Laplace operator

Coordinate expressions [mathjax]

Two dimensions:

$$
\begin{aligned}\Delta f &=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}\\
\Delta f &=\frac1r\frac{\partial}{\partial r}\left(r\frac{\partial f}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}\\
&=\frac{\partial^2 f}{\partial r^2}+\frac1r\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2}
\end{aligned}
$$

Three dimensions:

$$
\begin{aligned}\Delta f =&\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}\\
\Delta f =&\frac{\partial^2 f}{\partial r^2}+\frac{2}{r} \frac{\partial f}{\partial r}\\
&+\frac{1}{r^2 \sin \theta}\left(\cos \theta \frac{\partial f}{\partial \theta}+\sin \theta \frac{\partial^2 f}{\partial \theta^2}\right)+\frac{1}{r^2 \sin ^2 \theta} \frac{\partial^2 f}{\partial \varphi^2}
\end{aligned}
$$

$latex N$ dimensions:

$$
\begin{aligned}
\Delta f &=\frac{1}{r^{N-1}}\frac{\partial}{\partial r}\left(r^{N-1}\frac{\partial f}{\partial r}\right)
+\frac{1}{r^2} \Delta_{S^{N-1}} f\\
&=\frac{\partial^2 f}{\partial r^2}+\frac{N-1}{r} \frac{\partial f}{\partial r}+\cdots
\end{aligned}
$$

where $latex \Delta_{S^{N-1}}$ is the Laplace-Beltrami operator on the $latex (N-1)$-sphere, known as the spherical Laplacian.

Euclidean invarian

The Laplacian is invariant under all Euclidean transformations: rotations and translations. In two dimensions, i.e., $$ \Delta \big( f(x\cos\theta-y\sin\theta+a, x\sin\theta+y\cos\theta+b) \big)= (\Delta f)(x\cos\theta-y\sin\theta+a, x\sin\theta+y\cos\theta+b) $$ for all \theta, a and $latex b$ .

In arbitrary dimensions,
$$ \Delta (f\circ\rho)=(\Delta f)\circ \rho,\quad \Delta (f\circ\tau)=(\Delta f)\circ \tau, $$

where \rho is a rotation and \tau is a translation. (More generally, this remains true when \rho is an orthogonal transformation such as reflectino.)

In fact, the algebra of all scalar linear differential operators with constant coefficientes, that commute with all Euclidean transformations, is the polynomial algebra generated by Laplace operator.