Laplacian

The Laplacian is a differential operator of the form
$\sum _{i}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}$
where $x_{i}$ are Cartesian (that is, rectangular) co-ordinates. The Laplacian is usually denoted by the symbol $\Delta$ or written as the gradient squared $\nabla ^{2}$.

In cylindrical co-ordinates, the Laplacian takes the form
${\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}{\bigl (}\rho {\frac {\partial }{\partial \rho }}{\bigr )}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$

In spherical co-ordinates, the Laplacian is
${\frac {1}{\rho ^{2}}}{\frac {\partial }{\partial \rho }}{\bigl (}\rho ^{2}{\frac {\partial }{\partial \rho }}{\bigr )}+{\frac {1}{\rho ^{2}\mathrm {sin} \theta }}{\frac {\partial }{\partial \theta }}{\bigl (}\mathrm {sin} \theta {\frac {\partial }{\partial \theta }}{\bigr )}+{\frac {1}{\rho ^{2}\mathrm {sin} ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}$