Suppose I have a vector field $\mathbf A(x,y,z)$ such that $$ \mathbf A(\mathbf r)=A_x(\mathbf r)\mathbf{\hat e}_x+A_y(\mathbf r)\mathbf{\hat e}_y+A_z(\mathbf r)\mathbf{\hat e}_z $$ where $\mathbf{r}=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z=(x,y,z)$.
I know $\mathbf A(\mathbf r)$ is a function $\mathbb{R}^3 \rightarrow \mathbb{R}^3$.
But what is $\mathbf r$ itself? Is it a function $\mathbb{R}^1 \rightarrow \mathbb{R}^3$?
Thanks!
No, $\mathbf{r}$ is just the independent variable in this context. You could have $\mathbf{A}(\mathbf{r}(t))$, then $\mathbf{r}$ would be a function from $\mathbb{R}^1$ to $\mathbb{R}^3$. (Similarly, it would be better to say that $\mathbf{A}$ is a function from $\mathbb{R}^3$ to $\mathbb{R}^3$ while $\mathbf{A}(\mathbf{r})$ is just a vector in $\mathbb{R}^3$, but this type of abuse of notation is quite common.)