Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

In fact any scalar product of two first-order tensors (vectors) is a zero-order

tensor (scalar), as might be expected since it can be written in a coordinate-free

way asu·v.

By considering the components of the vectorsuandvwith respect to two Cartesian

coordinate systems (related by a rotation), show that the scalar productu·vis invariant

under rotation.

In the original (unprimed) system the scalar product is given in terms of components by
uivi(summed overi), and in the rotated (primed) system by

u′iv′i=LijujLikvk=LijLikujvk=δjkujvk=ujvj,

where we have used the orthogonality relation(26.6). Since the resulting expression in the
rotated system is the same as that in the original system, the scalar product is indeed
invariant under rotations.

The above result leads directly to the identification of many physically im-

portant quantities as zero-order tensors. Perhaps the most immediate of these is

energy, either as potential energy or as an energy density (e.g.F·dr,eE·dr,D·E,

B·H,μ·B), but others, such as the angle between two directed quantities, are

important.

As mentioned in the first paragraph of this chapter, in most analyses of physical

situations it is a scalar quantity (such as energy) that is to be determined. Such

quantities areinvariantunder a rotation of axes and so it is possible to work with

the most convenient set of axes and still have confidence in the results.

Complementing the way in which a zero-order tensor was obtained from two

first-order tensors, so a first-order tensor can be obtained from a zero-order tensor

(i.e. a scalar). We show this by taking a specific example, that of the electric field

E=−∇φ; this is derived from a scalar, the electrostatic potentialφ, and has

components

Ei=−

∂φ

∂xi

. (26.14)

Clearly, Eis a first-order tensor, but we may prove this more formally by

considering the behaviour of its components (26.14) under a rotation of the

coordinate axes, since the components of the electric fieldEi′are then given by

Ei′=

(

−

∂φ

∂xi

)′

=−

∂φ′

∂x′i

=−

∂xj

∂x′i

∂φ

∂xj

=LijEj, (26.15)

where (26.5) has been used to evaluate∂xj/∂x′i. Now (26.15) is in the form

(26.9), thus confirming that the components of the electric field do behave as the

components of a first-order tensor.