My class notes define a vector field $F$ to be a vector $$F=(f_1(t),f_2(t),f_3(t)).$$ In actuality, is this actually just the section of a rank $3$ vector bundle on $\Bbb R^1$?
In proper generality, if we have a vector bundle of rank $r$ on a topological space $X$, then any section of this bundle is called a vector field on $X$, is that correct?
Just trying to recall something I'd done in more generality, really loosely, in the past.
Additionally, my notes say that $$\frac{dF}{dt}=(\frac{d}{dt}f_1,\frac{d}{dt}f_2,\frac{d}{dt}f_3).$$
What does $\frac{d}{dt}$ correspond to, as a map on sections of a bundle?