Question

Questão de aplicação de formula de Green, alguém pode me ajudar?

Answer 1

$\displaystyle W=\oint_{\Gamma}\dfrac{dx-dy}{x+y}\\ \\ \\ W=\oint_{\Gamma}\dfrac{1}{x+y}dx+\left(-\dfrac{1}{x+y}\right)dy\\ \\ \\ W=\iint_{\Omega}\dfrac{\partial}{\partial x}\left(-\dfrac{1}{x+y}\right)-\dfrac{\partial}{\partial y}\left(\dfrac{1}{x+y}\right)\; dx\,dy\\ \\ \\ W=2\iint_{\Omega}\dfrac{1}{(x+y)^2}\; dx\,dy\\ \\ \\ \text{onde }\partial\Omega=\Gamma\\ \\ \text{Uma transformacao util \'e: }\\ \\ \\ x(u,v)=\dfrac{u+v}{2}\;\;,\;\; y(u,v)=\dfrac{u-v}{2} \\ \\ \\ \text{e a nova region e: }$

$H=\{(u,v):-1\leq u\leq 1\;,\; -1\leq v\leq 1\}\\ \\ \\ \displaystyle W=\iint_H\dfrac{1}{u^2}\;du\,dv\\ \\ \\ W=\int_{-1}^{1}\int_{-1}^{1}\dfrac{1}{u^2}\;du\,dv\\ \\ \\ \boxed{W=-4}$