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.
 | Work In Progress In particular: It is appreciated that this is a special case of the more general Poisson's equation, but until we are in a position to present a full exposition of Poisson's equation in the general context of differential geometry, it is probably best to leave this as the main definition. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
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.
Also see
Source of Name
This entry was named for Siméon-Denis Poisson.
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $7$. The Classification of Vector Fields: $\text {(ii)}$