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Green's Theorem

  1. Dec 11, 2012 #1

    MacLaddy

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

    Use Green's Theorem to evaluate [itex]\int_c(x^2ydx+xy^2dy)[/itex], where c is the positively oriented circle, [itex]x^2+y^2=9[/itex]

    2. Relevant equations

    [tex]\int\int_R (\frac{\delta g}{\delta x}-\frac{\delta f}{\delta y})dA[/tex]


    3. The attempt at a solution

    I have found [itex]\frac{\delta g}{\delta x}-\frac{\delta f}{\delta y}[/itex] to be [itex]y^2-x^2[/itex]

    My hangup is moving forward. My integral will look like this,

    [itex]\int\int_R [y^2-x^2]dA[/itex]

    however, since the region is a circle I am integrating over should I convert this to polar? If I do, will my values in the integral be [itex]rcos^2\theta - rsin^2\theta[/itex], or since it's basically just a line integral, will it be [itex]3cos^2\theta - 3sin^2\theta[/itex]?

    This setup is confusing me. Any help is appreciated. Am I integrating just the perimeter of the circle, or the entire thing?

    Thanks,
    Mac
     
    Last edited: Dec 11, 2012
  2. jcsd
  3. Dec 11, 2012 #2

    SammyS

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    Some typos / errors corrected below:
    Should be

    [itex]\displaystyle \int\int_R (\frac{\delta g}{\delta x}-\frac{\delta f}{\delta y})dA[/itex]
    Following line corrected:
    The r's should be squared.

    [itex]r^2\cos^2(\theta) - r^2\sin^2(\theta)[/itex]
    What integral do you get ?
     
    Last edited: Dec 11, 2012
  4. Dec 11, 2012 #3

    MacLaddy

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    Sorry, this is confusing me. I fixed this [itex]\int\int_R (\frac{\delta g}{\delta f}-\frac{\delta f}{\delta y})dA[/itex] to be [itex]\int\int_R (\frac{\delta g}{\delta x}-\frac{\delta f}{\delta y})dA[/itex] this, but it may have been while you were typing. Let me know if there are still errors that I am missing.

    Well, if I convert to polar, and from what you are saying, it should be
    [tex]\displaystyle \int_0^{2\pi}\int_0^3[r^2 sin^2\theta - r^2 cos^2\theta]rdrd\theta[/tex]
    Do I still use the additional r in the Jacobian when I am doing it this way? Or am I completely off base?

    Thanks for the help.
    Mac
     
    Last edited: Dec 11, 2012
  5. Dec 11, 2012 #4

    SammyS

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    Yes, you must have corrected it while I was typing.
    Of course you use the Jacobian. dxdy → rdrdθ .
     
  6. Dec 11, 2012 #5

    MacLaddy

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    Groovy, thank you very much for the help. My mind was definitely glitching on that one.

    I evaluated the integral and it equaled 0, so I suppose there is no net rotation.

    Mac
     
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