Mathematical Methods for Physics and Engineering : A Comprehensive Guide

TENSORS

A useful application of (26.30) is in obtaining alternative expressions for vector

quantities that arise from the vector product of a vector product.

Obtain an alternative expression for∇×(∇×v).

As shown in the previous example,∇×(∇×v) can be expressed in tensor form as

[∇×(∇×v)]i=ijkklm

∂^2 vm

∂xj∂xl

=(δilδjm−δimδjl)

∂^2 vm

∂xj∂xl

=

∂

∂xi

(

∂vj

∂xj

)

−

∂^2 vi

∂xj∂xj

=[∇(∇·v)]i−∇^2 vi,

where in the second line we have used the identity (26.30). This result has already
been mentioned in chapter 10 and the reader is referred there for a discussion of its
applicability.

By examining the various possibilities, it is straightforward to verify that, more

generally,

ijkpqr=

∣

∣

∣

∣

∣

∣

δip δiq δir

δjp δjq δjr

δkp δkq δkr

∣

∣

∣

∣

∣

∣

(26.34)

and it is easily seen that (26.30) is a special case of this result. From (26.34) we

can derive alternative forms of (26.30), for example,

ijkilm=δjlδkm−δjmδkl. (26.35)

The pattern of subscripts in these identities is most easily remembered by noting

that the subscripts on the firstδon the RHS are those that immediately follow

(cyclically, if necessary) the common subscript, herei,ineach-term on the LHS;

the remaining combinations ofj, k, l, mas subscripts in the otherδ-terms on the

RHS can then be filled in automatically.

Contracting (26.35) by settingj=l(say) we obtain, sinceδkk= 3 when using

the summation convention,

ijkijm=3δkm−δkm=2δkm,

and by contracting once more, settingk=m, we further find that

ijkijk=6. (26.36)

26.9 Isotropic tensors

It will have been noticed that, unlike most of the tensors discussed (except for

scalars),δijandijkhave the property that all their components have values

that are the same whatever rotation of axes is made, i.e. the component values