Suppose $f:\mathbb{R}^3\to\mathbb{R}$. The Laplacian of $f$ in spherical coordinates is $\nabla^2 f(r, \phi, \theta) = \frac 1 {r^2} \frac {\partial} {\partial r} \left ( r^2 \frac {\partial f} {\partial r} \right ) + \frac 1 {r^2 \sin \phi} \frac {\partial} {\partial \phi} \left ( \sin \phi \frac {\partial f} {\partial \phi} \right ) + \frac 1 {r^2 \sin^2 \phi} \frac {\partial^2 f} {\partial \theta^2}$.
Suppose $g:\mathbb{R}^2\to\mathbb{R}$. The Laplacian of $g$ in polar coordinates is $\nabla^2 g(r, \theta) = \frac 1 r \frac {\partial} {\partial r} \left ( r \frac {\partial g} {\partial r} \right ) + \frac 1 {r^2} \frac {\partial^2 g} {\partial \theta^2}$.
Is there a way to derive the polar version from the spherical one?