Suppose F=(3x-3y)i + (x+2y)j. Use Stoke's Theorem to find the circulation of F around the circle C of radius 4, centered at the origin in the xy plane, oriented clockwise, as viewed from the positive z axis.

Question

tiffany_rhodes · Answer

I believe the curl of F to be (0,0,4) but I'm not sure what to do after that.

phi · Answer

I think they want you to use

$$ \oint \overrightarrow{F}\cdot d\overrightarrow{r}= \int\int_R curl(\overrightarrow{F}) \ dA$$

phi · Answer

Stoke's thm is for 3-D, and I posted Green's thm (for 2D) which we can use here
either way, it becomes
$$ 4 \int \int_R dA $$
were 4 is the curl, and you now need to find the area of the region (not hard, it's a circle)

phi · Answer

if we use Stokes thm $$ \oint \overrightarrow{F}\cdot d\overrightarrow{r}= \int\int_S curl(\overrightarrow{F}) \cdot \hat{n} \ dS $$ here we integrate over any surface inclosed by the circle... if we use the interior of the circle (with normal <0,0,1> ) , we get <0,0,4> * <0,0,1> = 4 and we integrate over the surface of the circle i.e. again, it is 4 * area of the circle

tiffany_rhodes · Answer

Okay, thanks! I think I was confused because I thought I had to parameterize the circle. So basically you find the curl of the vector field and dot product that with the unit normal vector, and then multiple by the area of the circle (which in this case was 16*pi)?