Coarea formula
# 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 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 $R^n$ 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 change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer (Federer 1959), and for BV functions by Fleming & Rishel (1960).

A precise statement of the formula is as follows. Suppose that Ω is an open set in $\mathbb {R} ^{n}$ and u is a real-valued Lipschitz function on Ω. Then, for an $L^1$ function g,

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

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

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

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

    $\int _{\Omega }g(x)|J_{k}u(x)|\,dx=\int _{\mathbb {R} ^{k}}\left(\int _{u^{-1}(t)}g(x)\,dH_{n-k}(x)\right)\,dt$

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

    $|J_{k}u(x)|=\left({\det \left(Ju(x)Ju(x)^{\intercal }\right)}\right)^{1/2}.$

## Applications

* Taking u(x) = |x − $x_0|$ gives the formula for integration in spherical coordinates of an integrable function f:

       $\int _{\mathbb {R} ^{n}}f\,dx=\int _{0}^{\infty }\left\{\int _{\partial B(x_{0};r)}f\,dS\right\}\,dr.$
* Combining the coarea formula with the isoperimetric inequality gives a proof of the Sobolev inequality for $W^{1,1}$ with best constant:

       $\left(\int _{\mathbb {R} ^{n}}|u|^{\frac {n}{n-1}}\right)^{\frac {n-1}{n}}\leq n^{-1}\omega _{n}^{-{\frac {1}{n}}}\int _{\mathbb {R} ^{n}}|\nabla u|$
    where $\omega _{n}$ is the volume of the unit ball in $\mathbb {R} ^{n}.$
