Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

Physical examples involving second-order tensors will be discussed in the later

sections of this chapter, but we might note here that, for example, magnetic

susceptibility and electrical conductivity are described by second-order tensors.

26.6 The algebra of tensors

Because of the similarity of first- and second-order tensors to column vectors and

matrices, it would be expected that similar types of algebraic operation can be

carried out with them and so provide ways of constructing new tensors from old

ones. In the remainder of this chapter, instead of referring to theTij(say) as the

componentsof a second-order tensorT, we may sometimes simply refer toTij

as the tensor. It should always be remembered, however, that theTijare in fact

just the components ofTin a given coordinate system and thatTij′refers to the

components of thesametensorTin a different coordinate system.

The addition and subtraction of tensors follows an obvious definition; namely

that ifVij···kandWij···kare (the components of) tensors of the same order, then

their sum and difference,Sij···kandDij···krespectively, are given by

Sij···k=Vij···k+Wij···k,

Dij···k=Vij···k−Wij···k,

for each set of valuesi,j,...,k.ThatSij···kandDij···kare the components of

tensors follows immediately from the linearity of a rotation of coordinates.

It is equally straightforward to show that if theTij···kare the components of

a tensor, then so is the set of quantities formed by interchanging the order of (a

pair of) indices, e.g.Tji···k.

IfTji···kis found to be identical withTij···kthenTij···kis said to besymmetric

with respect to its first two subscripts (or simply ‘symmetric’, for second-order

tensors). If, however,Tji···k=−Tij···kfor every element then it is anantisymmetric

tensor. An arbitrary tensor is neither symmetric nor antisymmetric but can always

be written as the sum of a symmetric tensorSij···kand an antisymmetric tensor

Aij···k:

Tij···k=^12 (Tij···k+Tji···k)+^12 (Tij···k−Tji···k)

=Sij···k+Aij···k.

Of course these properties are valid for any pair of subscripts.

In (26.20) in the previous section we had an example of a kind of ‘multiplication’

of two tensors, thereby producing a tensor of higher order – in that case two

first-order tensors were multiplied to give a second-order tensor. Inspection of

(26.21) shows that there is nothing particular about the orders of the tensors

involved and it follows as a general result that the outer product of anNth-order

tensor with anMth-order tensor will produce an (M+N)th-order tensor.