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.
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