Concept question about stokes theorem?

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SUMMARY

This discussion centers on the application of Stokes' Theorem, specifically the integral form ∫∫(curlF ° n)dS. It clarifies that one can replace the differential area element dS with a simpler surface that shares the same bounds, facilitating easier integration. Additionally, it confirms that if the vector field F is a curl and the surface S is closed, the integral evaluates to zero. To determine if F is a curl, one must check if it is path-independent, which is equivalent to verifying that its divergence is zero.

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madcattle
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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!
 
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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
 
Thanks a lot, that was really helpful.
 

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