Let $M$ be a $n$-dimensional Riemannian manifold with metric $g$ in radial coordinates $$g=dr^2 + r^2 g_{ij} d\theta_i d\theta_j$$ then $$\Delta r = \frac{n-1}{r} + \frac{\partial}{\partial r} \log \sqrt{G}$$ where $G=\sqrt{\det g_{ij}}$ for $1\leq i,j\leq n-1$.
I am unsure how to prove this identity using the definition of the Hessian, so any comments/suggestions will be much appreciated.