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 integrals over 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 Rn 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.
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| - Coarea formula (en)
- Formule de la co-aire (fr)
- Formula di coarea (it)
- Формула коплощади (ru)
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| - La formule de la co-aire est un théorème de théorie géométrique de la mesure qui exprime l'intégrale du jacobien d'une fonction sur ℝn comme l'intégrale de la mesure de Hausdorff de ses ensembles de niveau. Elle généralise le théorème de Fubini. Elle joue un rôle décisif dans l'approche moderne des problèmes isopérimétriques. Pour les fonctions lisses, la formule est un résultat d'analyse à plusieurs variables qui résulte d'un simple changement de variable. Elle a été généralisée aux fonctions lipschitziennes par Herbert Federer puis aux fonctions à variation bornée par Fleming et Rishel. (fr)
- In matematica, più precisamente nell'ambito della teoria della misura, la formula di coarea permette di calcolare l'integrale del gradiente di una funzione in termini dell'integrale dei suoi insiemi di livello.Tale formula viene spesso utilizzata per problemi isoperimetrici. (it)
- Формула коплощади — интегральная формула, связывающая интеграл по области и интеграл по поверхностям уровней данной функции или отображения.Принцип Кавальери является частным случаем формулы коплощади. Для справедливости формулы коплощади функция и её область определения должны удовлетворять некоторым свойствам. Наиболее простой случай — гладкая функция, заданная на открытой области .Также она верна для липшицевых и соболевских функций. (ru)
- 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 integrals over 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 Rn 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. (en)
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| - 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 integrals over 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 Rn 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 change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer, and for BV functions by . A precise statement of the formula is as follows. Suppose that Ω is an open set in and u is a real-valued Lipschitz function on Ω. Then, for an L1 function g, where Hn−1 is the (n − 1)-dimensional Hausdorff measure. In particular, by taking g to be one, this implies 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 taking on values in where k ≤ n. In this case, the following identity holds where Jku is the k-dimensional Jacobian of u whose determinant is given by (en)
- La formule de la co-aire est un théorème de théorie géométrique de la mesure qui exprime l'intégrale du jacobien d'une fonction sur ℝn comme l'intégrale de la mesure de Hausdorff de ses ensembles de niveau. Elle généralise le théorème de Fubini. Elle joue un rôle décisif dans l'approche moderne des problèmes isopérimétriques. Pour les fonctions lisses, la formule est un résultat d'analyse à plusieurs variables qui résulte d'un simple changement de variable. Elle a été généralisée aux fonctions lipschitziennes par Herbert Federer puis aux fonctions à variation bornée par Fleming et Rishel. (fr)
- In matematica, più precisamente nell'ambito della teoria della misura, la formula di coarea permette di calcolare l'integrale del gradiente di una funzione in termini dell'integrale dei suoi insiemi di livello.Tale formula viene spesso utilizzata per problemi isoperimetrici. (it)
- Формула коплощади — интегральная формула, связывающая интеграл по области и интеграл по поверхностям уровней данной функции или отображения.Принцип Кавальери является частным случаем формулы коплощади. Для справедливости формулы коплощади функция и её область определения должны удовлетворять некоторым свойствам. Наиболее простой случай — гладкая функция, заданная на открытой области .Также она верна для липшицевых и соболевских функций. (ru)
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