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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 Uniqueness theorem for Poisson's equation ： ウィキペディア英語版 Uniqueness theorem for Poisson's equation The uniqueness theorem for Poisson's equation states that the equation has a unique gradient of the solution for a large class of boundary conditions. In the case of electrostatics, this means that if an electric field satisfying the boundary conditions is found, then it is the complete electric field. ==Proof== In Gaussian units, the general expression for Poisson's equation in electrostatics is :$\backslash mathbf\backslash cdot(\backslash epsilon\backslash mathbf\backslash varphi)=\; -4\backslash pi\backslash rho\_f$ Here $\backslash varphi$ is the electric potential and $\backslash mathbf=-\backslash mathbf\backslash varphi$ is the electric field. The uniqueness of the gradient of the solution (the uniqueness of the electric field) can be proven for a large class of boundary conditions in the following way. Suppose that there are two solutions $\backslash varphi\_$ and $\backslash varphi\_$. One can then define $\backslash phi=\backslash varphi\_-\backslash varphi\_$ which is the difference of the two solutions. Given that both $\backslash varphi\_$ and $\backslash varphi\_$ satisfy Poisson's Equation, $\backslash phi$ must satisfy :$\backslash mathbf\backslash cdot(\backslash epsilon\; \backslash mathbf\backslash phi)=\; 0$ Using the identity :$\backslash nabla\; \backslash cdot\; (\backslash phi\; \backslash epsilon\; \backslash ,\; \backslash nabla\; \backslash phi\; )=\backslash epsilon\; \backslash ,\; (\backslash nabla\; \backslash phi\; )^2\; +\; \backslash phi\; \backslash nabla\; \backslash cdot\; (\backslash epsilon\; \backslash ,\; \backslash nabla\; \backslash phi\; )$ And noticing that the second term is zero one can rewrite this as :$\backslash mathbf\backslash cdot(\backslash phi\backslash epsilon\; \backslash mathbf\backslash phi)=\; \backslash epsilon\; (\backslash mathbf\backslash phi)^2$ Taking the volume integral over all space specified by the boundary conditions gives :$\backslash int\_V\; \backslash mathbf\backslash cdot(\backslash phi\backslash epsilon\; \backslash mathbf\backslash phi)\; d^3\; \backslash mathbf=\; \backslash int\_V\; \backslash epsilon\; (\backslash mathbf\backslash phi)^2\; \backslash ,\; d^3\; \backslash mathbf$ Applying the divergence theorem, the expression can be rewritten as :$\backslash sum\_i\; \backslash int\_\; (\backslash phi\backslash epsilon\; \backslash mathbf\backslash phi)\; \backslash cdot\; \backslash mathbf=\; \backslash int\_V\; \backslash epsilon\; (\backslash mathbf\backslash phi)^2\; \backslash ,\; d^3\; \backslash mathbf$ Where $S\_i$ are boundary surfaces specified by boundary conditions. Since $\backslash epsilon\; >\; 0$ and $(\backslash mathbf\backslash phi)^2\; \backslash ge\; 0$, then $\backslash mathbf\backslash phi$ must be zero everywhere (and so $\backslash mathbf\backslash varphi\_\; =\; \backslash mathbf\backslash varphi\_$) when the surface integral vanishes. This means that the gradient of the solution is unique when :$\backslash sum\_i\; \backslash int\_\; (\backslash phi\backslash epsilon\; \backslash ,\; \backslash mathbf\backslash phi)\; \backslash cdot\; \backslash mathbf\; =$ 0 The boundary conditions for which the above is true are: # Dirichlet boundary condition: $\backslash varphi$ is well defined at all of the boundary surfaces. As such $\backslash varphi\_1=\backslash varphi\_2$ so at the boundary $\backslash phi\; =\; 0$ and correspondingly the surface integral vanishes. # Neumann boundary condition: $\backslash mathbf\backslash varphi$ is well defined at all of the boundary surfaces. As such $\backslash mathbf\backslash varphi\_1=\backslash mathbf\backslash varphi\_2$ so at the boundary $\backslash mathbf\backslash phi=0$ and correspondingly the surface integral vanishes. # Modified Neumann boundary condition (also called Robin boundary condition - conditions where boundaries are specified as conductors with known charges): $\backslash mathbf\backslash varphi$ is also well defined by applying locally Gauss's Law. As such, the surface integral also vanishes. # Mixed boundary conditions (a combination of Dirichlet, Neumann, and modified Neumann boundary conditions): the uniqueness theorem will still hold. The boundary surfaces may also include boundaries at infinity (describing unbounded domains) - for these as well the uniqueness theorem holds if the surface integral vanishes, which is the case (for example) when at large distances the integrand decays faster than the surface area grows.v1-v2=0 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「Uniqueness theorem for Poisson'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.