I am currently trying to understand the definition of differentiation of scalar fields. In doing so, I have been considering the question in the title.
I have the following definition in my lecture notes: A vector field, $\underline{f}$, is a map $\underline{f}:\mathbb{R}^n \to \mathbb{R}^n$ such that $\underline{x} \mapsto \underline{f}(\underline{x})$.
Since $\underline{f}$ is a function on $\mathbb{R}^n$, I assume it must have $n$ components such that $\underline{f} = \begin{pmatrix} f_1 \\ f_2 \\ \vdots \\ f_n \end{pmatrix}$. So does this mean that $\underline{f}(\underline{x}) = \begin{pmatrix} f_1(x_1) \\ f_2(x_2) \\ \vdots \\ f_n(x_n) \end{pmatrix}$, when the $f_i$ are real functions $f_i: \mathbb{R} \to \mathbb{R}$?
No, each component $f_i$ may depend on all the variables $x_1,\dots,x_n$. In other words, it's $f_i \colon \mathbb{R}^n \to \mathbb{R}$, not $f_i \colon \mathbb{R} \to \mathbb{R}$.
(And by the way, your phrasing “Since $\underline{f}$ is a function on $\mathbb{R}^n$ [...]” is misleading. The reason for $\underline{f}$ having $n$ components is that the codomain is $\mathbb{R}^n$, not that the domain is $\mathbb{R}^n$.)