Mathematical Tools for Physics - Department of Physics - University

12—Tensors 309

Examine the component form of the basic representation theorem for linear functionals, as in
Eqs. (12.18) and (12.19).

f(~v) =A~.~v for all ~v.

Claim: A~=~eif(~ei) =~eif(~ei) (12.34)

The proof of this is as before: write~vin terms of components and compute the scalar productA~.~v.

~v=vi~ei. Then A~.~v=

(

~ejf(~ej)

)

.

(

vi~ei

)

=vif(~ej)δij

=vif(~ei) =f(vi~ei) =f(~v).

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 (12.35)

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 (12.36)

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.
All previous statements concerning the symmetry properties of tensors are unchanged because
they were made in a way independent of basis, though it’s easy to see that the symmetry properties