How Do You Calculate the Flux of a Vector Field Through a Parametric Surface?

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SUMMARY

The discussion focuses on calculating the flux of the vector field F = [x, y, z] through the parametric surface defined by S:[u cos v, u sin v, u^2] = r(u, v), with parameters ranging from 0 ≤ u ≤ 2 and 0 ≤ v ≤ 2π. Participants emphasize the importance of using the divergence theorem and the Jacobian for transforming coordinates. The correct approach involves computing the surface integral of the vector field over the defined surface, utilizing the dot product of the vector field and the differential area vector.

PREREQUISITES
  • Understanding of vector fields and surface integrals
  • Familiarity with parametric equations and cylindrical coordinates
  • Knowledge of the divergence theorem in vector calculus
  • Ability to compute the Jacobian for transformations
NEXT STEPS
  • Study the divergence theorem and its applications in vector calculus
  • Learn how to compute surface integrals for vector fields
  • Explore the concept of the Jacobian and its role in coordinate transformations
  • Practice problems involving parametric surfaces and flux calculations
USEFUL FOR

Students in calculus and vector analysis, particularly those studying physics or engineering, will benefit from this discussion as it addresses the calculation of flux through parametric surfaces.

lazyluke
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Homework Statement


1. The expression F = [x,y,z] defines a vector field. Given the parametric representation of a surface S:[u cos v, u sin v, u^2] = r (u,v), where the parameters cover the ranges 0 ≤ u ≤ 2 and 0 ≤ v ≤ 2π, calculate the flux F through the surface S.

Homework Equations


How do i start this problem?

The Attempt at a Solution


I know the transformation equations for the cylindrical coordinates, and:
flux=Int(E dot dA=Int(divergence of E d tao
Im not sure if i should find the jackobian, but if so how do i work with the parametric representation vector r??

P.S. sry about spelling
 
Last edited:
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Am i even in the right department with those questions?
 

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