Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

but since|L|=±1 we may rewrite this as

ijk=|L|LilLjmLknlmn.

From this expression, we see that althoughijkbehaves as a tensor under proper

rotations, as discussed in section 26.8, it should properly be regarded as a third-

order Cartesianpseudotensor.

Ifbjandckare the components of vectors, show that the quantitiesai=ijkbjckform

the components of a pseudovector.

In a new coordinate system we have

a′i=′ijkb′jc′k

=|L|LilLjmLknlmnLjpbpLkqcq

=|L|Lillmnδmpδnqbpcq

=|L|Lillmnbmcn

=|L|Lilal,

from which we see immediately that the quantitiesaiform the components of a pseu-
dovector.

The above example is worth some further comment. If we denote the vec-

tors with componentsbjandckbybandcrespectively then, as mentioned in

section 26.8, the quantitiesai=ijkbjckare the components of the real vector

a=b×c,provided that we are using a right-handed Cartesian coordinate system.

However, in a coordinate system that is left-handed the quantititesa′i=′ijkb′jc′k

arenotthe components of the physical vectora=b×c, which has, instead, the

components−a′i. It is therefore important to note the handedness of a coordinate

system before attempting to write in component form the vector relationa=b×c

(which is true without reference to any coordinate system).

It is worth noting that, although pseudotensors can be useful mathematical

objects, the description of the real physical world must usually be in terms of

tensors (i.e. scalars, vectors, etc.).§For example, the temperature or density of a

gas must be a scalar quantity (rather than a pseudoscalar), since its value does

not change when the coordinate system used to describe it is inverted through

the origin. Similarly, velocity, magnetic field strength or angular momentum can

only be described by a vector, and not by a pseudovector.

At this point, it may be useful to make a brief comment on the distinction

betweenactiveandpassivetransformations of a physical system, as this difference

often causes confusion. In this chapter, we are concerned solely with passive trans-

§In fact the quantum-mechanical description of elementary particles, such as electrons, protons and

neutrons, requires the introduction of a new kind of mathematical object called aspinor,whichis

not a scalar, vector, or more general tensor. The study of spinors, however, falls beyond the scope

of this book.