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Green's Theorem

  1. Nov 8, 2007 #1
    1. The problem statement, all variables and given/known data

    (Q) Find the outward flux of the field

    F=(3xy-x/(1+y^2 ))i+(e^x+tan^(-1)⁡y )j

    across the cardioid r=a(1+cos⁡θ), a>0.

    2. Relevant equations

    div F = (∂M )/∂x+∂N/∂y

    3. The attempt at a solution

    I could easily set up the double integral which is:

    ∬▒3 r^2 sin⁡θ dA

    However, I am unsure as to how to determine the limits to be used.
    Please help. Thank-you.
  2. jcsd
  3. Nov 8, 2007 #2


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    Staff Emeritus
    Science Advisor

    ?? The limits of integration ARE the cardioid, of course: [itex]\theta[/itex] going from 0 to [itex]2\pi[/itex], r from 0 to a(1+ cos([itex]\theta[/itex])). Since that is given in polar coordinates, it might be best to convert div F to polar coordinates.
  4. Nov 8, 2007 #3
    Yes but.....

    Usually, the limits of r ranges from a constant value to another constant value of r. Why do we, in this case simply plug in the equation of the cardioid?

    Thanks a lot for your help and support!!!!:smile:
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