Laplacian (operator)

The Laplacian is defined as

\nabla^2 = \nabla \cdot \nabla.

In cartesian coordinates, $\nabla^2 \mathbf F = \frac{\partial^2 F_x}{\partial x^2} + \frac{\partial^2 F_y}{\partial y^2} + \frac{\partial^2 F_z}{\partial z^2}$

In spherical coordinates,

\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial}{\partial r}\right) + \frac{1}{r^2\sin\theta} \frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right) + \frac{1}{r^2 \sin^2\theta} \left(\frac{\partial^2}{\partial \phi^2}\right).