Consider the following DPE system:
$\left\{ \begin{array}{rcl} g_x - f_y& = &1-x^2 \\ h_x - f_z &= &3x^2 \\ h_y - g_z &=& -1 \ \end{array}\right .$
This comes from trying to prove that a certain 2-form is exact (and of course they never told us about PDEs before), and maybe the point is that I am not approaching the problem in the right way, but if so, is there any reasonable way to solve that DPE system? It is to note that I attempted Schwarz theorem, which took me to the (ridiculous) point where $g_x - f_y = 0$, so $1-x^2 = 0$, which is absurd since the 2-form is defined on the whole space.
Any ideas?
For any interest, the 2-form is, of course, $\omega = (1-x^2)dx \wedge dy + 3x^2 dx \wedge dz - dy \wedge dz$ and, in order for it to be exact, i am supposing that there is some $\eta = fdx + gdy + hdz \in \Omega^1(\mathbb{R}^3)$ such that $d\eta = \omega$.