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Flux through two parametrized surfaces

  1. May 17, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the flux through a parametrized surface does not depend on the choice of parametrization. Suppose that the surface [tex]\sigma[/tex] has two parametrizations, r(s,t) for (s,t) in the region R of st-space, and also r(u,v) for (u,v) in the region T of uv-space, and suppose that the two parametrizations are related by a change of variables: u = u(s,t), v = v(s,t). Suppose that the Jacobian determinant [tex]\frac{\partial(u,v)}{\partial(s,t)}[/tex] is positive at every point (s,t) in R Use the change of variables formula for double integrals to show that computing the flux integral [tex]\Phi=\int\int Fnds [/tex] parametrization gives the same result.


    3. The attempt at a solution
    So if the Jacobian determinant is positive at every point in R, what does that mean for T? That it is positive as well?
    Have I used to the change of variables formula to show that computing the flux using either parametrization gives the same result?

    Thanks for your help.

    fluxproblem.jpg
     
    Last edited: May 17, 2010
  2. jcsd
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