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December 27th, 2013, 11:06 AM   #1
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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?
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December 27th, 2013, 04:15 PM   #2
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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 is a differential form defined on a differentiable manifold then (http://en.wikipedia.org/wiki/Stokes'_theorem). That is, the integral of the differential form, over region is the anti-derivative, integrated (i.e. evaluated) over the boundary of the region .

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" on that boundary is just evaluating at the two endpoints and subtracting. In the case that is a two dimensional surface in three dimension, is the closed path boundary of the surface.
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December 27th, 2013, 07:14 PM   #3
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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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