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How do you compute the circulation of this fluid (path integrals)

  1. Feb 27, 2013 #1
    1. The problem statement, all variables and given/known data
    A fluid as velocity field F(x, y, z) = (xy, yz, xz). Let C denote the unit circle in the xy-plane. Compute the circulation, and interpret your answer.


    2. Relevant equations



    3. The attempt at a solution
    Since the unit circle is a closed loop, I assumed that ∫ F * dr = 0
    (the ∫ symbol is supposed to have a circle)
    However, when I attempt to verify whether or not F = [itex]\nabla[/itex] f, the two are not equal, suggesting that ∫ F * dr does not= 0
    At this point I'm at a loss of how to go about calculating the circulation. Could someone please help?
     
  2. jcsd
  3. Feb 27, 2013 #2

    Dick

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    Homework Helper

    You just want to calculate the integral. Pick a parametrization of the unit circle, like r(t)=(cos(t),sin(t),0) and work it out.
     
  4. Feb 27, 2013 #3
    I would rephrase this to "suggesting that ∫ F * dr might not= 0" Note that IF the vector field is conservative then every closed path integral must be zero; however, the opposite is not necessarily true. There are many cases when the field is not conservative but a given closed path integral is still zero (and I suspect that this problem will end up being one of those cases).

    Good Luck!
     
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