Okay.
1.Those Gamma's components are not zero...Not in the general case,anyway...
2.I'll use the column-semicolumn notation (though we physicst are not really fond of it...)
In the following,"g" is the determinant of the metric tensor:
g_{,i}=g \ g^{kl} g_{kl,i} (1)
(1):This is the rule as how to differentiate the determinant of a matrix...
A^{i}_{;i}=A^{i}_{,i}+\Gamma^{i} \ _{ij}A^{j}(2)
(2):The covariant divergence (the one u're interested in).
\Gamma^{i} \ _{ij} =\frac{1}{2}g^{ki}(g_{kj,i}+g_{ik,j}-g_{ji,k})<br />
=\frac{1}{2}g^{ki}g_{ki,j}=\frac{1}{2g}g_{,j} (3)
In getting (3) I made use of the definition of 2-nd rank Christoffel symbols (mannifold with both connection & metric) and of relation (1).
Use (3) and (2) and the fact that:
g=(\sqrt{g})^{2} (4)
to get your result.
Report any problems...
Daniel.
