C:\Book\Booktex\C1s2.DVI
35 1.2 TENSOR CONCEPTS AND TRANSFORMATIONS § ~ For e1, e2, e3 independent orthogonal unit vectors (base vectors), we may write any vector A as ~ b b b A = A1 e1 + A2 e2 + A3 e3 ~ where (A1,A2,A3) are the coordinates of A relativeb to theb baseb vectors chosen. These components are the projection of A~ onto the base vectors and A~ =(A~ e ) e +(A~ e ) e +(A~ e ) e . · 1 1 · 2 2 · 3 3 ~ ~ ~ Select any three independent orthogonal vectors,b b (E1, Eb 2, bE3), not necessarilyb b of unit length, we can then write ~ ~ ~ E1 E2 E3 e1 = , e2 = , e3 = , E~ E~ E~ | 1| | 2| | 3| and consequently, the vector A~ canb be expressedb as b A~ E~ A~ E~ A~ E~ A~ = · 1 E~ + · 2 E~ + · 3 E~ . ~ ~ 1 ~ ~ 2 ~ ~ 3 E1 E1 ! E2 E2 ! E3 E3 ! · · · Here we say that A~ E~ · (i) ,i=1, 2, 3 E~ E~ (i) · (i) ~ ~ ~ ~ are the components of A relative to the chosen base vectors E1, E2, E3. 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 (E1, E2, E3) which are not necessarily orthogonal, nor of unit length. In order to represent the vector A~ in terms of these vectors we must find components (A1,A2,A3) such that ~ 1 ~ 2 ~ 3 ~ A = A E1 + A E2 + A E3. This can be done by taking appropriate projections and obtaining three equations and three unknowns from which the components are determined.
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