Vector Calculus: Integral Theorems

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SUMMARY

The discussion centers on the application of the Divergence and Stokes Theorems in solving a vector calculus problem involving flux calculations through a sphere. The user successfully computed the magnetic field using curl but encountered difficulties while attempting to resolve the flux through the top hemisphere and the entire sphere, resulting in complex expressions involving trigonometric and exponential functions. The user concluded that the Divergence Theorem yielded a flux of zero, indicating a need for further exploration of Stokes' Theorem to simplify the calculations. The lecturer emphasized the importance of selecting the appropriate theorem to facilitate easier computation.

PREREQUISITES
  • Understanding of Divergence Theorem and Stokes' Theorem
  • Proficiency in vector calculus and parametrization techniques
  • Familiarity with curl and line integrals
  • Basic knowledge of trigonometric and exponential functions
NEXT STEPS
  • Study the application of Stokes' Theorem in vector fields
  • Review examples of flux calculations using the Divergence Theorem
  • Practice parametrization techniques for complex surfaces
  • Explore computational tools for visualizing vector fields and integrals
USEFUL FOR

Students and educators in mathematics, particularly those focusing on vector calculus, as well as anyone seeking to deepen their understanding of integral theorems and their applications in solving complex problems.

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Homework Statement



Question 3 part b and c


Homework Equations



Divergence and Stokes Theorems. Knowledge of parametrization ect ect



The Attempt at a Solution



I got the B field by using curl. However any attempt to resolve the flux through the top hemisphere or even the sphere as a whole just gives me a horrible mess filled with cos's, sines and exponentials.

I used divergence theorem and calculated that the flux = 0. This cannot be write otherwise there would be no need for part c. Stokes theorem is the one that gives me the horrible mess.

My lecturer says that I have to use the right theorem, and ds will produce 1 easy component to calculate. I've tried everything but literally it's impossible.

Can someone shine the light on what theorem to use?

I tried to just do a line integral since the sphere is bounded by x^2 + y^2 = 1... Again I get a horrible mess that I cannot integrate.

I would show my proper attempt but your latex reference is too longwinded and difficult to use. It would be better if you reprogrammed it to work like the equation editor on Microsoft Word.
 

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