Smooth Coarea Formula

In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.

Let be smooth Riemannian manifolds of respective dimensions . Let be a smooth surjection such that the pushforward (differential) of is surjective almost everywhere. Let a measurable function. Then, the following two equalities hold:

where is the normal Jacobian of, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.

Note that from Sard's lemma, almost every point is a regular point of and hence the set is a Riemannian submanifold of, so the integrals in the right-hand side of the formulas above make sense.