# Green's Theorm

• Apr 26th 2006, 06:46 PM
comp_engr
Green's Theorm
Does anyone know the proof for Green's theorm over a rectangle?
Thanks for any help
• Apr 27th 2006, 02:28 AM
Rebesques
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.