Thread: Verify Green's Theorem Over Unit Circle

1. Verify Green's Theorem Over Unit Circle

I only need a kick in the right direction here. I've done a few of these and I get them just that this one they are asking to verify green's theorem for the vector field F=<-x^2y,xy^2> over the a unit circle. Ive done a few that give me the actual function but this time they give me a vector field... How should I begin?

2. Originally Posted by caliguy
I only need a kick in the right direction here. I've done a few of these and I get them just that this one they are asking to verify green's theorem for the vector field F=<-x^2y,xy^2> over the a unit circle. Ive done a few that give me the actual function but this time they give me a vector field... How should I begin?
Note that $\oint \mathbf{F}\cdot d\mathbf{x}=\oint F_1(x,y)\,dx+F_2(x,y)\,dy$ where $\mathbf{F}(x,y)=\left$. Thus, we can write the integral in differential form as follows:

$\oint_C -x^2y\,dx+xy^2\,dy$, where $C: x^2+y^2=1$

Can you take it from here?

3. quick question, when parametrizing the circle dA= r drd(theta) correct? for some reason my book leaves it as d(theta)

4. Originally Posted by caliguy
quick question, when parametrizing the circle dA= r drd(theta) correct? for some reason my book leaves it as d(theta)
When you apply green's theorem, the integral becomes $\iint\limits_R x^2+y^2\,dA$.

In this case, I would recommend using polar (like you started to do). Note that the limits of integration are $0\leq r\leq 1$ and $0\leq\theta\leq2\pi$.

So, $\oint_C -x^2y\,dx+xy^2\,dy=\int_0^{2\pi}\int_0^1r^3\,dr\,d\ theta$.

5. ok for the integral side I got pi/2 but for the partial derivative side i got the integral of (x^2+y^2) dA, this doesn't seem right to me...

EDIT: ok now I got (2pi/3) for the partial derivative side after integrating it..

6. I got pi/2 as both answers in the problem. Is this the correct answer?