Editing Siméon Denis Poisson (section)

==== Poisson's equation ====
[[File:Front cover of Griffiths' Electrodynamics.jpg|thumb|Poisson's equations for electricity (top) and magnetism (bottom) in SI units on the front cover of [[Introduction to Electrodynamics|an undergraduate textbook]].]]
In the theory of potentials, [[Poisson's equation]], 

: <math> \nabla^2 \phi = - 4 \pi \rho, \; </math>

is a well-known generalization of [[Laplace's equation]] of the second order [[partial differential equation]] <math> \nabla^2 \phi = 0</math> for [[potential]] <math>\phi</math>. 

If <math> \rho(x, y, z) </math> is a [[continuous function]] and if for <math> r \rightarrow \infty </math> (or if a point 'moves' to [[Extended real number line|infinity]]) a function <math> \phi </math> goes to 0 fast enough, the solution of Poisson's equation is the [[Newtonian potential]]

:<math> \phi = - {1\over 4 \pi} \iiint \frac{\rho (x, y, z)}{ r} \, dV, \; </math>

where <math> r </math> is a distance between a volume element <math> dV </math>and a point <math> P </math>. The integration runs over the whole space.

Poisson's equation was first published in the ''Bulletin de la société philomatique'' (1813).<ref name="EB1911" /> Poisson's two most important memoirs on the subject are ''Sur l'attraction des sphéroides'' (Connaiss. ft. temps, 1829), and ''Sur l'attraction d'un ellipsoide homogène'' (Mim. ft. l'acad., 1835).<ref name="EB1911" />

Poisson discovered that [[Laplace's equation]] is valid only outside of a solid. A rigorous proof for masses with variable density was first given by [[Carl Friedrich Gauss]] in 1839. Poisson's equation is applicable in not just gravitation, but also electricity and magnetism.<ref>{{Cite book|last=Kline|first=Morris|title=Mathematical Thought from Ancient to Modern Times|publisher=Oxford University Press|year=1972|isbn=0-19-506136-5|location=United States of America|pages=682–4|chapter=28.4: The Potential Equation and Green's Theorem}}</ref>