I am a physicist and was reading about differential forms in Classical Mechanics. Now, I thought that a two-form is a smooth map $\omega : M \rightarrow \Lambda(T^*M)$ so that a point $p$ on the manifold is mapped smoothly to a k-form in $T_p^*M \wedge ...\wedge T_p^*M.$
But now I found the notation $\omega(p,v)(q,w)$ in these notes (sorry they are not online) for $p,q \in M$ and $v \in T_pM$, $w \in T_qM$ respectively.
This is of course something different than what I assumed, because here we are talking about two points of the manifold.
Is this non-sense or am I understanding the definition not correctly?
Since the tangent bundle of a manifold $M$ is the disjoint union of tangent spaces $$TM=\bigcup_{p \in M}T_pM$$ usually a tangent vector is denoted by $(p,v) \in TM$, where $p \in M$ is a point and $v \in T_pM$ is a vector. This notation has the purpose to remember that the vector $v$ is tangent to $M$ in the point $p$.
Now, it seems that the correct notation would be $\omega(p)(v,w)$ or $\omega(p,v)(p,w)$, since $\omega(p,v)(q,w)$ does not make much sense for $p \neq q$.