On Sun, 29 Jun 2008 13:23:42 -0700, Jerry Comisar
<jcomisar@[EMAIL PROTECTED]
> wrote:
>Hello,
>
>I have been studying Synge and Schild's book "Tensor Calculus". I am
>stuck on Equation 3.104. I do not see why the second pair of partial
>derivatives cancel out.
>
>Can anyone please help me with this?
>
>Thanks.
>
>Jerry Comisar
>Instructor
>UC Santa Cruz Extension
The reason they do not cancel is there is a glaring error in this
calculation. The calculation consists of taking two covariant
derivatives. Since we know the first covariant derivative of a first
rank covariant tenser ( a covariant vector ) produces a second rank
(covariant) tensor, the second covariant derivate expression should
have three terms. Let Sr;m = Tr,m - G(i,rm)Ti, where a semi-colon
indicates a covariant derivative, a comma represents a partial with
respect to the index/letter following, small letters indicate indices,
and G(i,rm) represents the Christoffel symbol. Then,
Tr;m;n = (Sr;m),n - G(j,rn)Sj;m - G(j,rm)Sj;n.
Doing this correctly with m and n interchanged leads to the correct
cancellations.
Kevin
|
