1 - Cartesian Tensor Analysis

Summary

Introduction

In this chapter we present an elementary introduction to Cartesian tensor analysis in a three-dimensional Euclidean point space or a two-dimensional subspace. A Euclidean point space is the space of position vectors of points. The term vector is used in the sense of classical vector analysis, and scalars and polar vectors are zeroth- and first-order tensors, respectively. The distinction between polar and axial vectors is discussed later in this chapter. A scalar is a single quantity that possesses magnitude and does not depend on any particular coordinate system, and a vector is a quantity that possesses both magnitude and direction and has components, with respect to a particular coordinate system, which transform in a definite manner under change of coordinate system. Also vectors obey the parallelogram law of addition. There are quantities that possess both magnitude and direction but are not vectors, for example, the angle of finite rotation of a rigid body about a fixed axis.

A second-order tensor can be defined as a linear operator that operates on a vector to give another vector. That is, when a second-order tensor operates on a vector, another vector, in the same Euclidean space, is generated, and this operation can be illustrated by matrix multiplication. The components of a vector and a second-order tensor, referred to the same rectangular Cartesian coordinate system, in a three-dimensional Euclidean space, can be expressed as a (3 × 1) matrix and a (3 × 3) matrix, respectively.