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 $$\frac{1}{\rho^{2}}\frac{\partial}{\partial \rho}\bigl(\rho^{2}\frac{\partial}{\partial\rho}\bigr)+\frac{1}{\rho^{2}\mathrm{sin}\theta}\frac{\partial}{\partial\theta}\bigl(\mathrm{sin}\theta\frac{\partial}{\partial\theta}\bigr)+\frac{1}{\rho^{2}\mathrm{sin}^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}$$