Im struggling to understand how to apply Green's theorem in the case where you have a hole in a region which is traversed in the same direction as the exterior.
For a workable example I want to evaluate $\displaystyle \oint_{C}{{{y^3}\,dx - {x^3}\,dy}}$. I want to consider 2 circles where C1 is given by $x^{2}+y^{2}=4$ , C2 is given by $x^{2}+y^{2}=1$, and D is the region in between.
If we consider the case where both C1 and C2 are positively oriented we can break the region into 2, offset the new boundaries created, and ultimately end up with $\begin{align*}\oint_{{{C_1}}}{{Pdx + Qdy}} + \oint_{{{C_2}}}{{Pdx + Qdy}}= \oint_{C}{{Pdx + Qdy}}\end{align*}$. When we do the integral we do it (in polar coordinates) as the integral we would have done without there being a hole. Only, we change the limits in the inner integral to end at 1 as would be expected.
$\begin{align*} \int_{{\,0}}^{{\,2\pi }}{{\int_{{\,1}}^{{\,2}}{{{- 3r^3}\,dr}}\,d\theta }}=- \frac{{45\pi }}{2}\end{align*}$
What happens when we have both C1 and C2 oriented in the counterclockwise direction? (my guess is that we can just change the sign of the C2 integral portion but I havent been able to verify this from searching around) Ideally an explanation using this example in applied terms and in general terms would be amazing.