Stokes theorem for manifolds with corners? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:06:28Z http://mathoverflow.net/feeds/question/12920 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12920/stokes-theorem-for-manifolds-with-corners Stokes theorem for manifolds with corners? Agusti Roig 2010-01-25T09:57:23Z 2010-02-01T09:47:21Z <p>Maybe this is an elementary question, but I'm unable to find the appropriate reference for it. The <em>Stokes theorem</em> tells us that, for a $n+1$-dimensional manifold $M$ with boundary $\partial M$ and any differentiable $n$-form $\omega$ on $M$, we have $\int_{\partial M} \omega = \int_M d\omega$.</p> <p>But Stokes theorem is also true, say, for a cone <code>$M = \{(x,y,z) \in \mathbb{R}^3 \ \vert\ \ x^2 + y^2 = z^2, 0\leq z \leq 1 \}$</code>, or a square in the plane, <code>$M =\{(x,y) \in \mathbb{R}^2 \ \vert\ 0 \leq x, y\leq 1 \}$</code> which are not manifolds. So my questions are:</p> <ol> <li>Are these cone and square examples of what I think are called "manifold with corners"?</li> <li>If this is so, where can I find a reference for a version of Stokes' theorem for manifolds with corners?</li> <li>If "manifold with corners" is not, which is the appropriate setting (and a reference) for a Stokes' theorem that includes those examples?</li> </ol> <p>Any hints will be appreciated.</p> <p>EDIT: Since thanking individually everyone would be too long, let me edit my question to acknowledge all of your answers. Thank you very much: I've found what I was looking for and more.</p> http://mathoverflow.net/questions/12920/stokes-theorem-for-manifolds-with-corners/12926#12926 Answer by Christian Blatter for Stokes theorem for manifolds with corners? Christian Blatter 2010-01-25T12:02:54Z 2010-01-25T12:02:54Z <p>Triangulate your manifold $M$ so that $\partial M$ is triangulated as well. Then prove Stokes' theorem for diffeomorphic images of a standard simplex, as in de Rham's "Variétées différentiables".</p> http://mathoverflow.net/questions/12920/stokes-theorem-for-manifolds-with-corners/12927#12927 Answer by Orbicular for Stokes theorem for manifolds with corners? Orbicular 2010-01-25T12:04:09Z 2010-02-01T09:47:21Z <p>The most general form of Stokes' theorem I know of is proved in the book Partial Differential Equations 1. Foundations and Integral Representations by Friedrich Sauvigny.<br /> The aim in the book is to provide a version of the divergence theorem which holds also in cases where the boundary has certain singularities (as you described: the singular boundary has to have zero capacity). As a precursor they also prove the Stokes' theorem (they credit the proof to E. Heinz!).<br /> Note that this is much more general than manifolds with corners, it encompasses your cone as well!</p> http://mathoverflow.net/questions/12920/stokes-theorem-for-manifolds-with-corners/12929#12929 Answer by bavajee for Stokes theorem for manifolds with corners? bavajee 2010-01-25T12:58:27Z 2010-01-25T12:58:27Z <p>You could take a look at Ch. XXIII paragraph 6 in Lang's Real and Functional Analysis entitled "Stokes' Theorem with Singularities". This version works for the cone too, I think. I have not read it myself, though. </p> http://mathoverflow.net/questions/12920/stokes-theorem-for-manifolds-with-corners/12931#12931 Answer by Lars for Stokes theorem for manifolds with corners? Lars 2010-01-25T13:15:10Z 2010-01-25T13:15:10Z <p>John Lee's excellent book "Introduction to smooth manifolds" has a chapter on manifolds with corners, in which he proves Stokes' theorem for them.</p> http://mathoverflow.net/questions/12920/stokes-theorem-for-manifolds-with-corners/12995#12995 Answer by Marty for Stokes theorem for manifolds with corners? Marty 2010-01-26T00:39:27Z 2010-01-26T00:39:27Z <p>If you are looking for an online reference, you can check out <a href="http://math.stanford.edu/~conrad/diffgeomPage/handouts.html" rel="nofollow">Brian Conrads course notes on differential geometry</a>. Near the bottom of that page, you can find the <a href="http://math.stanford.edu/~conrad/diffgeomPage/handouts/stokescorners.pdf" rel="nofollow">handout with Stokes theorem</a> for manifolds with corners.</p>