Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

Rotate byπ/2 about theOx 3 -axis:L 12 =−1,L 21 =1,L 33 = 1, the otherLij=0.

(d)T 111 =(−1)×(−1)×(−1)×T 222 =−T 222 ,

(e)T 112 =(−1)×(−1)× 1 ×T 221 ,

(f)T 221 =1× 1 ×(−1)×T 112 ,

(g)T 123 =(−1)× 1 × 1 ×T 213.

Relations (a) and (d) show that elements with all subscripts the same are zero. Relations
(e), (f) and (b) show that all elements with repeated subscripts are zero. Relations (g) and
(c) show thatT 123 =T 231 =T 312 =−T 213 =−T 321 =−T 132.
In total,Tijkdiffers fromijkby at most a scalar factor, but sinceijk(and henceλijk)
has already been shown to be an isotropic tensor,Tijkmust be the most general third-order
isotropic Cartesian tensor.

Using exactly the same procedures as those employed forδijandijk,itmaybe

shown that the only isotropic first-order tensor is the trivial one with all elements

zero.

26.10 Improper rotations and pseudotensors

So far we have considered rigid rotations of the coordinate axes described by

an orthogonal matrixLwith|L|= +1, (26.4). Strictly speaking such transfor-

mations are calledproper rotations. We now broaden our discussion to include

transformations that are still described by an orthogonal matrixLbut for which

|L|=−1; these are calledimproper rotations.

This kind of transformation can always be considered as aninversionof the

coordinate axes through the origin represented by the equation

x′i=−xi, (26.38)

combined with a proper rotation. The transformation may be looked upon

alternatively as one that changes an initially right-handed coordinate system into

a left-handed one; any prior or subsequent proper rotation will not change this

state of affairs. The most obvious example of a transformation with|L|=−1is

the matrix corresponding to (26.38) itself; in this caseLij=−δij.

As we have emphasised in earlier chapters, any real physical vectorvmay be

considered as a geometrical object (i.e. an arrow in space), which can be referred

to independently of any coordinate system and whose direction and magnitude

cannot be altered merely by describing it in terms of a different coordinate system.

Thus the components ofvtransform asv′i=Lijvjunderallrotations (proper and

improper).

We can define another type of object, however, whose components may also

be labelled by a single subscript but which transforms asv′i=Lijvjunder proper

rotations and asvi′=−Lijvj(note the minus sign) under improper rotations. In

this case, theviare not strictly the components of a true first-order Cartesian

tensor but instead are said to form the components of a first-order Cartesian

pseudotensororpseudovector.