Cartesian Tensors An Introduction

This undergraduate text provides an introduction to the theory of Cartesian tensors, defining tensors as multilinear functions of direction, and simplifying many theorems in a manner that lends unity to the subject. The author notes the importance of the analysis of the structure of tensors in terms of spectral sets of projection operators as part of the very substance of quantum theory. He therefore provides an elementary discussion of the subject, in addition to a view of isotropic tensors and spinor analysis within the confines of Euclidean space. The text concludes with an examination of tensors in orthogonal curvilinear coordinates. Numerous examples illustrate the general theory and indicate certain extensions and applications. 1960 ed.

G. Temple, F. R. S.: Sedleian Professor of Natural Philosophy in the University of Oxford

Preface

1

(2)

Vectors, Bases and Orthogonal Transformations

Introduction

3

(1)

The geometrical theory of vectors

3

(2)

Bases

5

(2)

The summation convention

7

(1)

The components of a vector

7

(2)

Transformations of base

9

(1)

Properties of the transformation matrix T

10

(1)

The orthogonal group

11

(2)

Examples

13

(3)

The Definition of a Tensor

Introduction

16

(1)

Geometrical examples of multilinear functions of direction

16

(2)

Examples of multilinear functions of direction in rigid dynamics

18

(2)

The stress tensor in continuum dynamics

20

(3)

Formal definition of a tensor

23

(2)

The angular velocity tensor

25

(2)

The Algebra of Tensors

Introduction

27

(1)

Addition and scalar multiplication

27

(1)

Outer multiplication

28

(1)

Spherical means of tensors and contraction

28

(2)

Symmetry and antisymmetry

30

(1)

Antisymmetric tensors of rank 2

31

(1)

Products of vectors

32

(1)

The Chapman--Cowling notation

32

(2)

The Calculus of Tensors

Introduction

34

(1)

The differentiation of tensors

34

(1)

Derived tensors

35

(1)

The strain tensor

36

(3)

The rate of strain tensor

39

(1)

The momentum equations for a continuous medium

39

(3)

The Structure of Tensors

Introduction

42

(1)

Projection operators

42

(2)

Definition of eigenvalues and eigenvectors

44

(3)

Existence of eigenvalues and eigenvectors

47

(3)

The secular equation

50

(3)

Isotropic Tensors

Introduction

53

(1)

Definition of isotropic tensors

53

(2)

Isotropic tensors in two dimensions

55

(2)

Isotropic tensors of rank 2 in three dimensions

57

(1)

Isotropic tensors of rank 3 in three dimensions

58

(1)

Isotropic tensors of rank 4 in three dimensions

59

(1)

The stress-strain relations for an isotropic elastic medium

60

(1)

The constitutive equations for a viscous fluid

61

(2)

Spinors

Introduction

63

(1)

Isotropic vectors

63

(1)

The isotropic parameter

64

(2)

Spinors

66

(2)

Spinors and vectors

68

(1)

The Clifford algebra

69

(2)

The inner automorphisms of the Clifford algebra

71

(2)

The spinor manifold

73

(3)

Tensors in Orthogonal Curvilinear Coordinates

Introduction

76

(1)

Curvilinear orthogonal coordinates

76

(2)

Curvilinear components of tensors

78

(1)

Gradient, divergence and curl in orthogonal curvilinear coordinates

79

(3)

The strain tensor in orthogonal curvilinear coordinates

82

(3)

The three index symbols

85

(1)

The divergence of the stress tensor in curvilinear coordinates

86

(5)

Index

91

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Cartesian Tensors An Introduction

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