I need help with this problem:
Verify Green's Theorem in the plane where $S$ is the annulus $\{(x,y)\in\mathbb{R^2}|a^2\leq x^2+y^2\leq b^2\}$ and
- $F(x,y)=\left(\frac{-y}{\sqrt{x^2+y^2}},\frac{x}{\sqrt{x^2+y^2}}\right)$
- $F(x,y)=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$
- $F(x,y)=\left(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right)$
I was able to compute the line integral $\int_{\partial S^+} F\cdot d\mathbf{r}$, but I'm having problems with the double integral $\iint_S\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA$.
My problem is that I don't know which limist should I use. I know that I could use polar coordinates, but this problem is from a chapter before change of variables, so I think I'm not supposed to solve it like that.
I just need help with getting the limits of integration right, since I think it is easy to compute the double integral after that.
Split the integral in four parts:
$$\iint_S\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA=$$
$$=\int_{-b}^{-a}\int_{-\sqrt{b^2-x^2}}^{\sqrt{b^2-x^2}}\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dydx+\int_{-a}^{a}\int_{\sqrt{a^2-x^2}}^{\sqrt{b^2-x^2}}\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dydx+$$
$$+\int_{a}^{b}\int_{-\sqrt{b^2-x^2}}^{\sqrt{b^2-x^2}}\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dydx+\int_{-a}^{a}\int_{-\sqrt{b^2-x^2}}^{-\sqrt{a^2-x^2}}\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dydx$$
Maybe this can work too:
$$\int_{-b}^{b}\int_{-\sqrt{b^2-x^2}}^{\sqrt{b^2-x^2}}\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dydx-\int_{-a}^{a}\int_{-\sqrt{a^2-x^2}}^{\sqrt{a^2-x^2}}\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dydx$$