Overview
A vector field is a function from a Euclidian space, typically {% \mathbb{R}^3 %} to a vector space {% V %}.
{% F: \mathbb{R}^3 \rightarrow V %}
The concept of a vector can be extended by choosing the domain to be a set that can map to a Euclidian space,
in which case, the vector field is defined over a
manifold. When this is done, the vector spaces typically
used are
operators
on the manifold.