# Definition:Laplace's Equation

## Equation

Laplace's equation is a second order PDE of the form:

$\dfrac {\partial^2 \psi} {\partial x^2} + \dfrac {\partial^2 \psi} {\partial y^2} + \dfrac {\partial^2 \psi} {\partial z^2} = 0$

or:

$\nabla^2 \psi = 0$

where $\nabla^2$ denotes the Laplacian operator.

## Also known as

Laplace's equation is also known as the equation of continuity.

Some sources render it as Laplace equation.

## Examples

### Electric Force in Free Space

Let $R$ be a region of free space.

Let $\mathbf V$ be an electric force in a given electric field over $R$.

Then $\mathbf V$ satisfies Laplace's equation.

### Irrotational Motion of Incompressible Fluid

Let $B$ be a body of incompressible fluid.

Let $\mathbf V$ be the vector field which describes the irrotational motion of $B$.

Then $\mathbf V$ satisfies Laplace's equation.

## Source of Name

This entry was named for Pierre-Simon de Laplace.