# 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 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 simple 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 Rn, and u is a real-valued Lipschitz function on Ω. Then, for an L1 function g,

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

where Hn − 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 Ω ⊂ Rn, taking on values in Rk 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 Jku is the k-dimensional Jacobian of u.

## Applications

• Taking u(x) = |x − x0| gives the formula for integration in spherical coordinates of an integrable function ƒ:
${\displaystyle \int _{\mathbb {R} ^{n}}f\,dx=\int _{0}^{\infty }\left\{\int _{\partial B(x_{0};r)}f\,dS\right\}\,dr.}$
${\displaystyle \left(\int _{\mathbb {R} ^{n}}|u|^{n/(n-1)}\right)^{\frac {n-1}{n}}\leq n^{-1}\omega _{n}^{-1/n}\int _{\mathbb {R} ^{n}}|\nabla u|}$
where ωn is the volume of the unit ball in Rn.