Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

Scalars behave differently under transformations, however, since they remain

unchanged. For example, the value of the scalar product of two vectorsx·y

(which is just a number) is unaffected by the transformation from the unprimed

to the primed basis. Different again is the behaviour of linear operators. If a

linear operatorAis represented by some matrixAin a given coordinate system

then in the new (primed) coordinate system it is represented by a new matrix,

A′=S−^1 AS.

In this chapter we develop a general formulation to describe and classify these

different types of behaviour under a change of basis (or coordinate transfor-

mation). In the development, the generic nametensoris introduced, and certain

scalars, vectors and linear operators are described respectively as tensors of ze-

roth, first and second order (theorder–orrank– corresponds to the number of

subscripts needed to specify a particular element of the tensor). Tensors of third

and fourth order will also occupy some of our attention.

26.3 Cartesian tensors

We begin our discussion of tensors by considering a particular class of coordinate

transformation – namely rotations – and we shall confine our attention strictly

to the rotation of Cartesian coordinate systems. Our object is to study the prop-

erties of various types of mathematical quantities, and their associated physical

interpretations, when they are described in terms of Cartesian coordinates and

the axes of the coordinate system are rigidly rotated from a basise 1 ,e 2 ,e 3 (lying

along theOx 1 ,Ox 2 andOx 3 axes) to a new onee′ 1 ,e′ 2 ,e′ 3 (lying along theOx′ 1 ,

Ox′ 2 andOx′ 3 axes).

Since we shall be more interested in how the components of a vector or linear

operator are changed by a rotation of the axes than in the relationship between

the two sets of basis vectorseiande′i, let us define the transformation matrixL

as the inverse of the matrixSin (26.2). Thus, from (26.2), the components of a

position vectorx, in the old and new bases respectively, are related by

x′i=Lijxj. (26.4)

Because we are considering only rigid rotations of the coordinate axes, the

transformation matrixLwill be orthogonal, i.e. such thatL−^1 =LT.Therefore

the inverse transformation is given by

xi=Ljix′j. (26.5)

The orthogonality ofLalso implies relations among the elements ofLthat

express the fact thatLLT=LTL=I. In subscript notation they are given by

LikLjk=δij and LkiLkj=δij. (26.6)

Furthermore, in terms of the basis vectors of the primed and unprimed Cartesian