In the flows chapter of Lee's Introduction to Smooth Manifolds (2e), Lee states on p. 210 that if $\theta:\mathbb{R}\times M\rightarrow M$ is a smooth global flow, then the vector field defined pointwise by:
$$V_p=\theta^{(p)'}(0)=\lim_{t\rightarrow0}\frac{d}{dt}\theta(t,p)$$
is a rough vector field, and the infinitesimal generator of $\theta$. Then, the proceeding proposition states that for a smooth global flow, the infinitesimal generator is a smooth vector field, the proof of which I understand. Am I missing something here? I feel like these statements are contradictory to one another.
Best Answer
It seems the point he is making is that a priori the time derivative of a smooth flow will produce an assignment of a vector to each point, but that assignment may fail to even to be continuous. Then right after in the proposition you refer to he addresses this matter, and verifies that said assignment will be continuous, and indeed smooth. (In this terminology any smooth vector field is rough, but not vice versa.)
I should note that it is also common to think of "rough" as "measurable" (or "random"); especially in the context of differential geometry, objects rougher than measurable are rarely used.