Mathematical Tools for Physics

12—Tensors 374

The basis reciprocal to the reciprocal basis is the direct basis.
Now to use these things: Expand the vector~uin terms of the direct basis and~vin terms of the reciprocal
basis.
~u=ui~ei and ~v=vj~ej. Then ~u.~v= (ui~ei).(vj~ej)
=uivjδji
=uivi=u^1 v 1 +u^2 v 2 +u^3 v 3.

Notation: The superscript on the components (ui) will refer to the components in the direct basis (~ei); the
subscripts (vj) will come from the reciprocal basis (~ej). You could also have expanded~u in terms of the
reciprocal basis and~vin the direct basis, then

~u.~v=uivi=uivi (32)

Summation Convention
At this point I modify the previously established summation convention: Like indices in a given term are to be
summed when one is a subscript and one is a superscript. Furthermore the notation is designed so that this is the
only kind of sum that should occur. If you find a term such asuivithen this means that you made a mistake.
The scalar product now has a simple form in terms of components (at the cost of introducing an auxiliary
basis set). Now for further applications to vectors and tensors.
Terminology: The components of a vector in the direct basis are called the contravariant components of
the vector:vi. The components in the reciprocal basis are called* the covariant components:vi.
Examine the component form of the basic representation theorem for linear functionals, as in Eqs. ( 25 ) and
( 26 ).
f(~v) =A~.~v for all ~v.
Claim: A~=~eif(~ei) =~eif(~ei)

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).

* These terms are of more historical than mathematical interest.