Mathematical Tools for Physics

12—Tensors 375

Analogous results hold for the expression ofA~in terms of the direct basis.
You can see how the notation forced you into considering this expression forA~. The summation convention
requires one upper index and one lower index, so there is practically no other form that you could even consider
in order to representA~.
The same sort of computations will hold for tensors. Start off with one of second rank. Just as there were
covariant and contravariant components of a vector, there will be covariant and contravariant components of a
tensor.T(~u, ~v)is a scalar. Express~uand~vin contravariant component form:

~u=ui~ei and ~v=vj~ej. Then T(~u, ~v) =T(ui~ei, vj~ej)

=uivjT(~ei, ~ej)

=uivjTij

The numbersTijare called the covariant components of the tensorT.
Similarly, write~uand~vin terms of covariant components:

~u=ui~ei and ~v=vj~ej. Then T(~u, ~v) =T(ui~ei, vj~ej)

=uivjT(~ei, ~ej)

=uivjTij

AndTijare the contravariant components ofT. It is also possible to have mixed components:

T(~u, ~v) =T(ui~ei, vj~ej)

=uivjT(~ei, ~ej)

=uivjTij

As before, from the bilinear functional, a linear vector valued function can be formed such that

T(~u,~v) =~u.T(~v) and T(~v) =~eiT(~ei,~v)

=~eiT(~ei,~v)

For the proof of the last two lines, simply write~uin terms of its contravariant or covariant components respectively.