Cartesian Tensors

An Introduction

By: G. TEMPLE

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Paperback | 9 September 2004

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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 edition.

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Preface	p. 1
Vectors, Bases and Orthogonal Transformations
Introduction	p. 3
The geometrical theory of vectors	p. 3
Bases	p. 5
The summation convention	p. 7
The components of a vector	p. 7
Transformations of base	p. 9
Properties of the transformation matrix T	p. 10
The orthogonal group	p. 11
Examples	p. 13
The Definition of a Tensor
Introduction	p. 16
Geometrical examples of multilinear functions of direction	p. 16
Examples of multilinear functions of direction in rigid dynamics	p. 18
The stress tensor in continuum dynamics	p. 20
Formal definition of a tensor	p. 23
The angular velocity tensor	p. 25
The Algebra of Tensors
Introduction	p. 27
Addition and scalar multiplication	p. 27
Outer multiplication	p. 28
Spherical means of tensors and contraction	p. 28
Symmetry and antisymmetry	p. 30
Antisymmetric tensors of rank 2	p. 31
Products of vectors	p. 32
The Chapman-Cowling notation	p. 32
The Calculus of Tensors
Introduction	p. 34
The differentiation of tensors	p. 34
Derived tensors	p. 35
The strain tensor	p. 36
The rate of strain tensor	p. 39
The momentum equations for a continuous medium	p. 39
The Structure of Tensors
Introduction	p. 42
Projection operators	p. 42
Definition of eigenvalues and eigenvectors	p. 44
Existence of eigenvalues and eigenvectors	p. 47
The secular equation	p. 50
Isotropic Tensors
Introduction	p. 53
Definition of isotropic tensors	p. 53
Isotropic tensors in two dimensions	p. 55
Isotropic tensors of rank 2 in three dimensions	p. 57
Isotropic tensors of rank 3 in three dimensions	p. 58
Isotropic tensors of rank 4 in three dimensions	p. 59
The stress-strain relations for an isotropic elastic medium	p. 60
The constitutive equations for a viscous fluid	p. 61
Spinors
Introduction	p. 63
Isotropic vectors	p. 63
The isotropic parameter	p. 64
Spinors	p. 66
Spinors and vectors	p. 68
The Clifford algebra	p. 69
The inner automorphisms of the Clifford algebra	p. 71
The spinor manifold	p. 73
Tensors in Orthogonal Curvilinear Coordinates
Introduction	p. 76
Curvilinear orthogonal coordinates	p. 76
Curvilinear components of tensors	p. 78
Gradient, divergence and curl in orthogonal curvilinear coordinates	p. 79
The strain tensor in orthogonal curvilinear coordinates	p. 82
The three index symbols	p. 85
The divergence of the stress tensor in curvilinear coordinates	p. 86
Index	p. 91
Table of Contents provided by Ingram. All Rights Reserved.

Cartesian Tensors

An Introduction

At a Glance

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