Laplacian

Let $(x_1,\mathrm{\dots },x_n)$ be Cartesian coordinates for some open set $\mathrm{\Omega }$in ${ℝ}^n$.Then the Laplacian differential operator $\mathrm{\Delta }$ is defined as

$$\mathrm{\Delta }=\frac{{\partial }^2}{\partial x_1^2}+\mathrm{\cdots }+\frac{{\partial }^2}{\partial x_n^2}.$$

In other words, if $f$ is a twice differentiable function $f:\mathrm{\Omega }\to ℂ$, then

$$\mathrm{\Delta }f=\frac{{\partial }^{2}f}{\partial x_1^2}+\mathrm{\cdots }+\frac{{\partial }^{2}f}{\partial x_n^2}.$$

A coordinate independent definition of the Laplacianis $\mathrm{\Delta }=\nabla \cdot \nabla$, i.e., $\mathrm{\Delta }$ is the composition ofgradient 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¡http://planetmath.org/?method=l2h&from=collab&id=76&op=getobj”¿ to see an article that derives the Laplacian in spherical coordinates.