I am confused about the definition of a differential form on a manifold.
The definition I have comes from Bott and Tu and is as follows:
A differential form, $\omega$, on a manifold $M$ is a collection of forms $\omega_U$ for $U$ in the atlas defining $M$, which are compatible in the following sense:
$i^*\omega_U=j^*\omega_V$ where $i,j$ are the inclusion maps.
I am confused as to what exactly $\omega_u$ is. Is it the pull back by a chart of a form on Euclidean space?
Moreover, how can I fail the compatibility criterion? It seems to me like it should always be true.
I think I simple example would really help me understand but I can't find one.
Thanks for your help,