## Encyclopedia > Laplace operator

Article Content

# 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:

$\nabla^2 \phi = \nabla \cdot ( \nabla \phi )$

It is also written as Δ.

In two dimensional Cartesian coordinates, the Laplacian is:

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

In three:

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

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 DB ... DB is a French automobile maker; see DB (car) DB is the abbreviation for Deutsche Bahn, the major German railway company DB is the abbreviation of Dominion ...