Stokes' Theorem says the curve integral of any surface S simply equals the counter-clockwise circulation around its boundary-curve C.(adsbygoogle = window.adsbygoogle || []).push({});

How can this be right? Let's say you have a hemisphere surface S with centre in origo, and its shadow on the xy plane. Both surfaces will have C as their boundary curve, and so according to the theorem they will have equal circulation. However, what if the curl increases with z? Then more curl will go through the higher surfaces than the bottom, and thus the curl-integral of surface S will not equal the circulation around C.

In addition, can somebody explain to me why the formula for curl, ∇xF, is intuitively pleasing? Ie, why does it make sense?

Thanks :)

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# Doubts about Stokes' theorem

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