Smooth coarea formula

In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains. Let [math]\displaystyle{ \scriptstyle M,\,N }[/math] be smooth Riemannian manifolds of respective dimensions [math]\displaystyle{ \scriptstyle m\,\geq\, n }[/math]. Let [math]\displaystyle{ \scriptstyle F:M\,\longrightarrow\, N }[/math] be a smooth surjection such that the pushforward (differential) of [math]\displaystyle{ \scriptstyle F }[/math] is surjective almost everywhere. Let [math]\displaystyle{ \scriptstyle\varphi:M\,\longrightarrow\, [0,\infty) }[/math] a measurable function. Then, the following two equalities hold:

[math]\displaystyle{ \int_{x\in M}\varphi(x)\,dM = \int_{y\in N}\int_{x\in F^{-1}(y)}\varphi(x)\frac{1}{N\!J\;F(x)}\,dF^{-1}(y)\,dN }[/math]

[math]\displaystyle{ \int_{x\in M}\varphi(x)N\!J\;F(x)\,dM = \int_{y\in N}\int_{x\in F^{-1}(y)} \varphi(x)\,dF^{-1}(y)\,dN }[/math]

where [math]\displaystyle{ \scriptstyle N\!J\;F(x) }[/math] is the normal Jacobian of [math]\displaystyle{ \scriptstyle F }[/math], i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.

Note that from Sard's lemma almost every point [math]\displaystyle{ \scriptstyle y\,\in\, N }[/math] is a regular point of [math]\displaystyle{ \scriptstyle F }[/math] and hence the set [math]\displaystyle{ \scriptstyle F^{-1}(y) }[/math] is a Riemannian submanifold of [math]\displaystyle{ \scriptstyle M }[/math], so the integrals in the right-hand side of the formulas above make sense.

References

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