Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.8 THE TENSORSδijANDijk

Write the following as contracted Cartesian tensors:a·b,∇^2 φ,∇×v,∇(∇·v),∇×(∇×v),

(a×b)·c.

The corresponding (contracted) tensor expressions are readily seen to be as follows:

a·b=aibi=δijaibj,

∇^2 φ=

∂^2 φ

∂xi∂xi

=δij

∂^2 φ

∂xi∂xj

,

(∇×v)i=ijk

∂vk

∂xj

,

[∇(∇·v)]i=

∂

∂xi

(

∂vj

∂xj

)

=δjk

∂^2 vj

∂xi∂xk

,

[∇×(∇×v)]i=ijk

∂

∂xj

(

klm

∂vm

∂xl

)

=ijkklm

∂^2 vm

∂xj∂xl

,

(a×b)·c=δijcijklakbl =iklciakbl.

An important relationship between the-andδ- tensors is expressed by the

identity

ijkklm=δilδjm−δimδjl. (26.30)

To establish the validity of this identity between two fourth-order tensors (the

LHS is a once-contracted sixth-order tensor) we consider the various possible

cases.

The RHS of (26.30) has the values

+1 ifi=landj=m=i, (26.31)

−1ifi=mandj=l=i, (26.32)

0 for any other set of subscript valuesi, j, l, m. (26.33)

In each product on the LHSkhas the same value in both factors and for a

non-zero contribution none ofi, l, j, mcan have the same value ask. Since there

are only three values, 1, 2 and 3, that any of the subscripts may take, the only

non-zero possibilities arei=landj=mor vice versa but not all four subscripts

equal (since then eachfactor is zero, as it would be ifi=jorl=m). This

reproduces (26.33) for the LHS of (26.30) and also the conditions (26.31) and

(26.32). The values in (26.31) and (26.32) are also reproduced in the LHS of

(26.30) since

(i) ifi=landj=m,ijk=lmk=klmand, whetherijkis +1 or−1, the

product of the two factors is +1; and

(ii) ifi=mandj=l,ijk=mlk=−klmand thus the productijkklm(no

summation) has the value−1.

This concludes the establishment of identity (26.30).