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Completeness for Tensor Basic Vectors
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Phil's working note, from the files for the May 2015 update of his curvilinear tensor document. It checks that expanding a vector V in the basis e_n with coefficients given by the reciprocal vectors E_n is consistent, which requires a completeness sum over E_n and e_n to equal a Kronecker delta. The proof uses the relations (e_n)_c = S_cn and (E_n)_k = R_ni g_ik, and he says the result is written up at the end of Section 6.2.
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Completeness for Tensor Basic Vectors
Consider this expansion, all in developmental notation
V = Σn V'n en where En V = V'n
First of all, how do I know the above equation pair is valid? Well dot the left one into V
Em V = Em [Σn V'n en] = Σn V'n Em en = Σn V'n δmn = Vm
Then
Vi = Σn (En V) (en)i = Σn ( Σab ab(En)aVb ) (en)i
= Σb Vb { Σan ab(En)a (en)i }
For this to be true, we must have
Σan ba (En)a (en)i = δb,i
or
Σan ba (En)a (en)c = δb,c
or
ba (En)a (en)c = δb,c (*) // implied sum on a and n, not on b and c
Is this true for general Picture A ?
I do know these facts,
(en)c = Scn (3.2.6)
(En)k = g'niSki = Rnigik (6.14)
and I rewrite the last as
(En)a = g'niSai = Rnigia
Then let's see if (*) is true or not:
ba (En)a (en)c = ba Rni gia Scn = Scn Rni gia ab
= [SRg]ab = [1 1 ]ab = δa,b QED
A better way to write things is
(n)a (en)c = δa,c (*) // implied sum on n
and this is the way I have now written things up at the end of Section 6.2,