Cartesian Tensors An Introduction

by Temple, G.

• ISBN13: 9780486439082
• ISBN10: 0486439089
• Format: Paperback
• Copyright: 2004-09-09
• Publisher: Dover Publications

Summary

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.

Author Biography

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

Table of Contents

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