Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.5 SECOND- AND HIGHER-ORDER CARTESIAN TENSORS

(ii)The gradient of a vector.Supposevirepresents the components of a vector;

let us consider the quantities generated by forming the derivatives of eachvi,

i=1, 2 ,3, with respect to eachxj,j=1, 2 ,3, i.e.

Tij=

∂vi

∂xj

.

These nine quantities form the components of a second-order tensor, as can be

seen from the fact that

Tij′=

∂v′i

∂x′j

=

∂(Likvk)

∂xl

∂xl

∂x′j

=Lik

∂vk

∂xl

Ljl=LikLjlTkl.

In coordinate-free language the tensorTmay be written asT=∇vand hence

gives meaning to the concept of the gradient of a vector, a quantity that was not

discussed in the chapter on vector calculus (chapter 10).

A test of whether any given set of quantities forms the components of a second-

order tensor can always be made by direct substitution of thex′iin terms of the

xi, followed by comparison with the right-hand side of (26.16). This procedure is

extremely laborious, however, and it is almost always better to try to recognise

the set as being expressible in one of the forms just considered, or to make

alternative tests based on the quotient law of section 26.7 below.

Show that theTijgiven by

T=[Tij]=

(

x^22 −x 1 x 2

−x 1 x 2 x^21

)

(26.24)

are the components of a second-order tensor.

Again we consider a rotationθabout thee 3 -axis. Carrying out the direct evaluation first
we obtain, using (26.7),

T 11 ′ =x′ 22 =s^2 x^21 − 2 scx 1 x 2 +c^2 x^22 ,

T 12 ′ =−x′ 1 x 2 ′=scx^21 +(s^2 −c^2 )x 1 x 2 −scx^22 ,

T 21 ′ =−x′ 1 x 2 ′=scx^21 +(s^2 −c^2 )x 1 x 2 −scx^22 ,

T 22 ′ =x′ 12 =c^2 x^21 +2scx 1 x 2 +s^2 x^22.

Now, evaluating the right-hand side of (26.16),

T 11 ′=ccx 22 +cs(−x 1 x 2 )+sc(−x 1 x 2 )+ssx^21 ,

T 12 ′=c(−s)x^22 +cc(−x 1 x 2 )+s(−s)(−x 1 x 2 )+scx^21 ,

T 21 ′=(−s)cx^22 +(−s)s(−x 1 x 2 )+cc(−x 1 x 2 )+csx^21 ,

T 22 ′=(−s)(−s)x^22 +(−s)c(−x 1 x 2 )+c(−s)(−x 1 x 2 )+ccx^21.

After reorganisation, the corresponding expressions are seen to be the same, showing, as
required, that theTijare the components of a second-order tensor.
The same result could be inferred much more easily, however, by noting that theTij
are in fact the components of the outer product of the vector (x 2 ,−x 1 ) with itself. That
(x 2 ,−x 1 ) is indeed a vector was established by (26.12) and (26.13).