Applying Divergence Theorem to Stokes' Theorem

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The discussion centers on the application of the Divergence Theorem and Stokes' Theorem, questioning the reasoning behind the apparent contradiction when applying both theorems together. It highlights that Stokes' Theorem relates a line integral to a surface integral of the curl of a vector field, while the Divergence Theorem connects a surface integral of a vector field to a triple integral of its divergence. The confusion arises when attempting to equate the triple integral of the divergence of the curl of a vector field, which is always zero, with the results from Stokes' Theorem. The key point made is the importance of ensuring that the surface in Stokes' Theorem is closed, as this impacts the validity of the application of both theorems. Understanding the conditions under which each theorem applies is crucial for resolving the issue.
schaefera
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Not really a homework problem, just me wondering about this: why is there a problem here?

Say you want to use the divergence theorem in conjunction with Stokes' theorem. So, from Stokes' you know: Line integral (F*T ds)= Surface integral (curl(F)*n)dS.

And you know that Surface integral(F*n)dS= Triple integral (div(F) dV))

But then, if you try to apply that to Stokes' you get: Triple integral (div(curl(F)) dV) which has to be 0, because div(curl(F))=0, right?

What's wrong with my reasoning?
 
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I think you should check if one of the theorem states the surface as a "closed" one.
 

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