Surface integral over a cylinder

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SUMMARY

The discussion focuses on the calculation of a surface integral over a cylinder, specifically addressing the confusion surrounding the solution that yields a result of 36π. Key points include the evaluation of the double integral over surface S1, which is determined to be zero, and the divergence of the vector field, which is consistently 3 regardless of the z-coordinate. The application of Stokes' Theorem is also mentioned, highlighting the relationship between the volume integral and the divergence, reinforcing that the divergence does not directly lead to the surface integral result of 36π.

PREREQUISITES
  • Understanding of surface integrals in multivariable calculus
  • Familiarity with Stokes' Theorem and its applications
  • Knowledge of divergence in vector calculus
  • Ability to interpret and evaluate double integrals
NEXT STEPS
  • Study the derivation and application of Stokes' Theorem in vector calculus
  • Learn about surface integrals and their geometric interpretations
  • Explore divergence and its implications in three-dimensional vector fields
  • Review examples of double integrals over various surfaces
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Students and educators in mathematics, particularly those studying multivariable calculus, as well as professionals working with vector fields and surface integrals.

faen
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Homework Statement



Here's a picture of the question:

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Here's the solution:

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The Attempt at a Solution




I can't really make complete sense of some things around this... Like how did the integral become 36pi based on what is written in the solution? Why is z equal to 0.. What is the double integral over surface S1 and why is it zero? How is the divergence 3 if z is zero? Also if i use stokes theorem and find the volum integral, it would equal to 3 times the volume of the sylinder since the divergence is 3. But it doesn't equal to 36 pi... So if somebody could enlighten me around these questions i'd really appreciate it :)
 
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