I'm studying smooth manifolds with Lee's book. He defines a $k$-form on a manifold $M$ as a section $M \to \Lambda^k M$ (where $\Lambda^k M = \bigsqcup_{p\in M} \Lambda^k T_pM$ is the smooth vector bundle of alternating $k$-tensors).
Now he defines the wedge product of such forms pointwise by $(\omega \wedge \eta)_p = \omega_p \wedge \eta_p$. I don't see why this depends differentiably on $p$.
Kind regards, Sebastian