Laplace operator

In vector calculus, the Laplace operator or Laplacian is a differential operator[?]. It is 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:

<math>\nabla^2 \phi = \nabla \cdot ( \nabla \phi )</math>

It is also written as Δ.

In two dimensional Cartesian coordinates, the Laplacian is:

<math>\nabla^2 = {\partial^2 \over \partial x^2 } +

{\partial^2 \over \partial y^2 } </math>

In three:

<math>\nabla^2 =

{\partial^2 \over \partial x^2 } + {\partial^2 \over \partial y^2 } + {\partial^2 \over \partial z^2 } </math>

It occurs, for example, in Laplace's equation.