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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 Coarea formula ： ウィキペディア英語版 Coarea formula In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of the integral of the level sets of another function. A special case is Fubini's theorem, which says under suitable hypotheses that the integral of a function over the region enclosed by a rectangular box can be written as the iterated integral over the level sets of the coordinate functions. Another special case is integration in spherical coordinates, in which the integral of a function on R''n'' is related to the integral of the function over spherical shells: level sets of the radial function. The formula plays a decisive role in the modern study of isoperimetric problems. For smooth functions the formula is a result in multivariate calculus which follows from a simple change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer , and for by . A precise statement of the formula is as follows. Suppose that Ω is an open set in R''n'', and ''u'' is a real-valued Lipschitz function on Ω. Then, for an L1 function ''g'', :$\backslash int\_\backslash Omega\; g(x)\; |\backslash nabla\; u(x)|\backslash ,\; dx\; =\; \backslash int\_^\backslash infty\; \backslash left(\backslash int\_g(x)\backslash ,dH\_(x)\backslash right)\backslash ,dt$ where ''H''''n'' − 1 is the (''n'' − 1)-dimensional Hausdorff measure. In particular, by taking ''g'' to be one, this implies :$\backslash int\_\backslash Omega\; |\backslash nabla\; u|\; =\; \backslash int\_^\backslash infty\; H\_(u^(t))\backslash ,dt,$ and conversely the latter equality implies the former by standard techniques in Lebesgue integration. More generally, the coarea formula can be applied to Lipschitz functions ''u'' defined in Ω ⊂ R''n'', taking on values in R''k'' where ''k'' < ''n''. In this case, the following identity holds :$\backslash int\_\backslash Omega\; g(x)\; |J\_k\; u(x)|\backslash ,\; dx\; =\; \backslash int\_\; \backslash left(\backslash int\_g(x)\backslash ,dH\_(x)\backslash right)\backslash ,dt$ where ''J''''k''''u'' is the ''k''-dimensional Jacobian of ''u''. ==Applications== * Taking ''u''(''x'') = |''x'' − ''x''0| gives the formula for integration in spherical coordinates of an integrable function ƒ: ::$\backslash int\_f\backslash ,dx\; =\; \backslash int\_0^\backslash infty\backslash left\backslash \backslash ,dr.$ * Combining the coarea formula with the isoperimetric inequality gives a proof of the Sobolev inequality for ''W''1,1 with best constant: ::$\backslash left(\backslash int\_\; |u|^\backslash right)^\}\backslash le\; n^\backslash omega\_n^\backslash int\_|\backslash nabla\; u|$ :where ωn is the volume of the unit ball in R''n''. 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「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.