Talk:Smooth coarea formula

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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

${\begin{aligned}\int _{M}fv_{g}&=\int _{0}^{\sup f}\mathrm {vol} _{n}f^{-1}([t,+\infty [)\,dt\qquad {\text{and}}\\\int _{M}|df|v_{g}&=\int _{0}^{\sup f}\mathrm {vol} _{n-1}(f^{-1}(t))\,dt.\end{aligned}}$

It appears that v_g 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. --KSmrq^T 01:43, 25 September 2007 (UTC)

Normal Jacobian

Normal Jacobian is not defined in the Jacobian-page. It should probably be defined here in an invariant way. I don't have an adequate reference at hand Lapasotka (talk) 02:05, 29 October 2010 (UTC)

Corrections

One needs M and N orientable so that the integrals are defined. Also, there is a more general (and usable) version for functions that are not neccessary positive, but compactly supported instead.

A reference is the book in the special case where F is a Riemannian submersion, from which the general case in the article easily follows. — Preceding unsigned comment added by Upmeier (talk • contribs) 11:49, 1 February 2017 (UTC)

References