# Laplacian

Let $(x_{1},\ldots,x_{n})$ be Cartesian coordinates for some open set $\Omega$ in $\mathbb{R}^{n}$. 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 $f:\Omega\to\mathbb{C}$, 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.