Difference between Vector Functions and Vector Field

Question

I understand that a vector function is a function that has a domain $\mathbb{R}^n$ and range on $\mathbb{R}^m$ so it takes vectors and gives vectors right? So what is a vector field?And how can I visualize them?

Answer 1

A vector field $F|_S\colon S\to\mathbb{R}^n$ is an assignment of $n$-dimensional real vectors to points in a subset of $\mathbb{R}^n$ so it's really just a vector-valued function $F\colon\mathbb{R}^n\to\mathbb{R}^n$ restricted to a subdomain $S\subset\mathbb{R}^n$. Note that the domain and codomain of $F$ have the same dimension, but $S$ can possibly be a lower-dimensional subspace (for instance a proper linear subspace or a sphere).

(Note: there is a more sophisticated definition of a vector field which is defined in terms of tangent spaces and tangent bundles, but the above definition more than likely suffices for introductory calculus courses.)