Green's Theorm

**comp_engr** · April 26th 2006, 05:46 PM

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

**Rebesques** · April 27th 2006, 01:28 AM

Let the rectangle $R$ be $a\leq x\leq b, \ c\leq y\leq d$ . Name the segments $(b,y), \ (a,y), \ c\leq y\leq d$ as $\gamma, \delta$ . Then for P,Q differentiable on a domain containing R,

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

which becomes

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

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

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

Proceed similarly for $\frac{\partial P}{\partial y}$ , noticing that dy=0 on the remaining two segments.

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