So the problem is in the title. The rules are that I can't split the circle into "rectangles" and I can't use pull-back. I tried to do something similar to the proof on unit squares. The problem is that to prove Green's theorem for unit squares you don't need to use any kind of coordinate transformation so when I try to use the same principle for circles I fail because I don't know how to proceed.
So, here's what I've got so far
$\omega=Pdx+Qdy, \space\space\gamma(\phi)=(s_x+R\cos\phi, s_y+R\sin\phi),\space\space\phi\in[0, 2\pi]$
The circle has the radius $R$ and is centered at $(s_x, s_y)$. Using this parametrization, $dx=-R\sin\phi,\space\space dy=R\cos\phi$
Now, the first integral in Green's theorem is the following
$\int_\gamma\omega=\int_0^{2\pi}P(\gamma(\phi))(-R\sin\phi)+\int_0^{2\pi}Q(\gamma(\phi))(R\cos\phi)$
As far as I know, I can't calculate anything further than that without knowing $P$ and $Q$.
So, the second integral is supposed to be something like this
$\int_Dd\omega=\int\int_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy$
I have no idea where to go next. Since the parametrization for the interior of the circle $D$ would be something like $g(r, \phi)=(s_x+r\cos\phi, s_y+r\sin\phi)$, I don't know what $dxdy$ means. Also, I'm pretty sure I can't just write $\frac{\partial Q}{\partial x}$ and be fine with it since I'm not using $x,y$.
I'm pretty sure most of what I've wrote doesn't make sense but I'm still very fuzzy on the whole "differential form" thing. If someone could help me out, I'd appreciate it.