Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 330

Appendix A: Matrices and Tensors

A.1 Introduction and Rationale

The purpose of this appendix is to present the notation and most of the mathematical
techniques that will be used in the body of the text. The audience is assumed to have
been through several years of college level mathematics that included the differential
and integral calculus, differential equations, functions of several variables, partial
derivatives, and an introduction to linear algebra. Matrices are reviewed briefly and
determinants, vectors, and tensors of order two are described. The application of this
linear algebra to material that appears in undergraduate engineering courses on
mechanics is illustrated by discussions of concepts like the area and mass moments
of inertia, Mohr's circles and the vector cross and triple scalar products. The solutions
to ordinary differential equations are reviewed in the last two sections. The notation,
as far as possible, will be a matrix notation that is easily entered into existing
symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad
etc. The desire to represent the components of three-dimensional fourth order tensors
that appear in anisotropic elasticity as the components of six-dimensional second
order tensors and thus represent these components in matrices of tensor components
in six dimensions leads to the nontraditional part of this appendix. This is also one of
the nontraditional aspects in the text of the topic, but a minor one. This is described in
Sect. A.11, along with the rationale for this approach.

A.2 Definition of Square, Column, and Row Matrices

An r by c matrix M is a rectangular array of numbers consisting of r rows and c
columns,