I'm trying to follow Moser's argument in symplectic geometry, and running into some troubles. Here is a picture (confusing parts are circled in red):
For the first one, what happens when $s+t$ is larger than $1$? Don't we leave the codomain $M \times I$?
For the second one, I don't understand what it means to take the derivative with respect to $t$ of a family of forms. Can somebody explain it in a clear, precise manner?
Thank you very much.
You're right that as written $\psi_t$ does not have codomain $M \times I$; if this worries you you can change it to $\psi_t : M \times I \to M \times 2I$ or similar. What's important is that $\psi_t$ is a local flow - the curves of varying $t$ are smooth and $\psi_0$ is the identity on $M \times I$.
As for differentiating a family of forms, note that the space of alternating $k$-tensors at a point is a finite-dimensional vector space; so the usual derivative $$\frac{d\omega_t}{dt} = \lim_{h \to 0} \frac{\omega_{t+h} - \omega_t}{h}$$ makes sense. You can then differentiate a family of differential forms pointwise.