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thanks!

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- Thread starter joshmccraney
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- #1

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thanks!

- #2

Chestermiller

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thanks!

This is just the 3D version of [itex]\int_a^b{\frac{df}{dx}dx}=f(b)-f(a)[/itex]

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When you post in the differential geometry part of the forum, you get a differential geometry answer.

thanks!

Note first, though, that your "gradient theorem" is actually just a form of the Divergence Theorem. It turns out that all these integral theorems that you learn from multivariate calculus are actually results of a single theorem called the generalized Stokes' Theorem from differential geometry.

The intuition for your problem can be seen in 2 dimensions. From a topological standpoint, we can integrate over chains, and in the two dimensional case we'll look at a 2-chain and its boundary and see what integration over the two might be like. Look at the pictures below (Ignore the rainbows. Life in mathland is just that happy).

The first shows the oriented boundary of a region in two dimensional space. The second shows finer and finer oriented tilings of the region. Note that the interior arrows of each tiling go in opposite directions from their neighbors, "cancelling" each other out. The integral over the 2-chain would be equal to, in a sense, the contribution of the boundary.

Does this help?

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