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 December 27th, 2013, 11:06 AM #1 Senior Member   Joined: Nov 2013 Posts: 137 Thanks: 1 Fundamental theorem of calculus for surface integrals? Hellow! A simple question: if exist the fundamental theorem of calculus for line integrals not should exist too a fundamental theorem of calculus for surface integrals? I was searching about in google but I found nothing... What do you think? Such theorem make sense?
 December 27th, 2013, 04:15 PM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Fundamental theorem of calculus for surface integrals? That may be because it is incorporated into a much more general theorem- the generalized Stokes' theorem: if $d\omega$ is a differential form defined on a differentiable manifold $\Omega$ then $\int_{\Omega} d\omega= \int_{\partial\Omega} \omega$ (http://en.wikipedia.org/wiki/Stokes'_theorem). That is, the integral of the differential form, $d\omega$ over region $\Omega$ is the anti-derivative, $\omega$ integrated (i.e. evaluated) over the boundary of the region $\partial\Omega$. The reason why we have specific expressions for an integral on the real number line or on a path in two or three dimensions is that the boundary is just the two endpoints and in "integrating" $\omega$ on that boundary is just evaluating at the two endpoints and subtracting. In the case that $\Omega$ is a two dimensional surface in three dimension, $\partial\Omega$ is the closed path boundary of the surface.
 December 27th, 2013, 07:14 PM #3 Senior Member   Joined: Nov 2013 Posts: 137 Thanks: 1 Re: Fundamental theorem of calculus for surface integrals? But, anyway, exist a geometric explanation for the fundamental theorem of calculus for surface integral, a geometric explanation analogous to t.f.c. for line integral?

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