26.7 The quotient law
An operation that produces the opposite effect – namely, generates a tensorof smaller rather than larger order – is known ascontractionand consists of
making two of the subscripts equal and summing over all values of the equalised
subscripts.
Show that the process of contraction of anNth-order tensor produces another tensor, of
orderN− 2.LetTij···l···m···kbe the components of anNth-order tensor, then
Tij′···l···m···k=LipLjq···Llr···Lms···Lkn
︸ ︷︷ ︸
NfactorsTpq···r···s···n.Thus if, for example, we make the two subscriptslandmequal and sum over all values
of these subscripts, we obtain
Tij′···l···l···k=LipLjq···Llr···Lls···LknTpq···r···s···n
=LipLjq···δrs···LknTpq···r···s···n
= LipLjq···Lkn
︸ ︷︷ ︸
(N−2) factorsTpq···r···r···n,showing thatTij···l···l···kare 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 thetrace of the corresponding matrix. The traceTiiitself 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 tensorlanguage as forming the outer productTij=uivjof two first-order tensorsuand
vand then contracting the second-order tensorTso formed, to giveTii=uivi,a
scalar (invariant under a rotation of axes).
As yet another example of a familiar operation that is a particular case of acontraction, we may note that the multiplication of a column vector [ui]bya
matrix [Bij] to produce another column vector [vi],
Bijuj=vi,can be looked upon as the contractionTijjof the third-order tensorTijkformed
from the outer product ofBijanduk.
26.7 The quotient lawThe 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 thatBandCare tensors and also
that
Apq···k···mBij···k···n=Cpq···mij···n, (26.25)