TENSORS
does this imply that theApq···k···malso form the components of a tensorA?Here
A,BandCare respectively ofMth,Nth and (M+N−2)th order and it should be
noted that the subscriptkthat has been contracted may be any of the subscripts
inAandBindependently.
Thequotient lawfor tensors states that if (26.25) holds in all rotated coordinateframes then theApq···k···mdo indeed form the components of a tensorA.Toprove
it for generalMandNis no more difficult regarding the ideas involved than to
show it for specificMandN, but this does involve the introduction of a large
number of subscript symbols. We will therefore take the caseM=N= 2, but
it will be readily apparent that the principle of the proof holds for generalM
andN.
We thus start with (say)ApkBik=Cpi, (26.26)whereBikandCpiare arbitrary second-order tensors. Under a rotation of coor-
dinates the setApk(tensor or not) transforms into a new set of quantities that
we will denote byA′pk. We thus obtain in succession the following steps, using
(26.16), (26.17) and (26.6):
A′pkB′ik=Cpi′ (transforming (26.26)),
=LpqLijCqj (sinceCis a tensor),
=LpqLijAqlBjl (from (26.26)),
=LpqLijAqlLmjLnlBmn′ (sinceBis a tensor),
=LpqLnlAqlBin′ (sinceLijLmj=δim).Nowkon the left andnon the right are dummy subscripts and thus we maywrite
(A′pk−LpqLklAql)B′ik=0. (26.27)SinceBik, and henceB′ik, is an arbitrary tensor, we must have
A′pk=LpqLklAql,showing that theA′pkare given by the general formula (26.18) and hence that
theApkare 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
ApkBki=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 isgenerally 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