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be smooth

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.

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