Laplacian

The n-dimensional Laplacian operator in Cartesian coordinates is defined by

$\Delta u = \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2}$

It is an important differential operator which occurs in many equations of mathematical physics and is usually associated with dissipative effects. Some of the important equations are

• Laplace equation
$\Delta u = 0$
• Poisson equation
$\Delta u = f$

Solutions of these equations are very smooth and in most cases are infinitely differentiable (when the associated data of the problem are sufficiently smooth).

The Laplacian operator is invariant under coordinate rotation.

The Laplace operator is also denoted as $\nabla^2$ since it is the divergence of the gradient operator

$\Delta = \nabla^2 = \nabla \cdot \nabla$

Laplacian in cylindrical coordinates

If $(x,r,\phi)$ are cylindrical coordinates, then the Laplacian of a scalar field variable $u$ is

$\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \phi^2}$

Laplacian in spherical coordinates

If $(r,\theta,\phi)$ are spherical coordinates, then the Laplacian of a scalar field variable $u$ is

$\Delta u = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial u}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 u}{\partial \phi^2}$