12—Tensors 308answer; it will be the same as ever. It will however be in a neater form and hence easier to manipulate.)
The reciprocal basis is defined by the equation
~ei.~ej=δji=
{
1 ifi=j
0 ifi 6 =j
(12.32)
The~ej’s are vectors. The index is written as a superscript to distinguish it from the original basis,~ej.
~e^1
~e 1
~e 2
~e^2
To elaborate on the meaning of this equation,~e^1 is perpendicular to the plane defined by~e 2
and~e 3 and is therefore more or less in the direction of~e 1. Its magnitude is adjusted so that the scalar
product
~e^1 .~e 1 = 1.
The “direct basis” and “reciprocal basis” are used in solid state physics and especially in describing
X-ray diffraction in crystallography. In that instance, the direct basis is the fundamental lattice of the
crystal and the reciprocal basis would be defined from it. The reciprocal basis is used to describe the
wave number vectors of scattered X-rays.
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)
=uivjδij
=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~uin terms
of the reciprocal basis and~vin the direct basis, then
~u.~v=uivi=uivi (12.33)
Summation Convention
At this point, 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 compo-
nents of the vector:vi. The components in the reciprocal basis are called* the covariant components:
vi.
* These terms are of more historical than mathematical interest.