If I have just these three equations \begin{align*} x = uv\cos\phi,\quad y = uv\sin\phi,\quad z = \frac{1}{2}(u^2-v^2) \end{align*} I'm asked to find the divergence $\nabla_\mu V^\mu$ and Laplacian $\nabla_\mu\nabla^\mu f$.
I just really need so guidence because $V^\mu$ is a vector and I don't even know what vector to use to then calculate
\begin{align*} \nabla_\mu V^\mu = \partial_\mu V^\mu + \Gamma^\nu_{\mu\lambda}V^\lambda\text{ where }\partial_\mu V^\mu = \frac{\partial V^\mu}{\partial X^i} \end{align*} I can't even find how to calculate this Laplacian creature.
In local coordinates, the divergent of a vector field $V$ is: $$\textrm{div}( V)=\frac{1}{\sqrt{\det (g_{\mu\nu}})}\partial_\mu\left(\sqrt {\det (g_{\mu\nu})}V^{\nu}\right).$$
For the laplacian, remember that $\Delta f=\textrm{div}(\nabla f)$.