a function over a region of three dimensional space that gives at each point a vector

an assignment of a vector to each point in a space (or to each point of what mathematicians would call a manifold ), which is just a function that maps points in the space to vectors

a tangent bundle section of its tangent bundle

a vector valued function of position in space

a section of a vector bundle. More specifically, a vector field can mean a section of the tangent bundle.

See field.

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.