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Math Help - Green's Formula, polar co-ordinates

  1. #1
    pkr
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    Green's Formula, polar co-ordinates



    where
    C is the ellipse x^2/a^2 + y^2/b^2 = 1
    a,b>0



    So basically I get upto II(-2)dxdy, with the elipse as limits, then converting to polar co-ordinates i'm unsure on the limits, but I have a vague idea it is sqrt(ab), can anyone confirm this? Would be appreciated, also can anyone tell me the best way be able to post all the different mathematical symbols
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  2. #2
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    Let \bold{F}(x,y) = (M(x,y),N(x,y)) where M(x,y) = x+y and N(x,y) = x - y.

    Green's theorem says \oint_C \bold{F} = \iint_D \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = \iint_D (1 - 1) = 0

    However, if you had \bold{F}(x,y) = (N(x,y),M(x,y)) then:
    \oint_C F = \iint_D 2 = 2\text{area}(D) = 2\pi ab

    Because the area of the ellipse D is given by \pi ab.
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  3. #3
    pkr
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    Where you have N(x,y)= x-y, should it not be = y-x

    as there was a "-" in the original formula?
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  4. #4
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    Quote Originally Posted by pkr View Post
    Where you have N(x,y)= x-y, should it not be = y-x

    as there was a "-" in the original formula?
    Yes, I copied down your problem in correctly.
    However, all the steps necessary to solve it are still there.
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