Limitations of the divergence theorem

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SUMMARY

The discussion centers on evaluating the surface integral of the vector field F=<0, y, -z> over the surface defined by y=x^2+y^2 for y between 0 and 1, using the divergence theorem. The user calculated the divergence of F, resulting in div F = 0, leading to the conclusion that both the triple integral and surface integral yield a result of 0. The user expresses skepticism about the simplicity of this conclusion and questions whether any limitations of the divergence theorem were overlooked.

PREREQUISITES
  • Divergence theorem
  • Vector calculus
  • Surface integrals
  • Triple integrals
NEXT STEPS
  • Evaluate the surface integral directly to confirm the result.
  • Study the conditions under which the divergence theorem applies.
  • Explore examples of vector fields with non-zero divergence.
  • Review potential limitations of the divergence theorem in specific scenarios.
USEFUL FOR

Students in calculus, particularly those studying vector calculus, as well as educators and anyone interested in the practical applications and limitations of the divergence theorem.

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Homework Statement



Evaluate the surface integral F * dr, where F=<0, y, -z> and the S is y=x^2+y^2 where y is between 0 and 1.

Homework Equations



Divergence theorem

The Attempt at a Solution



I just got out of my calculus final, and that was a problem on it. I used the divergence theorem and got div F = 0 + 1 - 1 =0

Therefore the triple integral and therefore the surface integral will return 0. Is this true? I thought it was too easy, I can't see why it wouldn't be true.

Is there a limitation I glossed over?
 
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Don't think so, that should work. Try evaluating the surface integral and seeing if it also returns 0.
 

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