Laplace operator

In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the second partial derivatives of a dependent variable.

This corresponds to div(grad φ), hence the use of the symbol del to represent it:

$\Delta \phi =\nabla ^{2}\phi =\nabla \cdot (\nabla \phi )$

It is also written as Δ.

In two-dimensional Cartesian coordinates, the Laplacian is:

$\Delta =\nabla ^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}$

In three:

$\Delta =\nabla ^{2}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}+{\partial ^{2} \over \partial z^{2}}$

In spherical coordiantes:

$\nabla ^{2}t={1 \over r^{2}}{\partial \over \partial r}(r^{2}{\partial t \over \partial r})+{1 \over r^{2}sin\theta }{\partial \over \partial \theta }(sin\theta {\partial t \over \partial \theta })+{1 \over r^{2}sin^{2}\theta }{\partial ^{2}t \over \partial \phi ^{2}}$

The Laplacian is linear:

$\nabla ^{2}(f+g)=\nabla ^{2}f+\nabla ^{2}g$

This might be true, but it's a bit late at night to be thinking clearly enough to be sure:

$\nabla ^{2}(fg)=(\nabla ^{2}f)g+2(\nabla f)\cdot (\nabla g)+f(\nabla ^{2}g)$

It occurs in Laplace's equation and Poisson's equation.