# Definition:Poisson's Differential Equation

## Equation

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

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

where $\phi$ is a function which is not identically zero

or:

$\nabla^2 \psi = \phi$

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

## Examples

### General Scenario

Let $R$ be a region of ordinary space.

Let $R$ contain a spatial distribution of sources of flux.

Let $\mathbf V$ be the vector field arising from the sources in $R$.

Then $\mathbf V$ satisfies Poisson's differential equation.

### Electric Field of Volume Distribution of Charges

Let $R$ be a region of ordinary space.

Let $R$ contain a volume distribution of particles bearing electric charge.

Let $\mathbf V$ be the vector field arising from these electric charge.

Then $\mathbf V$ satisfies Poisson's differential equation.

### Gravitational Force inside Mass

Let $B$ be a body situated in ordinary space.

$B$ consists of a volume distribution of particles with mass.

Let $\mathbf V$ be the vector field arising from these electric charge.

Then $\mathbf V$ satisfies Poisson's differential equation.

## Source of Name

This entry was named for Siméon-Denis Poisson.