# 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 **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 simple change of variables. More general forms of the formula for Lipschitz functions were first established by Herbert Federer Template:Harv, and for *BV* functions by Template:Harvtxt.

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

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

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 ⊂ **R**^{n}, taking on values in **R**^{k} where *k* < *n*. In this case, the following identity holds

where *J*_{k}*u* is the *k*-dimensional Jacobian of *u*.

## Applications

- Taking
*u*(*x*) = |*x*−*x*_{0}| gives the formula for integration in spherical coordinates of an integrable function ƒ:

- Combining the coarea formula with the isoperimetric inequality gives a proof of the Sobolev inequality for
*W*^{1,1}with best constant:

- where ω
_{n}is the volume of the unit ball in**R**^{n}.

## See also

## References

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