KNOWPIA
WELCOME TO KNOWPIA

**Poisson's equation** is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.^{[1]}^{[2]}

Poisson's equation is

In three-dimensional Cartesian coordinates, it takes the form

When identically, we obtain Laplace's equation.

Poisson's equation may be solved using a Green's function:

In the case of a gravitational field **g** due to an attracting massive object of density *ρ*, Gauss's law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity:

Since the gravitational field is conservative (and irrotational), it can be expressed in terms of a scalar potential *ϕ*:

Substituting this into Gauss's law,

If the mass density is zero, Poisson's equation reduces to Laplace's equation. The corresponding Green's function can be used to calculate the potential at distance r from a central point mass m (i.e., the fundamental solution). In three dimensions the potential is

One of the cornerstones of electrostatics is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the electric potential φ for a given charge distribution .

The mathematical details behind Poisson's equation in electrostatics are as follows (SI units are used rather than Gaussian units, which are also frequently used in electromagnetism).

Starting with Gauss's law for electricity (also one of Maxwell's equations) in differential form, one has

Assuming the medium is linear, isotropic, and homogeneous (see polarization density), we have the constitutive equation

Substituting this into Gauss's law and assuming that ε is spatially constant in the region of interest yields

The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field,

Solving Poisson's equation for the potential requires knowing the charge density distribution. If the charge density is zero, then Laplace's equation results. If the charge density follows a Boltzmann distribution, then the Poisson–Boltzmann equation results. The Poisson–Boltzmann equation plays a role in the development of the Debye–Hückel theory of dilute electrolyte solutions.

Using Green's function, the potential at distance r from a central point charge Q (i.e., the fundamental solution) is

The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the Coulomb gauge is used. In this more general context, computing φ is no longer sufficient to calculate **E**, since **E** also depends on the magnetic vector potential **A**, which must be independently computed. See Maxwell's equation in potential formulation for more on φ and **A** in Maxwell's equations and how Poisson's equation is obtained in this case.

If there is a static spherically symmetric Gaussian charge density

This solution can be checked explicitly by evaluating ∇^{2}*φ*.

Note that for r much greater than σ, the erf function approaches unity, and the potential *φ*(*r*) approaches the point-charge potential,

Surface reconstruction is an inverse problem. The goal is to digitally reconstruct a smooth surface based on a large number of points *p _{i}* (a point cloud) where each point also carries an estimate of the local surface normal

The goal of this technique is to reconstruct an implicit function *f* whose value is zero at the points *p _{i}* and whose gradient at the points

In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary to find a good discretization of the vector field **V**. The basic approach is to bound the data with a finite-difference grid. For a function valued at the nodes of such a grid, its gradient can be represented as valued on staggered grids, i.e. on grids whose nodes lie in between the nodes of the original grid. It is convenient to define three staggered grids, each shifted in one and only one direction corresponding to the components of the normal data. On each staggered grid we perform trilinear interpolation on the set of points. The interpolation weights are then used to distribute the magnitude of the associated component of *n _{i}* onto the nodes of the particular staggered grid cell containing

For the incompressible Navier–Stokes equations, given by

The equation for the pressure field is an example of a nonlinear Poisson equation:

**^**Jackson, Julia A.; Mehl, James P.; Neuendorf, Klaus K. E., eds. (2005),*Glossary of Geology*, American Geological Institute, Springer, p. 503, ISBN 9780922152766**^**Poisson (1823). "Mémoire sur la théorie du magnétisme en mouvement" [Memoir on the theory of magnetism in motion].*Mémoires de l'Académie Royale des Sciences de l'Institut de France*(in French).**6**: 441–570. From p. 463:*"Donc, d'après ce qui précède, nous aurons enfin:**selon que le point M sera situé en dehors, à la surface ou en dedans du volume que l'on considère."*(Thus, according to what preceded, we will finally have:*M*is located outside, on the surface of, or inside the volume that one is considering.)*V*is defined (p. 462) as*M*are denoted by and denotes the value of (the charge density) at*M*.**^**Calakli, Fatih; Taubin, Gabriel (2011). "Smooth Signed Distance Surface Reconstruction" (PDF).*Pacific Graphics*.**30**(7).- ^
^{a}^{b}Kazhdan, Michael; Bolitho, Matthew; Hoppe, Hugues (2006). "Poisson surface reconstruction".*Proceedings of the fourth Eurographics symposium on Geometry processing (SGP '06)*. Eurographics Association, Aire-la-Ville, Switzerland. pp. 61–70. ISBN 3-905673-36-3.

- Evans, Lawrence C. (1998).
*Partial Differential Equations*. Providence (RI): American Mathematical Society. ISBN 0-8218-0772-2. - Mathews, Jon; Walker, Robert L. (1970).
*Mathematical Methods of Physics*(2nd ed.). New York: W. A. Benjamin. ISBN 0-8053-7002-1. - Polyanin, Andrei D. (2002).
*Handbook of Linear Partial Differential Equations for Engineers and Scientists*. Boca Raton (FL): Chapman & Hall/CRC Press. ISBN 1-58488-299-9.

- "Poisson equation",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Poisson Equation at EqWorld: The World of Mathematical Equations
- Poisson's equation on PlanetMath.