Correct to write $\vec{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$? Suppose I have some vector field
\begin{align}
\vec{F}\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)&=G\textbf{i}+H\textbf{j}+T\textbf{k}.\tag{1}
\end{align}
Would it be correct for me to say
\begin{align}
\mathbb{R}^3\overset{\vec{F}}{\longrightarrow}\mathbb{R}^3\;?\tag{2}
\end{align}
 A: Writing
$$
\mathbb{R}^3\overset{\vec{F}}{\longrightarrow}\mathbb{R}^3
$$
I would think that $\vec{F}$ is a function with domain $\mathbb{R}^3$ and it looks like you have a function with domain $\mathbb{R}$. So the notation isn't good. I also don't think it is a good idea to write $\vec{F}_t: \mathbb{R}^3 \to \mathbb{R}^3$ because this makes it look like as if for each fixed $t$ you get a function with domain $\mathbb{R}^3$. So I would write, for example, $G" \mathbb{R}^3 \to \mathbb{R}$. Again writing $G_t$ makes it look like you have a function for each fixed $t$.
I am guessing you want
$$
\mathbb{R}\overset{\vec{F}}{\longrightarrow}\mathbb{R}^3.
$$
And I am guessing that the $x, y$, and $z$ are then functions from $\mathbb{R}$ to $\mathbb{R}$. If you want to have the functions $G, H$ and $T$, then these would be functions from $\mathbb{R}^3$ to $\mathbb{R}$.
A: It looks like your vector field is also parameterized by time, so writing $\vec{F}_t:\mathbb{R}^3\to\mathbb{R}^3$ might be better.
For more notation: each of the component for the vector field are functions $G_t,H_t,T_t:\mathbb{R}^3\to\mathbb{R}$.
