Question

Use o Teorema de Green (verique as hipoteses!)
para calcular x^2ydx -xy^2dy, onde C é o círculo x^2 +y^2=4 orientado no sentido anti-horário

Answer 1

As funçoes $x^2y$ e $xy^2$ são de clase $C^{\infty}(\mathbb R^2)$ 
 
          $\displaystyle \oint_{C}x^2y\,dx-xy^2\,dy=\iint\limits_{D}\dfrac{\partial (-xy^2)}{\partial x}-\dfrac{\partial (x^2y)}{\partial y}\; dx\,dy\\ \\ \\ \text{Onde }D=\{(x,y):x^2+y^2\leq 4\}\\ \\ \\ \oint_{C}x^2y\,dx-xy^2\,dy=\iint\limits_{D}-y^2-x^2\;dx\,dy$
     
        $\displaystyle \oint_{C}x^2y\,dx-xy^2\,dy=-\iint\limits_{H} r^2\cdot r\, dr\, d\theta\\ \\ \text{Onde }H=\{(r,\theta): 0\leq r\leq 2\;,\; 0\leq \theta \leq 2\pi\}\\ \\ \\ \oint_{C}x^2y\,dx-xy^2\,dy=-\iint\limits_{H} r^3\; dr\, d\theta\\ \\ \\ \oint_{C}x^2y\,dx-xy^2\,dy=-\int\limits_{0}^{2\pi}\int\limits_{0}^{2} r^3\,dr\,d\theta\\ \\ \\ \oint_{C}x^2y\,dx-xy^2\,dy=-2\pi\int\limits_{0}^{2} r^3\,dr\\ \\ \\ \boxed{\oint_{C}x^2y\,dx-xy^2\,dy=-8\pi}$