Calculating flux through a surface area

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SUMMARY

The discussion focuses on calculating the flux of a vector field F through a surface defined by the parameterization r(u,v) for 0 < u < 2 and 0 < v < 2π. The user successfully computed the derivatives dr/du and dr/dv, and determined the normal vector N using the cross product. They expressed concerns about the complexity of their calculations due to the lengthy vector resulting from the cross product. The conversation suggests utilizing the Divergence Theorem or Stokes' Theorem as alternative methods to simplify the calculation process.

PREREQUISITES
  • Understanding of vector calculus concepts, specifically flux calculations.
  • Familiarity with parameterization of surfaces in three-dimensional space.
  • Knowledge of the Divergence Theorem and Stokes' Theorem.
  • Proficiency in performing vector operations such as cross products and dot products.
NEXT STEPS
  • Study the Divergence Theorem and its applications in flux calculations.
  • Learn Stokes' Theorem and how it relates to surface integrals.
  • Explore techniques for simplifying vector field calculations in three dimensions.
  • Practice parameterizing various surfaces and calculating their flux using different methods.
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Students and professionals in mathematics, physics, and engineering who are involved in vector calculus and surface integrals, particularly those seeking to optimize their calculations of flux through surfaces.

dan38
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S is a portion of a curve with r(u,v)
where 0 < u < 2 and 0 < v < 2pi

I'm meant to calculate Flux of the vector field F

My Calculations
First found dr/du
then dr/dv

Using the cross product, I found N = (- u cos (v) + 5 sin (v), -5 cos (v) - u sin(v), u)
Then I dot product with the given F (after putting it in terms of "u" and "v")
Then I do the relevant integration

I think my method is correct, however since my cross product yields an extremely long vector it makes all my following calculations really tedious.
So just want to check if there's a easy way to do this.
 
Last edited:
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There is no way to answer that without knowing what r(u,v) is. Some surfaces have "simple" differentials of surface area, some complicated ones. (What you give does NOT seem "extremely long" to me!)
 
Use the divergence theorem or Stokes' theorem.

BiP
 

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