Would anyone be willing to explain Stokes' Theorem to me?(adsbygoogle = window.adsbygoogle || []).push({});

I have managed to grasp the concepts of grad, div, curl, and what the text calls "green's theorem", but I cannot seem to grasp the geometric meaning of "stokes theorem." I've been trying to put the theorem together based on the following explanations from my electrodynamics text:

[tex]\int_{S}(\nabla\times\nu)\cdot da = \oint_{p}v\cdot dl[/tex]

"As always, the integral of a derivative (here, a curl) over a region (here, a patch of surface) is equal to the value of the function at the boundary (here, the perimeter of the patch). As in the case of green's theorem, the boundary term is itself an integral--specifically, a closed line integral."

"Geometrical interpretation: Recall that the curl measures the "twist" of the vectorsv; a region of high curl is a whirlpool--if you put a tiny paddlewheel there, it will rotate. Now, the integral of the curl over some surface (or, more precisely, the flux of the curl through the surface) represents the "total amount of swirl", and we can determine that swirl just as well by going around the edge and finding how much the flow is following the boundary. You may find this a rather forced interpretation of Stokes' theorem, but it's a helpful mnemonic, if nothing else."

Thanks for any help,

--Brian

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# Explanation of Stokes' Theorem

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