If $w_{t}$ is a family of $l$-forms on a manifold $M$, how is the $l$-form $\frac{\text{d}w_{t}}{\text{d}t}$ defined?
For example, if we have a family of $1$-forms, $p$ a point in $M$ and $v$ an element of $T_{p}M$, $\frac{\text{d}w_{t}}{\text{d}t}(p)(v)$ should be a real number, so such derivative has to be evaluated in some point. Am I wrong?
Thanks in advance.
It's just going back to the first definition. For each fixed $p\in M$,
$$\frac{ \omega_{t+h} (p) - \omega_t(p)}{h}$$
makes sense as $T^*_pM$ is a vector space. So by definition
$$ \frac{d\omega_t}{dt} (p)= \lim_{h\to 0}\frac{ \omega_{t+h} (p) - \omega_t(p)}{h}.$$
Basically you are thinking of, for each $p\in M$, $\omega_t (p)$ as a curve in $T_p^*M$.