Laplacian

Let $(x_1, \ldots, x_n)$ be Cartesian coordinates for some open set $\Omega$ in $\sR^n$ . Then the Laplacian differential operator $\Delta$ is defined as $$ \Delta = \frac{\partial^2 }{\partial x_1^2} + \cdots + \frac{\partial^2 }{\partial x_n^2}. $$ In other words, if $f$ is a twice differentiable function $f:\Omega\to \sC$ , then $$ \Delta f = \frac{\partial^2 f}{\partial x_1^2} + \cdots + \frac{\partial^2 f}{\partial x_n^2}. $$ A coordinate independent definition of the Laplacian is $\Delta = \nabla \cdot \nabla$ , i.e., $\Delta$ is the composition of gradient and codifferential.

A harmonic function is one for which the Laplacian vanishes.

Notes

An older symbol for the Laplacian is $\nabla^2$ - conceptually the scalar product of $\nabla$ with itself. This form is more favoured by physicists.

Derivation

Click here to see an article that derives the Laplacian in spherical coordinates.