Smooth coarea formula

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

Let $$\scriptstyle M,\,N$$ be smooth Riemannian manifolds of respective dimensions $$\scriptstyle m\,\geq\, n$$. Let $$\scriptstyle F:M\,\longrightarrow\, N$$ be a smooth surjection such that the pushforward (differential) of $$\scriptstyle F$$ is surjective almost everywhere. Let $$\scriptstyle\varphi:M\,\longrightarrow\, [0,\infty)$$ a measurable function. Then, the following two equalities hold:


 * $$\int_{x\in M}\varphi(x)\,dM = \int_{y\in N}\int_{x\in F^{-1}(y)}\varphi(x)\frac{1}{N\!J\;F(x)}\,dF^{-1}(y)\,dN$$


 * $$\int_{x\in M}\varphi(x)N\!J\;F(x)\,dM = \int_{y\in N}\int_{x\in F^{-1}(y)} \varphi(x)\,dF^{-1}(y)\,dN$$

where $$\scriptstyle N\!J\;F(x)$$ is the normal Jacobian of $$\scriptstyle F$$, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.

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