Definition:Gradient Operator/Real Cartesian Space/Region

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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:

 $\ds \nabla f$ $=$ $\ds \begin {bmatrix} \frac {\partial f} {\partial x_1} \\ \frac {\partial f} {\partial x_2} \\ \vdots \\ \frac {\partial f} {\partial x_n} \end {bmatrix}$