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Thread: Green's Theorm

  1. #1
    Apr 2006

    Green's Theorm

    Does anyone know the proof for Green's theorm over a rectangle?
    Thanks for any help
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  2. #2
    Super Member Rebesques's Avatar
    Jul 2005
    My house.
    Let the rectangle $\displaystyle R$be $\displaystyle a\leq x\leq b, \ c\leq y\leq d$. Name the segments $\displaystyle (b,y), \ (a,y), \ c\leq y\leq d $ as $\displaystyle \gamma, \delta$. Then for P,Q differentiable on a domain containing R,

    $\displaystyle \int\int_R \frac{\partial Q}{\partial x}dxdy=\int_c^d \bigg( \int_a^b \frac{\partial Q}{\partial x} dx \bigg)dy$

    which becomes

    $\displaystyle \int_c^d Q(b,y)dy-\int_c^d Q(a,y)dy=\int_{\gamma}Qdy+\int_{\delta}Qdy$

    and since dx=0 on these, the last equation is equal to

    $\displaystyle \int_{\gamma}(Pdx+Qdy)+\int_{\delta}(Pdx+Qdy).$

    Proceed similarly for $\displaystyle \frac{\partial P}{\partial y}$, noticing that dy=0 on the remaining two segments.
    Last edited by Rebesques; Apr 27th 2006 at 01:33 AM.
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