\begin{definition}[Definition:Gradient Operator/Real Cartesian Space/Region]
{{MissingLinks|Space of functions. Notation is already defined iirc}}
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.
Let $S \subseteq \R^n$.
Let $\sqbrk {X \to Y}$ be the space of functions from $X$ to $Y$.
Suppose that for all $\mathbf x \in S$, $\map {\nabla f} {\mathbf x}$ exists.
The '''gradient''' can then be defined as an operation acting on $f$:
:$\nabla: \mathbf F \to \sqbrk {S \to \R^n}$
:$\paren {f: \mathbf x \mapsto \map f {\mathbf x} } \mapsto \paren {\nabla f: \mathbf x \mapsto \map {\nabla f} {\mathbf x} }$
where:
:$\mathbf F =  \set {f \in \sqbrk {S \to \R}: \nabla f \text{ is defined} }$
That is:
{{begin-eqn}}
{{eqn | l = \nabla f
      | r = \begin {bmatrix} \frac {\partial f} {\partial x_1} \\ \frac {\partial f} {\partial x_2} \\ \vdots \\ \frac {\partial f} {\partial x_n} \end {bmatrix}
}}
{{end-eqn}}
{{proofread}}
\end{definition}