In a recent topic I've studied on complex analysis I had to study the differential system on the torus $\mathbb T^2:$ $\begin{cases}\frac{\partial}{\partial y}u-\frac{\partial}{\partial x}v=\sin(y)-\cos(x),\\\\ \frac{\partial}{\partial x} u+\frac{\partial}{\partial y}v=0,\end{cases}$ with the conditions $\int_0^{2\pi}\int_0^{2\pi}u(x,y)\mathrm d x\mathrm dy=\int_0^{2\pi}\int_0^{2\pi}v(x,y)\mathrm d x\mathrm dy=0.$
In particular it seemed to me that this system was explicitly solvable and to do so i relied on the inhomogeneous Cauchy Riemann equations (swapping the coordinates $x\leftrightarrow y) $ and i basically followed this link. Unfortunately my calculations didn't lead nowhere..
I am asking two things..
Is that way followable to finish the problem, and if so can you help me in finishing the proof?
More importantly: Are there smarter ways to do the problem?
Thanks in advance..
-Guido-
EDIT
I've got the following question related to the previous post so I'm writing it as an edit to this question.
The question is the following: prove that if $f$ is smooth, periodic and with zero average, then the solution to the system on $\mathbb T^2$
$\begin{cases}\frac{\partial}{\partial y}u-\frac{\partial}{\partial x}v=0,\\\\ \frac{\partial}{\partial x} u+\frac{\partial}{\partial y}v=f(x,y),\end{cases}$
satisfies
$\int_{0}^{2\pi}\int_0^{2\pi}\left(u(x,y)u'(x,y)+v(x,y)v'(x,y)\right)\mathrm dx\mathrm dy=0.$
Thanks for your patience.