Smooth coarea formula

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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.


  • Chavel, Isaac (2006) Riemannian Geometry. A Modern Introduction. Second Edition.