Understanding the Confusing Curl in R2 and R3: An Intuitive Explanation

[medwatt]

Hello,
Its been sometime since I touched calculus so some concepts seem to evade me. I understand all the related maths but can't seem to make an intuitive sense of the curl in this case.

Green's theorem relates the line integral of a closed curve to the double integral of the curl of the vector field in the k direction. Here curlF is already in the k direction. The intuitive way I understand this is curlF is perpendicular to the plane and points to a point in space. So the double integral is just integration of this these points over the domain. This integral is the same as the line integral over the boundary of the domain.

The thing is in R3 the curl of the vector field is in all directions. But according to Stoke's therem we should dot that with the vector normal to the plane. The thing is I'm confused why this should be so. It makes it look similar to the divergence theorem except that there there is no curl. So should curlF doted with the normal vector also measure some outward flow (flux) ?

 

[h6ss]

In Stokes' theorem, the flux through a surface S is exactly the same as the work through its boundary, i.e. the curve C.

It's hard to explain without being able to visually show it, so I will link you to this very good video that I think explains what you are asking about: https://www.youtube.com/watch?v=9iaYNaENVH4#t=122

 

[Serac]

Question: Have you studied the standard derivation of Green's theorem? That is, have you derived Stokes' theorem and then shown the special case? This might illuminate it for you, somewhat; it did for me, in any case.