A vector field on a smooth manifold is a section of the tangent bundle.
More intuitively it is a map that gives a tangent vector for every point in the manifold.
Definition
For a smooth manifold , a vector field is a continuous map
Types
If is a smooth map, we call it a smooth vector field (though these are usually what we study, so often they are just written as vector fields)
If is a vector field that isn’t continuous, we call it a rough vector field.
Notes
Given a vector field, we can split it up into the basis vectors so
Thus, evaluation of a vector field on a function gives