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Green's theorem

  1. Apr 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Use Green's Theorem to calculate the circulation of [tex] \vec{G} [/tex] around the curve, oriented counterclockwise. [tex] \vec{G} = 3y\vec{i} + xy\vec{j} [/tex] around the circle of radius 2 centered at the origin.

    2. Relevant equations



    3. The attempt at a solution

    [tex] \int_{-2}^{2}\int_{-\sqrt(4-y^2)}^{\sqrt(4-y^2)} y-3 dx dy [/tex]

    is this correct?
     
  2. jcsd
  3. Apr 30, 2009 #2

    HallsofIvy

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    No, it isn't. Your integrand should be
    [tex]\frac{\partial xy}{\partial y}- \frac{\partial 3y}{\partial x}[/tex]

    What you have is
    [tex]\frac{\partial xy}{\partial x}- \frac{\partial 3y}{\partial y}[/tex]

    Also, although your limits of integration are correct for Cartesian coordinates, I think the integral would be easier in polar coordinates.
     
  4. Apr 30, 2009 #3

    dx

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    Hm.. the curve is oriented counterclockwise, so shouldn't it be ∂x(xy) - ∂y(3y), i.e. the z-component of ∇ x G?
     
  5. Apr 30, 2009 #4

    HallsofIvy

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    Yes, sorry, my mind blew a fuse!
     
  6. Apr 30, 2009 #5
    so it is correct isn't it?
     
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