Proving Total Covariant/Total Partial Derivative EFE Derivation

[binbagsss]

I'm trying to prove that $\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\partial_{\mu}(\sqrt{-g}v^{\mu})$

So i have $\sqrt{-g}\bigtriangledown_{\mu}v^{\mu}=\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha})$ by just expanding out the definition of the covariant derivative.

Question

My text next makes the equality :
$\sqrt{-g}(\partial_{\mu}v^{mu}+\Gamma^{\mu}_{\mu \alpha}v^{\alpha}=\sqrt{-g}(\partial_{\mu}v^{mu}+(\partial_{\alpha} ln \sqrt{-g})v^{\alpha}\sqrt{-g}=\partial_{\mu}(v^{\mu}\sqrt{-g}$

I don't understand the last 2 equalities.

Particularity the second to last. I have no idea how you go from the connection term to the $ln$ term, if anyone could provide the identities I need or link me somewhere useful (had a google but couldn't find anything)

I think the last equality has used the product rule.
I'm unsure, if this is correct, of differentiating the $\sqrt{-g}$ to go from the last equality to the second from last, the identities I know that might be of use are:

$Tr (ln M) = In (det M)$
$\partial \sqrt{-g}=\frac{-1}{2}\sqrt{-g}g_{\mu \nu}g^{\nu \mu}$

Thanks in advance.

 

[strangerep]

$\partial \sqrt{-g}=\frac{-1}{2}\sqrt{-g}g_{\mu \nu}g^{\nu \mu}$

You need to generalize that last one to $\partial_\alpha$.

The details of Jacobi's formula for the derivative of the determinant might be helpful.

 

[binbagsss]

You need to generalize that last one to $\partial_\alpha$.

The details of Jacobi's formula for the derivative of the determinant might be helpful.

I've used this to attain the last identity given in the OP.
This is for the 2nd to 3rd equality right?
I'm still really unsure what to do, could anybody give me a hint?

Also any identites for the 1st and 2nd equality? I've never seen anything like the connection being expressed as something like that.
Thanks.

 

[strangerep]

There seems to be an unmatched parenthesis in your main equation in post #1.

For the step from 1st to 2nd expressions, study more carefully the meaning of "adjugate" that was mentioned in the Wiki entry on Jacobi's formula. Also, the relationships among "adjugate", "cofactor" and "inverse".

BTW, help sometimes arrives faster if one makes the effort to include a precise reference in one's OP, i.e., which textbook, and which equation or page number therein.