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Green's theorem

  • Thread starter -EquinoX-
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  • #1
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Homework Statement



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.

Homework Equations





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?
 

Answers and Replies

  • #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.
 
  • #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?
 
  • #4
HallsofIvy
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Yes, sorry, my mind blew a fuse!
 
  • #5
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so it is correct isn't it?
 

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