Does anyone know the proof for Green's theorm over a rectangle?

Thanks for any help

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- Apr 26th 2006, 05:46 PMcomp_engrGreen's Theorm
Does anyone know the proof for Green's theorm over a rectangle?

Thanks for any help - Apr 27th 2006, 01:28 AMRebesques
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.