Proving Epsilon and Covariant Derivatives with Christoffel Symbols

[re444]

Homework Statement

1) Show that $$\epsilon_{ijk,m}=0$$ and $$(\sqrt{g})_{,k}=0$$ . Where ' ,k ' , stands for covariant derivative and $$\epsilon$$ is the epsilon permutation symbol.

2)

where the {} is for christoffel symbol of the second kind.

Homework Equations

The Attempt at a Solution

In the case of problem 2, I've almost finished the solution except that the right side of my solution is multiplied by 3 ! I've started differentiating from $$e^{rst}J=e^{ijk}a^{r}_{i}a^{s}_{j}a^{t}_{k}$$ where $$a^{i}_{j}$$ is for $$\frac{\partial x^{i}}{\partial \bar x^{j}}$$ . Using the formula in the 'Relevant equations' section, I finally get to three times the desired solution.
Could someone please help and give me a hint?

thanks,

 

[lippyka]

For the first problem, I know that we have to use the epsilon symbol properties, but I'm not sure how to use it.