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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Green's function for the three-variable Laplace equation ： ウィキペディア英語版 Green's function for the three-variable Laplace equation In physics, the Green's function (or fundamental solution) for Laplace's equation in three variables is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form :$\backslash nabla^2u(\backslash mathbf)\; =\; f(\backslash mathbf)$ where $\backslash nabla^2$ is the Laplace operator in $\backslash mathbb^3$, $f(\backslash mathbf)$ is the source term of the system, and $u(\backslash mathbf)$ is the solution to the equation. Because $\backslash nabla^2$ is a linear differential operator, the solution $u(\backslash mathbf)$ to a general system of this type can be written as an integral over a distribution of source given by $f(\backslash mathbf)$: :$u(\backslash mathbf)\; =\; \backslash int\_\; G(\backslash mathbf,\backslash mathbf)f(\backslash mathbf)d\backslash mathbf\text{'}$ where the Green's function for Laplace's equation in three variables $G(\backslash mathbf,\backslash mathbf)$ describes the response of the system at the point $\backslash mathbf$ to a point source located at $\backslash mathbf$: :$\backslash nabla^2\; G(\backslash mathbf,\backslash mathbf)\; =\; \backslash delta(\backslash mathbf-\backslash mathbf)$ and the point source is given by $\backslash delta(\backslash mathbf-\backslash mathbf)$, the Dirac delta function. ==Motivation== One physical system of this type is a charge distribution in electrostatics. In such a system, the electric field is expressed as the negative gradient of the electric potential, and Gauss's law in differential form applies: :$\backslash mathbf\; =\; -\; \backslash mathbf\; \backslash phi(\backslash mathbf)$ :$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash frac$ Combining these expressions gives :$-\backslash mathbf^2\; \backslash phi(\backslash mathbf)\; =\; \backslash frac$ (Poisson's equation.) We can find the solution $\backslash phi(\backslash mathbf)$ to this equation for an arbitrary charge distribution by temporarily considering the distribution created by a point charge $q$ located at $\backslash mathbf$: :$\backslash rho(\backslash mathbf)\; =\; q\backslash delta(\backslash mathbf-\backslash mathbf)$ In this case, :$-\backslash frac\backslash mathbf^2\backslash phi(\backslash mathbf)\; =\; \backslash delta(\backslash mathbf-\backslash mathbf)$ which shows that $G(\backslash mathbf,\; \backslash mathbf)$ for $-\backslash frac\backslash nabla^2$ will give the response of the system to the point charge $q$. Therefore, from the discussion above, if we can find the Green's function of this operator, we can find $\backslash phi(\backslash mathbf)$ to be :$\backslash phi(\backslash mathbf)\; =\; \backslash int\_\; G(\backslash mathbf,\backslash mathbf)\backslash rho(\backslash mathbf)d\backslash mathbf\text{'}$ for a general charge distribution. 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「Green's function for the three-variable Laplace equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.