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In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. It is used, for instance, to describe the potential energy field caused by a given charge or mass density distribution. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson. ==Statement of the equation== Poisson's equation is : where is the Laplace operator, and ''f'' and ''φ'' are real or complex-valued functions on a manifold. Usually, ''f'' is given and ''φ'' is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2 and so Poisson's equation is frequently written as : In three-dimensional Cartesian coordinates, it takes the form : When we retrieve Laplace's equation. Poisson's equation may be solved using a Green's function; a general exposition of the Green's function for Poisson's equation is given in the article on the screened Poisson equation. There are various methods for numerical solution. The relaxation method, an iterative algorithm, is one example. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poisson's equation」の詳細全文を読む スポンサード リンク
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