Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.4 FIRST- AND ZERO-ORDER CARTESIAN TENSORS

(ii) Herev 1 =x 2 andv 2 =x 1. Following the same procedure,

v 1 ′=x′ 2 =−sx 1 +cx 2

v 2 ′=x′ 1 =cx 1 +sx 2.

But, by (26.9), for a Cartesian tensor we must have

v 1 ′=cv 1 +sv 2 =cx 2 +sx 1

v 2 ′=(−s)v 1 +cv 2 =−sx 2 +cx 1.

These two sets of expressions do not agree and thus the pair (x 2 ,x 1 ) is not a first-order
Cartesian tensor.
(iii)v 1 =x^21 andv 2 =x^22. As in (ii) above, considering the first component alone is
sufficient to show that this pair is also not a first-order tensor. Evaluatingv 1 ′directly gives

v′ 1 =x′ 12 =c^2 x^21 +2csx 1 x 2 +s^2 x^22 ,

whilst (26.9) requires that

v′ 1 =cv 1 +sv 2 =cx^21 +sx^22 ,

which is quite different.

There are many physical examples of first-order tensors (i.e. vectors) that will be

familiar to the reader. As a straightforward one, we may take the set of Cartesian

components of the momentum of a particle of massm,(mx ̇ 1 ,m ̇x 2 ,m ̇x 3 ). This set

transforms in all essentials as (x 1 ,x 2 ,x 3 ), since the other operations involved,

multiplication by a number and differentiation with respect to time, are quite

unaffected by any orthogonal transformation of the axes. Similarly, acceleration

and force are represented by the components of first-order tensors.

Other more complicated vectors involving the position coordinates more than

once, such as the angular momentum of a particle of massm, namelyJ=

x×p=m(x×x ̇), are also first-order tensors. That this is so is less obvious in

component form than for the earlier examples, but may be verified by writing

out the components ofJexplicitly or by appealing to the quotient law to be

discussed in section 26.7 and using the Cartesian tensorijkfrom section 26.8.

Having considered the effects of rotations on vector-like sets of quantities we

may consider quantities that are unchanged by a rotation of axes. In our previous

nomenclature these have been calledscalarsbut we may also describe them as

tensors of zero order. They contain only one element (formally, the number of

subscripts needed to identify a particular element is zero); the most obvious non-

trivial example associated with a rotation of axes is the square of the distance of

a point from the origin,r^2 =x^21 +x^22 +x^23. In the new coordinate system it will

have the formr′

2

=x′ 1

2

+x′ 2

2

+x′ 3

2

, which for any rotation has the same value as

x^21 +x^22 +x^23.