Applying Divergence Theorem to Stokes' Theorem

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SUMMARY

The discussion centers on the application of the Divergence Theorem in conjunction with Stokes' Theorem. It highlights the confusion arising from the expression of the triple integral of the divergence of the curl of a vector field, which equals zero, indicating a misunderstanding of the conditions under which these theorems apply. Specifically, the participant notes the necessity of ensuring that the surface involved is closed to properly utilize these theorems. This clarification is crucial for correctly applying vector calculus theorems in mathematical physics.

PREREQUISITES
  • Understanding of Stokes' Theorem and its applications in vector calculus
  • Familiarity with the Divergence Theorem and its mathematical implications
  • Knowledge of vector fields and their properties, particularly curl and divergence
  • Concept of closed surfaces in the context of multivariable calculus
NEXT STEPS
  • Study the conditions under which Stokes' Theorem is applicable
  • Explore the Divergence Theorem in detail, focusing on closed surfaces
  • Investigate examples of vector fields where curl and divergence are computed
  • Review advanced topics in vector calculus, such as differential forms and their applications
USEFUL FOR

Mathematics students, physicists, and engineers who are working with vector calculus and need to understand the interplay between Stokes' Theorem and the Divergence Theorem.

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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