Given a compact manifold $M$ and a smooth time-dependent exact 2-form $\omega_t$ on $M$, is it possible to find a smooth time-dependent 1-form $\mu_t$ on $M$ such that $\omega_t = d \mu_t$?
Related reference is in the Theorem 3.17. in Mcduff's Introduction to Symplectic Topology.
Specifically, I'd like to know the part of the setence "The inductive step is achieved by using the Mayer-Vietoris sequence."
I tried it by using bump functions.
Let $U, V$ be open sets on which it is true for compactly supported forms. Let $\omega_t$ be a compactly supported closed form on $U \cup V$. To prove it, I take two functions $\phi, \psi$, a partion of unity subordinate to $\{ U , V \}$. However $d (\phi \omega_t) = d\phi \wedge \omega_t$ is not a closed form.
So I'd like to know how to use this hint.
Thank you in advance.