Talk:Smooth coarea formula

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Articles for deletion This article was nominated for deletion on 2007-09-24. The result of the discussion was Keep.


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Gallot, Hulin, and Lafontaine, in Riemannian Geometry, 2nd edition (ISBN 978-0-387-52401-6) state Lemma 4.73, a "coarea formula".

Let ƒ be a smooth positive function on a compact Riemannian manifold (M,g). Then
\begin{align}
 \int_M f v_g &= \int_{0}^{\sup f} \mathrm{vol}_n f^{-1}([t,+\infty[) \, dt \qquad \text{and} \\
 \int_M |df|v_g &= \int_{0}^{\sup f} \mathrm{vol}_{n -1}(f^{-1}(t)) \, dt .
\end{align}

It appears that vg is a canonical measure, something like a Haar measure; and of course "[t,+∞[" means the same as "[t,+∞)". They cite Burago & Zalgaller, Geometric inequalities (ISBN 978-0-387-13615-8), p. 103 for a more general version. (Unfortunately, that page is not available at Amazon books.)

The theorem in this article is more general, but would seem to have the same flavor. Bad news: no location in the Chavel text is given, and a search for "coarea" in Amazon's version did not show this theorem. Good news: User Beltranc (talk · contribs) who created this article is most likely the same Carlos Beltrán (a recent PhD from Universidad de Cantabria, now a postdoc at University of Toronto) who coauthored a paper published this year in Mathematics of Computation (v.76,n.259,pp.1393–1424), and essentially this theorem does appear in that peer-reviewed paper citing a different source. --KSmrqT 01:43, 25 September 2007 (UTC)