## Vector Fields

Struggling a bit to understand what a "vector field" is. Is it a geometric
object? It is described as an operator, which is sort of confusing. Can it be
considered/visualized separately from any specific manifold function? Are
vector fields one-to-one with manifold functions?

Suspect this because we are using functional terminology, and trying to define
"vector field" as something with a type signature, aka in terms of what it
"takes" and "returns".

Remember, a manifold function maps (geometric) points on a manifold to real
numbers.

Didn't understand the "covariantly"/"contravariantly" bit in the "Coordinate
Transformations" section.

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Ok, so "vector field on manifold" is a mapping from points on the manifold to
vectors. It is often combined with a manifold function to give the directional
derivative of the function at a point, which is a real number. In the language
of differential geometry, it is an operator that takes a real-valued manifold
function, and a point, and returns a real value.

A "one-form field" is a geometric operator which works on a vector field to
define a mapping from points to real values.

Maybe I was over-thinking this?

TODO: look in Spivak to see definitions there

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Coordinate-basis one-form fields are "dual" to coordinate-basis vector fields.