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Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it?

This is the only definition I have and I don't really understand what it's saying.

Let $S$ be an oriented smooth surface that is bounded by a simple, closed, smooth boundary curve $C$ with positive orientation. Also let $\vec{F}$ be a vector field then,

$$\int\limits_C \vec{F} \cdot d\vec{r} = \iint\limits_S \mathrm{curl}\ \vec{F} \cdot d\vec{S}$$

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While I can't do it justice, this video done by Khan Academy was invaluable to me. I hope you find it useful as well:

Stokes' theorem intuition

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  • $\begingroup$ That helps, thank you! $\endgroup$ – user7000 May 15 '14 at 0:11
  • $\begingroup$ Yep, the best explanation you'll ever get on the internet is there :) $\endgroup$ – Lucas Zanella May 15 '14 at 0:13
  • $\begingroup$ That's a good video, but I don't think it explains why the quantity $\nabla \times F$ measures how much the vector field $F$ is "curling" at a given point. $\endgroup$ – littleO May 15 '14 at 0:42
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The spirit of this theorem is still about the meaning of $\nabla \times F$.

To realize this you can see the explanation in the textbook of Thomas Calculus, or see the following image of my brief note.

Circulation density (curl dot k) Circulation density (curl dot k)

Curl in 3D Curl in 3D

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