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 (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 ${\displaystyle \mathbb {R} ^{n}}$ and u is a real-valued Lipschitz function on Ω. Then, for an L¹ function g,

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

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

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

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

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

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

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