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 $scriptstylevarphi:M,longrightarrow,\; [0,infty]$ a measurable function. Then, the following two equalities hold:

:$int\_\{xin\; M\}varphi(x),dM\; =\; int\_\{yin\; N\}int\_\{xin\; F^\{-1\}(y)\}varphi(x)frac\{1\}\{N!J;F(x)\},dF^\{-1\}(y),dN$

:$int\_\{xin\; M\}varphi(x)N!J;F(x),dM\; =\; int\_\{yin\; N\}int\_\{xin\; F^\{-1\}(y)\}\; varphi(x),dF^\{-1\}(y),dN$

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 $scriptstyle\; y,in,\; Y$ 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.

References

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