翻訳と辞書 Words near each other ・ Smooth 70s ・ Smooth 91.5 ・ Smooth 95.3 ・ Smooth algebra ・ Smooth alligatorfish ・ Smooth as Satin ・ Smooth as the Wind ・ Smooth Assassin ・ Smooth beardtongue ・ Smooth breathing ・ Smooth butterfly ray ・ Smooth chameleon ・ Smooth Christmas ・ Smooth clam ・ Smooth clean surface ・ Smooth coarea formula ・ Smooth Collie ・ Smooth completion ・ Smooth Criminal ・ Smooth curve hull ・ Smooth East Midlands ・ Smooth Fox Terrier ・ Smooth functor ・ Smooth Glasgow ・ Smooth green snake ・ Smooth grouper ・ Smooth hammerhead ・ Smooth infinitesimal analysis ・ Smooth Island ・ Smooth Island (Antarctica) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Smooth coarea formula ： ウィキペディア英語版 Smooth coarea formula In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains. Let $\backslash scriptstyle\; M,\backslash ,N$ be smooth Riemannian manifolds of respective dimensions $\backslash scriptstyle\; m\backslash ,\backslash geq\backslash ,\; n$. Let $\backslash scriptstyle\; F:M\backslash ,\backslash longrightarrow\backslash ,\; N$ be a smooth surjection such that the pushforward (differential) of $\backslash scriptstyle\; F$ is surjective almost everywhere. Let $\backslash scriptstyle\backslash varphi:M\backslash ,\backslash longrightarrow\backslash ,\; ()$ a measurable function. Then, the following two equalities hold: :$\backslash int\_\backslash varphi(x)\backslash ,dM\; =\; \backslash int\_\backslash int\_\backslash varphi(x)\backslash frac\backslash ,dF^(y)\backslash ,dN$ :$\backslash int\_\backslash varphi(x)N\backslash !J\backslash ;F(x)\backslash ,dM\; =\; \backslash int\_\backslash int\_\; \backslash varphi(x)\backslash ,dF^(y)\backslash ,dN$ where $\backslash scriptstyle\; N\backslash !J\backslash ;F(x)$ is the normal Jacobian of $\backslash scriptstyle\; F$, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel. Note that from Sard's lemma almost every point $\backslash scriptstyle\; y\backslash ,\backslash in\backslash ,\; N$ is a regular point of $\backslash scriptstyle\; F$ and hence the set $\backslash scriptstyle\; F^(y)$ is a Riemannian submanifold of $\backslash scriptstyle\; M$, so the integrals in the right-hand side of the formulas above make sense. ==References== *Chavel, Isaac (2006) ''Riemannian Geometry. A Modern Introduction. Second Edition''. 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「Smooth coarea formula」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.