The quotient law

26.7 THE QUOTIENT LAW
An operation that produces the opposite effect – namely, generates a tensor
of smaller rather than larger order – is known as contraction and consists of
making two of the subscripts equal and summing over all values of the equalised
subscripts.
Show that the process of contraction of an Nth-order tensor produces another tensor, of
order N − 2.
Let Tij···l···m···k be the components of an Nth-order tensor, then
= Lip Ljq · · · Llr · · · Lms · · · Lkn Tpq···r···s···n .
Tij···l···m···k
8
9:
;
N factors
Thus if, for example, we make the two subscripts l and m equal and sum over all values
of these subscripts, we obtain
= Lip Ljq · · · Llr · · · Lls · · · Lkn Tpq···r···s···n
Tij···l···l···k
= Lip Ljq · · · δrs · · · Lkn Tpq···r···s···n
= Lip Ljq · · · Lkn Tpq···r···r···n ,
8
9:
;
(N − 2) factors
showing that Tij···l···l···k are the components of a (different) Cartesian tensor of order
N − 2. For a second-rank tensor, the process of contraction is the same as taking the
trace of the corresponding matrix. The trace Tii itself is thus a zero-order tensor
(or scalar) and hence invariant under rotations, as was noted in chapter 8.
The process of taking the scalar product of two vectors can be recast into tensor
language as forming the outer product Tij = ui vj of two first-order tensors u and
v and then contracting the second-order tensor T so formed, to give Tii = ui vi , a
scalar (invariant under a rotation of axes).
As yet another example of a familiar operation that is a particular case of a
contraction, we may note that the multiplication of a column vector [ui ] by a
matrix [Bij ] to produce another column vector [vi ],
Bij uj = vi ,
can be looked upon as the contraction Tijj of the third-order tensor Tijk formed
from the outer product of Bij and uk .
26.7 The quotient law
The previous paragraph appears to give a heavy-handed way of describing a
familiar operation, but it leads us to ask whether it has a converse. To put the
question in more general terms: if we know that B and C are tensors and also
that
Apq···k···m Bij···k···n = Cpq···mij···n ,
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(26.25)
TENSORS
does this imply that the Apq···k···m also form the components of a tensor A? Here
A, B and C are respectively of Mth, Nth and (M +N −2)th order and it should be
noted that the subscript k that has been contracted may be any of the subscripts
in A and B independently.
The quotient law for tensors states that if (26.25) holds in all rotated coordinate
frames then the Apq···k···m do indeed form the components of a tensor A. To prove
it for general M and N is no more difficult regarding the ideas involved than to
show it for specific M and N, but this does involve the introduction of a large
number of subscript symbols. We will therefore take the case M = N = 2, but
it will be readily apparent that the principle of the proof holds for general M
and N.
We thus start with (say)
Apk Bik = Cpi ,
(26.26)
where Bik and Cpi are arbitrary second-order tensors. Under a rotation of coordinates the set Apk (tensor or not) transforms into a new set of quantities that
we will denote by Apk . We thus obtain in succession the following steps, using
(26.16), (26.17) and (26.6):
Apk Bik =
=
=
=
=
Cpi
Lpq Lij Cqj
Lpq Lij Aql Bjl
Lpq Lij Aql Lmj Lnl Bmn
Lpq Lnl Aql Bin
(transforming (26.26)),
(since C is a tensor),
(from (26.26)),
(since B is a tensor),
(since Lij Lmj = δim ).
Now k on the left and n on the right are dummy subscripts and thus we may
write
(Apk − Lpq Lkl Aql )Bik = 0.
(26.27)
Since Bik , and hence Bik , is an arbitrary tensor, we must have
Apk = Lpq Lkl Aql ,
showing that the Apk are given by the general formula (26.18) and hence that
the Apk are the components of a second-order tensor. By following an analogous
argument, the same result (26.27) and deduction could be obtained if (26.26) were
replaced by
Apk Bki = Cpi ,
i.e. the contraction being now with respect to a different pair of indices.
Use of the quotient law to test whether a given set of quantities is a tensor is
generally much more convenient than making a direct substitution. A particular
way in which it is applied is by contracting the given set of quantities, having
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