Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.10 IMPROPER ROTATIONS AND PSEUDOTENSORS

O O

p

v

x 1

x 2

x 3

x′ 1

x′ 2

x′ 3

v′

p′

Figure 26.2 The behaviour of a vectorvand a pseudovectorpunder a

reflection through the origin of the coordinate systemx 1 ,x 2 ,x 3 giving the new

systemx′ 1 ,x′ 2 ,x′ 3.

It is important to realise that a pseudovector (as its name suggests) is not a

geometrical object in the usual sense. In particular, it shouldnotbe considered

as a real physical arrow in space, since its direction is reversed by animproper

transformation of the coordinate axes (such as an inversion through the origin).

This is illustrated in figure 26.2, in which the pseudovectorpis shown as a broken

line to indicate that it is not a real physical vector.

Corresponding to vectors and pseudovectors, zeroth-order objects may be

divided into scalars and pseudoscalars – the latter being invariant under rotation

but changing sign on reflection.

We may also extend the notion of scalars and pseudoscalars, vectors and pseu-

dovectors, to objects with two or more subscripts. For two subcripts, as defined

previously, any quantity with components that transform asTij′=LikLjlTklun-

derallrotations (proper and improper) is called a second-order Cartesian tensor.

If, however,Tij′=LikLjlTklunder proper rotations butTij′=−LikLjlTklunder

improper ones (which include reflections), then theTijare the components of

a second-order Cartesian pseudotensor. In general the components of Cartesian

pseudotensors of arbitary order transform as

Tij′···k=|L|LilLjm···LknTlm···n, (26.39)

where|L|is the determinant of the transformation matrix.

For example, from (26.29) we have that

|L|ijk=LilLjmLknlmn,