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August 3rd, 2014, 11:52 AM   #1
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Problem with Stoke's Theorem Intuition

Let's start with its statement:

The surface integral of a curl field is the same as the closed line integral of the corresponding vector field

It was explained to me that the reason behind this was that all the neighboring curls within each surface element inside the surface boundary would cancel leaving the boundary uncancelled





When I see the explanation being graphically depicted this way, it seems to me that this arguement hinges on the conditions:

1.) the curl fields within each surface element must be circular (for neighbors to be able to cancel) like this



and not like this



2.) the curl at each surface element must be equal in magnitude; otherwise not all the neighboring surface elements would cancel.



Based on the explanation i was given, it seems logical to me that (1) and (2) should be conditions for that explanation to be valid, but I'm most likely wrong. I just want to know why (1) and (2) don't naturally follow.


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August 3rd, 2014, 02:33 PM   #2
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Imagine that in an infinitesimal area of the surface, something like this happens. Do you see the arrows canceling each other in the common side?

pap1.jpg
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Last edited by ZardoZ; August 3rd, 2014 at 02:40 PM.
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August 3rd, 2014, 03:58 PM   #3
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i understand the rationale behind the whole cancellation concept. ie, i know how i'm supposed to treat the curls at each surface element, but i don't understand where those circular curls came from. Sure, a vector field can have a curl (curvature), but separate loops everywhere? wouldn't that make the field discontinuous?

Moreover, a curl vector points straight. If we are cancelling the interior by summing all the curls, then aren't we summing up the vector field and not the curl of the vector field?

For the sake of argument let's consider Green's thrm (2-D surface)
If we are summing up the curl vectors, well, we're summing up a bunch of parallel vectors, why is anything cancelling?
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August 3rd, 2014, 04:07 PM   #4
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Quote:
Originally Posted by iScience View Post
i understand the rationale behind the whole cancellation concept. ie, i know how i'm supposed to treat the curls at each surface element, but i don't understand where those circular curls came from. Sure, a vector field can have a curl (curvature), but separate loops everywhere? wouldn't that make the field discontinuous?

Moreover, a curl vector points straight. If we are cancelling the interior by summing all the curls, then aren't we summing up the vector field and not the curl of the vector field?

For the sake of argument let's consider Green's thrm (2-D surface)
If we are summing up the curl vectors, well, we're summing up a bunch of parallel vectors, why is anything cancelling?
I think that what you call "curls", are indicating the positive (left) or anti-clockwise orientation.
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