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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 Current (mathematics) ： ウィキペディア英語版 Current (mathematics) In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold ''M''. Formally currents behave like Schwartz distributions on a space of differential forms. In a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of ''M''. ==Definition== Let $\backslash Omega\_c^m(M)$ denote the space of smooth ''m''-forms with compact support on a smooth manifold $M$. A current is a linear functional on $\backslash Omega\_c^m(M)$ which is continuous in the sense of distributions. Thus a linear functional :$T\backslash colon\; \backslash Omega\_c^m(M)\backslash to\; \backslash mathbb$ is an ''m''-current if it is continuous in the following sense: If a sequence $\backslash omega\_k$ of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when $k$ tends to infinity, then $T(\backslash omega\_k)$ tends to 0. The space $\backslash mathcal\; D\_m(M)$ of ''m''-dimensional currents on $M$ is a real vector space with operations defined by :$(T+S)(\backslash omega):=\; T(\backslash omega)+S(\backslash omega),\backslash qquad\; (\backslash lambda\; T)(\backslash omega):=\backslash lambda\; T(\backslash omega).$ Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current $T\; \backslash in\; \backslash mathcal\_m(M)$ as the complement of the biggest open set $U\; \backslash subset\; M$ such that :$T(\backslash omega)\; =\; 0$ whenever $\backslash omega\; \backslash in\; \backslash Omega\_c^m(U)$ The linear subspace of $\backslash mathcal\; D\_m(M)$ consisting of currents with support (in the sense above) that is a compact subset of $M$ is denoted $\backslash mathcal\; E\_m(M)$. 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「Current (mathematics)」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.