文档介绍：35
§ TENSOR CONCEPTS AND TRANSFORMATIONS
For be1, be2, be3 independent orthogonal unit vectors (base vectors), we may write any vector A~ as
A~ = A1 be1 + A2 be2 + A3 be3
where (A1,A2,A3) are the coordinates of A~ relative to the base vectors chosen. ponents are the
projection of A~ onto the base vectors and
A~ =(A~ · be1) be1 +(A~ · be2) be2 +(A~ · be3) be3.
Select any three independent orthogonal vectors, (E~1, E~2, E~3), not necessarily of unit length, we can then
write
E~1 E~2 E~3
be1 = , be2 = , be3 = ,
|E~1| |E~2| |E~3|
and consequently, the vector A~ can be expressed as
! ! !
A~ · E~1 A~ · E~2 A~ · E~3
A~ = E~1 + E~2 + E~3.
E~1 · E~1 E~2 · E~2 E~3 · E~3
Here we say that
A~ · E~(i)
,i=1, 2, 3
E~(i) · E~(i)
are ponents of A~ relative to the chosen base vectors E~1, E~2, E~3. Recall that the parenthesis about
the subscript i denotes that there is no summation on this subscript. It is then treated as a free subscript
which can have any of the values 1, 2or3.
Reciprocal Basis
Consider a set of any three independent vectors (E~1, E~2, E~3) which are not necessarily orthogonal, nor of
unit length. In order to represent the vector A~ in terms of these vectors we must ponents (A1,A2,A3)
such that
1 2 3
A~ = A E~1 + A E~2 + A E~3.
This can be done by taking appropriate projections and obtaining three equations and three unknowns from
which ponents are determined. A much easier way to find ponents (A1,A2,A3) is to construct
1 2 3 1 2 3
a reciprocal basis (E~ , E~ , E~ ). Recall that two bases (E~1, E~2, E~3)and(E~ , E~ , E~ ) are said to be reciprocal
if they satisfy the condition
j j 1ifi = j
E~i · E~ = δi = .
0ifi =6 j
1 1 1 1 1
Note that E~2 · E~ = δ2 =0 and E~3 · E~ = δ3 = 0 so that the vector E~ is perpendicular to both the
vectors E~2 and E~3. (. A vector from one basis is orthogonal to two of the vectors from the other basis.)
1 −1
We can therefore write E~ = V E~2 × E~3 where V is a constan