42

Curvilinear co-ordinates．Differential elements42

Vector formulae44

Conformal transformation44

Instructive Exercises45

PART Ⅱ-TENSORS47

CHAPTER 5TENSORS AND THEIR TRANSFORMATION49

Introduction49

Two types of vector components50

Some notation53

Tensors of higher ranks56

Invariants57

Densities and tensor densities59

Contraction60

Instructive Exercises61

Appendix62

CHAPTER 6COVARIANT DIFFERENTIATION63

Introduction63

Differentiation of a tensor63

Curl of a vector64

The covariant derivatives65

Covariant differentiation68

Some examples69

A further illustration72

Parallel transfer75

An example77

Instructive Exercises79

CHAPTER 7THE METRIC TENSOR80

Introduction80

Metric tensor81

Contravariant metric tensor82

Raising and lowering indices84

Covariant vector components84

Three-index symbols85

Covariant derivatives of the metric tensor86

Instructive Exercises87

CHAPTER 8SOME GEOMETRY89

Introduction89

Curves89

Tangent vector90

Intrinsic derivative91

Principal normal and curvature92

Binormal and the torsion93

Geodesics96

Minimal distance98

Surfaces100

Surface metric101

Curves on a surface103

Curvature of a surface106

Curvature．Curved space106

Integrability107

The Riemann-Christoffel tensor108

Gaussian or total curvature of a surface110

Normal vector of the surface110

The third fundamental differential form112

Equations of Gauss and Codazzi113

Curves on a surface116

Principal curvatures of a surface117

Instructive Exercises119

CHAPTER 9TENSORS IN MATHEMATICAL PHYSICS121

Introduction121

Vectors121

Maxwell＇s equations125

Elasticity and fluid mechanics126

Stream function and stress function131

Instructive Exercises132

Appendix 1Suggestions for further reading134

Appendix 2Tensor and physical components136

Hints and Answers to the Exercises139

Index141