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Math Help - Verify Green's Theorem Over Unit Circle

  1. #1
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    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?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by caliguy View Post
    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<F_1(x,y),F_2(x,y)\right>. 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?
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  3. #3
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    quick question, when parametrizing the circle dA= r drd(theta) correct? for some reason my book leaves it as d(theta)
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by caliguy View Post
    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.
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  5. #5
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    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..
    Last edited by caliguy; December 5th 2009 at 10:02 PM.
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  6. #6
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    I got pi/2 as both answers in the problem. Is this the correct answer?
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