Correct to write $\vec{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$?

Question

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}

Answer 1

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 $\mathbb{R}$.

Answer 2

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 $G_t,H_t,T_t:\mathbb{R}^3\to\mathbb{R}$.