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I've been asked to show the following: For a vector field $V$ on a semi-Riemannian manifold with metric $g$ that $$Div \cdot V = \frac{1}{\sqrt{\det(g)}}\partial_i\left(\sqrt{\det(g)}V^i\right)$$ I know we're supposed to use Christoffel symbols as well as a few matrix formulas, but I'm not sure how to proceed. In particular, we were given that for a (invertible) matrix $M$ with some parameter $s$, that $$\frac{d}{ds}\det M(s)=\det M(s) \cdot tr\left(M(s)^{-1}\frac{d}{ds}M(s)\right)$$ and $$\frac{d}{ds}(M(s)^{-1})=-M(s)^{-1}M'(s)M(s)^{-1}$$ Any help would be greatly appreciated.

The definition of divergence that we were given was $$Div \cdot V = \nabla_{\partial_i}V^i = \partial_iV^i+\Gamma_{ij}^iV^i$$

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