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August 4th, 2014, 01:08 PM   #1
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Don't understand Stoke's Theorem (Revised)

Moderators: Couldn't find the edit/delete button for my last post on Stoke's Thrm. I've revised the question for clarity; could you please delete the other post. Thanks!


I'm trying to get an intuition on Stoke's Theorem, and all explanations I found use this same argument:
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On a given surface (image below), the curl field at each surface element cancels with the curl field of its neighbors, and the only curl fields that don't get cancelled are at the surface boundary.
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Main problem i have with this is, i don't understand what those curls are representing.
Say i have a vector field in R-2 with a non-zero curl. well, if i were to graph its curl field, i would be seeing a bunch of parallel vectors (the curl vector), not a bunch of circulating vectors. This makes me think that those circulating vectors aren't those of the curl field's, but rather those of the original vector field's.
On the other hand, Stoke's theorem mentions the sum of the curl field vectors, not those of the original vector field. See my conundrum?
So are the circulating vectors in the image referring to the original vector field's or the curl field's?


Thanks in advance
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August 6th, 2014, 04:31 AM   #2
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Those small curly arrows indicate the positive left-wise indication, they have nothing to do with what you are saying.
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August 6th, 2014, 06:13 AM   #3
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Quote:
Originally Posted by ZardoZ View Post
Those small curly arrows indicate the positive left-wise indication.
they're only purpose is to indicate direction? ignoring magnitude??



Consider the surface in this R-2 vector field



At various regions within the surface, the curvature (which is what the curl measures) varies. For instance, region A vs region B




the magnitude of the curl field (in [itex]\hat{z}[/itex]) is not uniform throughout the surface because of the changing curvature. So even if this vector field could somehow be represented as individual circulations, the curl value at any given point within the surface would not be the same as the curl values of its neighbors'. how then is the cancellation done?
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August 6th, 2014, 09:32 AM   #4
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I insist those arrows indicate the left positive orientation, look here.
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