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