Calculating Circulation of Vector G with Green's Theorem

[-EquinoX-]

Homework Statement

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

Homework Equations

The Attempt at a Solution

$$\int_{-2}^{2}\int_{-\sqrt(4-y^2)}^{\sqrt(4-y^2)} y-3 dx dy$$

is this correct?

 

[HallsofIvy]

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

What you have is
$$\frac{\partial xy}{\partial x}- \frac{\partial 3y}{\partial y}$$

Also, although your limits of integration are correct for Cartesian coordinates, I think the integral would be easier in polar coordinates.

 

[dx]

Hm.. the curve is oriented counterclockwise, so shouldn't it be ∂_x(xy) - ∂_y(3y), i.e. the z-component of ∇ x G?

 

[HallsofIvy]

Yes, sorry, my mind blew a fuse!

 

[-EquinoX-]

so it is correct isn't it?