exact and closed differential forms

Question

This exercise is taken from the Meyer-Hall-Offin book on Hamiltonian systems.

Let $Q(p,q)$ and $P(p,q)$ be smooth functions defined on an open set in $\mathbb{R}^2$. Consider the four differential forms $\Omega_1=PdQ-pdq,\ \Omega_2=PdQ+qdp,\ \Omega_3=QdP+pdq,\ \Omega_4 = QdP-qdp$. Show that $\Omega_i$ is closed (exact) iff $\Omega_j$ is closed (exact) for $i\neq j$.

I've been able to show the case "closed", but I don't know how to approach the second case. Any hints?

Thanks in advance!

Answer 1

Notice that for $\Omega_i$ it is always possible to write:

$$ \Omega_{i} = \sum_{j\neq i} a_j \Omega_j $$ for some constants $a_j$. For example: $\Omega_1 = \Omega_2 + \Omega_4 -\Omega_3 $. So if all $j\neq i$ forms are exact, so will be $\Omega_i$.