PlanetPhysics/Laplacian

The Laplacian is a vector differential operator. Like all vector operators, it is given in different forms in different coordinate systems. In general it is given by: $$ \nabla^2 f = \Delta f = \sum_i \frac{\partial f_i}{\partial x^2_i} $$ where the subscript $$i$$ refers to the different coordinate components of the vector $$f$$.

Laplacian in Cartesian coordinates
As usual with vector operators, the Cartesian form is the easiest to remember and apply. $$ \nabla^2 = {\partial \over \partial x^2} + {\partial \over \partial y^2} + {\partial \over \partial z^2} $$

Laplacian in spherical coordinates
$$ \nabla _{sph}^{2} = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r}\right) + \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} \left( sin \theta \frac{\partial}{\partial \theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2}{\partial \phi^2} $$

Laplacian in cylindrical coordinates
$$ \nabla ^2 = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} + \frac{\partial^2}{\partial z^2} $$