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.
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 generally, the coarea formula can be applied to Lipschitz functions u defined in