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(xμ).
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
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