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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 Darboux derivative ： ウィキペディア英語版 Darboux derivative The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. In a certain sense, it is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundamental theorem of calculus to higher dimensions, in a different vein than the generalization that is Stokes' theorem. ==Formal definition== Let $G$ be a Lie group, and let $\backslash mathfrak$ be its Lie algebra. The Maurer-Cartan form, $\backslash omega\_G$, is the smooth $\backslash mathfrak$-valued $1$-form on $G$ (cf. Lie algebra valued form) defined by :$\backslash omega\_G(X\_g)\; =\; (T\_g\; L\_g)^\; X\_g$ for all $g\; \backslash in\; G$ and $X\_g\; \backslash in\; T\_g\; G$. Here $L\_g$ denotes left multiplication by the element $g\; \backslash in\; G$ and $T\_g\; L\_g$ is its derivative at $g$. Let $f:M\; \backslash to\; G$ be a smooth function between a smooth manifold $M$ and $G$. Then the Darboux derivative of $f$ is the smooth $\backslash mathfrak$-valued $1$-form :$\backslash omega\_f\; :=\; f^$ * \omega_G, the pullback of $\backslash omega\_G$ by $f$. The map $f$ is called an integral or primitive of $\backslash omega\_f$. 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「Darboux derivative」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.