12—Tensors 307x
z
x
~k z
S~
~k
S~
surface→yisin̂
E~alongyˆ E~ to the right coming in
Simulation of a pattern of x’s seen through Calcite. See also Wikipedia: birefringence.
12.5 Non-Orthogonal Bases
The next topic is the introduction of more general computational techniques. These will lift the
restriction on the type of basis that can be used for computing components of various tensors. Until
now, the basis vectors have formed an orthonormal set
|ˆei|= 1, ˆei.ˆej= 0ifi 6 =j
Consider instead a more general set of vectors~ei. These must be independent. That is, in three
dimensions they are not coplanar. Other than this there is no restriction. Since by assumption the
vectors~eispan the space you can write
~v=vieˆi.
with the numbersvibeing as before the components of the vector~v.
NOTE: Here is a change in notation. Before, every index
was a subscript. (It could as easily have been a super-
script.) Now, be sure to make a careful distinction between
sub- and superscripts. They will have different meanings.Reciprocal Basis
Immediately, when you do the basic scalar product you find complications. If~u=uj~ej, then
~u.~v= (uj~ej).(vi~ei) =ujvi~ej.~ei.
But since the~ei aren’t orthonormal, this is a much more complicated result than the usual scalar
product such as
uxvy+uyvy+uzvz.
You can’t assume that~e 1 .~e 2 = 0any more. In order to obtain a result that looks as simple as this
familiar form, introduce an auxiliary basis: thereciprocal basis. (This trick will not really simplify the