# Green's theorem

1. Apr 30, 2009

### -EquinoX-

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

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

is this correct?

2. Apr 30, 2009

### 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.

3. Apr 30, 2009

### dx

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

4. Apr 30, 2009

### HallsofIvy

Yes, sorry, my mind blew a fuse!

5. Apr 30, 2009

### -EquinoX-

so it is correct isn't it?