The covariant derivatives of a four-vector is $$ \nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda} $$ It has the following identity:
$$ \nabla_{\mu}U^{\mu} = \frac{\partial_{\mu}(\sqrt{-g}U^{\mu})}{\sqrt{-g}} $$
where $g := \mbox{det}(g_{\mu\nu}$). Could someone show the derivation of the identity? I am deeply grateful for your response.
