I wanted to investigate the quite surprising point of view
that a vector can be represented as a differential operator. This is the subject
of section 2.3 Vectors again. At the bottom of page 66, it says

"

**Since a vector at a point can be thought of as a directional derivative operator along a path through that point**, it should be clear that a vector*field*defines a map from smooth functions to smooth functions all over the manifold, by taking a derivative at each point." In the next sentence he talks of a vector field Y having an action on a function f(x^{μ}). e.g. Y(f) in equation (2.20).
First of all I watched some Khan Academy videos on vector fields. It was good reminder.

Then I imagined a sphere containing two vector fields X and Y. X is shown in blue and circles along lines of latitude decreasing as the latitude reaches the poles. Y is shown in red and always goes from south to north, also decreasing nearer the poles. I also combined them to make a vector field

Z = X + Y / 2

which would swirl upwards.Read more at Commentary 2.3 Vector as differential operator.pdf