Does the definition of a vector field include maps from $\mathbb{R}^n \to \mathbb{R}^m$ or only maps from $\mathbb{R}^n \to \mathbb{R}^n$?

Question

The header pretty much has the whole question, can a vector field be a map between different spaces or do they have to be the same?

Answer 1

You probably came to know vector fields using an ad hoc defintion for some problem in Euclidean space. In general a vector field a more general object, a so called section in a vector bundle. Locally, a vector bundle is the topological product of topological $U$ space with a vector space $V$, $U\times V$. The dimensoin of $V$ need not be related to that of $U$ (if $U$ has a dimension assigned to it). (Globally, without going into the details here, it's glued together from such local objects)

A vector field is then a map $U\rightarrow U\times V$ which assigns a vector $v$ to some point $x$, i.e. $x \mapsto (x, v)$ -- the first factor is redundant (there are more elegant ways to say this).

The points related to your question are that

1. the dimension from which the vector part is taken can be different (that depends on what object you are actually looking at)
2. in simple examples the product structure is often ignored, instead of looking at a map $\mathbb{R}^n \rightarrow \mathbb{R}^n \times \mathbb{R}^n$ where the first factor is trivial anyway you ignore the first factor in the target and pretend that you just look at a map $\mathbb{R}^n \rightarrow \mathbb{R}^n$ and you think of the vector as somehow attached to its base point.

Answer 2

$\newcommand{\Reals}{\mathbf{R}}$Briefly, only maps $\Reals^{n} \to \Reals^{n}$.

In more detail, the tangent space to $\Reals^{n}$ at a point $p$ is an $n$-dimensional vector space $T_{p}\Reals^{n}$ that can be identified with $\Reals^{n}$ itself.

The collection $\bigcup_{p} T_{p}\Reals^{n}$ of all tangent spaces (together with some important extra structure) is the "total space of the tangent bundle". The tangent bundle $T\Reals^{n}$ is the projection $\pi:\bigcup_{p} T_{p}\Reals^{n} \to \Reals^{n}$ sending each tangent space $T_{p}\Reals^{n}$ to its "base point" $p$. In coordinates, $\pi(p, v) = p$ for all $p$ in $\Reals^{n}$ and all $v$ in $T_{p}\Reals^{n} \simeq \Reals^{n}$.

In fancy language, a vector field on $\Reals^{n}$ is a section (i.e, a right inverse) of $\pi$, i.e., a mapping $X:\Reals^{n} \to T\Reals^{n}$ such that $\pi \circ X(p) = p$ for all $p$. In words, a vector field associates a tangent vector $X(p)$ at $p$ to each point $p$ of $\Reals^{n}$.