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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!

JDoeDoe
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    The definition of a vector field is usually a bit more involved, see e.g. here: https://math.stackexchange.com/questions/2348634/does-the-definition-of-a-vector-field-include-maps-from-mathbbrn-to-mathb/2348645#2348645. The special case of a vector field on Euclidean space can be reduced to a map like your $G$. – Thomas Jul 07 '17 at 10:01

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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 $S \subset \mathbb{R}^n.$

Pawel
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