Definition:Gradient Operator/Real Cartesian Space/Region

Definition
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: