I was reading symplectic Moser's theorem and got confused about certain definition:
In the statement, it said we have a smooth family of symplectic forms $\omega_t$, how do we define the notion of smooth family? (i.e. what is the topology and diffferential strucuture we put on the space of forms?) also, in the proof using Moser's trick, we take the derivative of the form $\omega_t$, how to do this?
Intuitively I think when the form is represented in local coordinates, we have its coefficients are function with variable $t$ (also in variable $x$ which is the location of the point on the manifold). So by smooth family we are saying that the coefficients are smooth function in terms of $t$ and when we take the derivative we just take the partial derivative of the coefficients w.r.t. t thus the resulting derivative is also a form. Is that correct? But how to state this independent of local coordinates?
While it can be done, there's no need to topologize the space of forms when defining a smooth family.
Usually, a smooth family of maps between smooth manifolds $M$ and $N$ is defined to be a map $$ F:S\times M\to N $$ Where, depending on the context, the parameter space $S$ can be taken to be an interval, an open subset of $\mathbb{R}^n$ or even an arbitrary smooth manifold. For differential forms, this would be a smooth map $$ \omega:S\times M\to\Lambda^kT^*M $$ such that fixing the first argument $s\in S$ gives a differential form $\omega_s\in\Omega^kM$.
In the case that the parameter space is an interval, we can differentiate with respect to the parameter straightforwardly, since it makes sense add and subtract differential forms $$ \frac{d}{ds}\omega_s=\lim_{h\to 0}\frac{\omega_{s+h}-\omega_s}{h} $$ In the more general case, $\frac{d}{ds}\omega_s$ will be a map $TS\to\Omega^kM$.