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 functions by .
A precise statement of the formula is as follows. Suppose that Ω is an open set in
\Rn
\int\Omegag(x)|\nablau(x)|dx=\int\R
\left(\int | |
u-1(t) |
g(x)dHn-1(x)\right)dt
where Hn−1 is the (n − 1)-dimensional Hausdorff measure. In particular, by taking g to be one, this implies
\int\Omega|\nablau|=
infty | |
\int | |
-infty |
Hn-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\Rn,
\Rk
\int\Omegag(x)|Jku(x)|dx=
\int | |
\Rk |
\left(\int | |
u-1(t) |
g(x)dHn-k(x)\right)dt
where Jku is the k-dimensional Jacobian of u whose determinant is given by
|Jku(x)|=\left({\det\left(Ju(x)Ju(x)\intercal\right)}\right)1/2.
\int | |
\Rn |
fdx=
infty\left\{\int | |
\int | |
\partialB(x0;r) |
fdS\right\}dr.
\left(\int | |
\Rn |
| ||||
|u| |
| ||||
\right) |
\len-1
| ||||
\omega | ||||
n |
\int | |
\Rn |
|\nablau|
where
\omegan
\Rn.