Concept question about stokes theorem?

[madcattle]

Homework Statement

This is not actually a homework question, just a question I ran into while studying for my math final. When I am using stokes theorem:

∫∫(curlF ° n)dS

I have listed in my notes from lecture that there are time when it is applicable to replace dS with an easier surface to integrate over, but that has the same bounds. Can anyone explain what this means? Or where I might use this fact?

I also have one more question. I see that I also have written down that when F is a curl and S is closed, the integral always yields 0 as its answer when using stokes theorem. How do I tell when F is a curl? Does that just mean that I should check to see that it is independent of path?

Thanks for any clarification you can yield!

 

[sunjin09]

For example, if you want to calculate the line integral of a vector field around the unit circle in xy plane, rather than integrating its curl over the upper half unit sphere, you integrate over the unit disk in xy plane, a vector field is curl only if its divergence is 0

 

[madcattle]

Thanks a lot, that was really helpful.