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:

$\displaystyle \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.

$\displaystyle \nabla^2 = {\partial \over \partial x^2} + {\partial \over \partial y^2} + {\partial \over \partial z^2}
$

Laplacian in spherical coordinates

$\displaystyle \nabla _{sph}^{2} = \frac{1}{r^2} \frac{\partial}{\partial r}\lef...
...\theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2}{\partial \phi^2}
$

Laplacian in cylindrical coordinates

$\displaystyle \nabla ^2 = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{...
...1}{r^2} \frac{\partial^2}{\partial \theta^2} + \frac{\partial^2}{\partial z^2}
$



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by invisiblerhino.