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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 Laplace's equation ： ウィキペディア英語版 Laplace's equation In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as: : $\backslash nabla^2\; \backslash varphi\; =\; 0\; \backslash qquad\backslash mbox\backslash qquad\; \backslash Delta\backslash varphi\; =\; 0$ where ∆ = ∇2 is the Laplace operator and $\backslash varphi$ is a scalar function. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation. ==Definition== In three dimensions, the problem is to find twice-differentiable real-valued functions ''f'', of real variables ''x'', ''y'', and ''z'', such that In Cartesian coordinates : $\backslash Delta\; f\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; =\; 0.$ In cylindrical coordinates, :$\backslash Delta\; f=\backslash frac\; \backslash frac\; \backslash left(\; r\; \backslash frac\; \backslash right)\; +\; \backslash frac\; \backslash frac\; +\; \backslash frac\; =0$ In spherical coordinates, : $\backslash Delta\; f\; =\; \backslash frac\backslash frac\; \backslash left(\backslash rho^2\; \backslash frac\backslash right)\; +\; \backslash frac\; \backslash frac\; \backslash left(\backslash sin\backslash theta\; \backslash frac\backslash right)\; +\; \backslash frac\; \backslash frac\; =0.$ In curvilinear coordinates, : $\backslash Delta\; f\; =\backslash frac\backslash left(\backslash fracg^\backslash right)\; +\; \backslash frac\; g^\backslash Gamma^n\_\; =0,$ or : $\backslash Delta\; f\; =\; \backslash frac\backslash !\backslash left(\backslash sqrtg^\; \backslash frac\backslash right)\; =0,\; \backslash qquad\; (g=\backslash mathrm\backslash ).$ This is often written as : $\backslash nabla^2\; f\; =\; 0$ or, especially in more general contexts, : $\backslash Delta\; f\; =\; 0,$ where ∆ = ∇2 is the Laplace operator or "Laplacian" : $\backslash Delta\; f\; =\; \backslash nabla^2\; f\; =\backslash nabla\; \backslash cdot\; \backslash nabla\; f\; =\backslash operatorname\backslash operatorname\; f,$ where ∇ •  is the divergence operator (also symbolized "div") which maps vectors to scalars, and ∇  is the gradient operator (also symbolized "grad") which maps scalars to vectors. (hence, the Laplacian Δf ≝ div grad f, maps the scalar function f to a scalar magnitude; specifically it maps the vector grad (the partial derivatives ) of f to a scalar (function).) If the right-hand side is specified as a given function, ''h''(''x'', ''y'', ''z''), i.e., if the whole equation is written as : $\backslash Delta\; f\; =\; h$ then it is called "Poisson's equation". The Laplace equation is also a special case of the Helmholtz equation. Note: The delta symbol, Δ, is also commonly used to represent "a change in" some quantity, e.g. ∆Q ≝ Q + δ or ∆Q ≝ Q + εQ for some very small scalars δ or ε. Its use to represent the Laplacian should not be confused with this use. 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「Laplace's 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.