coarea formula

{{Short description|Mathematic 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 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 {{harv|Federer|1959}}, and for Bounded variation by {{harvtxt|Fleming|Rishel|1960}}.

A precise statement of the formula is as follows. Suppose that Ω is an open set in $\backslash R^n$ and u is a real-valued Lipschitz function on Ω. Then, for an L¹ function g,

:$\backslash int\_\backslash Omega\; g(x)\; |\backslash nabla\; u(x)|\backslash ,\; dx\; =\; \backslash int\_\{\backslash R\}\; \backslash left(\backslash int\_\{u^\{-1\}(t)\}g(x)\backslash ,dH\_\{n-1\}(x)\backslash right)\backslash ,dt$

where H_n−1 is the (n − 1)-dimensional Hausdorff measure. In particular, by taking g to be one, this implies

:$\backslash int\_\backslash Omega\; |\backslash nabla\; u|\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; H\_\{n-1\}(u^\{-1\}(t))\backslash ,dt,$

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 $\backslash Omega\; \backslash subset\; \backslash R^n,$ taking on values in $\backslash R^k$ where k ≤ n. In this case, the following identity holds

:$\backslash int\_\backslash Omega\; g(x)\; |J\_k\; u(x)|\backslash ,\; dx\; =\; \backslash int\_\{\backslash R^k\}\; \backslash left(\backslash int\_\{u^\{-1\}(t)\}g(x)\backslash ,dH\_\{n-k\}(x)\backslash right)\backslash ,dt$

where J_ku is the k-dimensional Jacobian of u whose determinant is given by

:$|J\_k\; u(x)|\; =\; \backslash left(\{\backslash det\backslash left(J\; u(x)\; J\; u(x)^\backslash intercal\backslash right)\}\backslash right)^\{1/2\}.$