# Stokes' Theorem Explanation

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,

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

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

• That helps, thank you! – user7000 May 15 '14 at 0:11
• Yep, the best explanation you'll ever get on the internet is there :) – Lucas Zanella May 15 '14 at 0:13
• 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. – littleO May 15 '14 at 0:42

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)

Curl in 3D