Is a vector field just a special case of a vector-valued function?

Question

For a vector-valued function we have $$ \mathbf{F}:\mathbb{R}^m\rightarrow\mathbb{R}^n $$

However, is it correct that a vector field $\mathbf{G}$ is just a special case then $m=n$? I.e. $$ \mathbf{G}:\mathbb{R}^n\rightarrow\mathbb{R}^n $$

Thanks!

Answer 1

That is basically correct. Sometimes instead of $G:\mathbb{R}^n \to \mathbb{R}^n$ people require only $G:S \to \mathbb{R}^n,$ where $S \subset \mathbb{R}^n.$