I am in trouble with the notation of vector fields in order to teach about vector fields, the difference between them and functions with codomain in $\mathbb R^n$, divergent, rotational and so on.
One often definition of a vector field is the following: given a subset $S$ of $\mathbb R^n$, a vector field is represented by a vector-valued function $V: S\rightarrow\mathbb R^n$ in standard Cartesian coordinates $(x_1,..,x_n)$ (or not).
But in the context of multivariable calculus and vector analysis there is no sinctatical difference between an usual function $V:S\rightarrow\mathbb R^n$ and a vector field $V: S\rightarrow\mathbb R^n$. Or at least, I could not find a standard notation to make this difference in the literature. In other words, we use the same notation $V:S\rightarrow\mathbb R^n$ for both objects, which generally cause some confusion for the beginners.
Of course, in the case $n=3$, is usual write a vector field $V:\mathbb R^3\rightarrow\mathbb R^3$ by $V(x)=f(x)\vec i+g(x)\vec j+h(x)\vec k$, with $f,g,h$ being functions from $\mathbb R^3$ into $\mathbb R$ (here $x=(x_1,x_2,x_3)$). But this notation is not standard and vector fields can be described in other basis different from the standard basis of $\mathbb R^n$.
Is there such notation available for multivariate calculus to make this distinction or will be I attached to the context involving a function (or a vector field) $V: S\rightarrow\mathbb R^n$?