In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.
Let M , N be smooth Riemannian manifolds of respective dimensions m ≥ n . Let F : M ⟶ N be a smooth surjection such that the pushforward (differential) of F is surjective almost everywhere. Let φ : M ⟶ [ 0 , ∞ ] a measurable function. Then, the following two equalities hold:
∫ x ∈ M φ ( x ) d M = ∫ y ∈ N ∫ x ∈ F − 1 ( y ) φ ( x ) 1 N J F ( x ) d F − 1 ( y ) d N ∫ x ∈ M φ ( x ) N J F ( x ) d M = ∫ y ∈ N ∫ x ∈ F − 1 ( y ) φ ( x ) d F − 1 ( y ) d N where N J F ( x ) is the normal Jacobian of F , i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.
Note that from Sard's lemma almost every point y ∈ N is a regular point of F and hence the set F − 1 ( y ) is a Riemannian submanifold of M , so the integrals in the right-hand side of the formulas above make sense.
