Definition:Gradient Operator/Real Cartesian Space/Region

Definition

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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:

\(\ds \nabla f\)

\(=\)

\(\ds \begin {bmatrix} \frac {\partial f} {\partial x_1} \\ \frac {\partial f} {\partial x_2} \\ \vdots \\ \frac {\partial f} {\partial x_n} \end {bmatrix}\)

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Sources

• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 13.6$

• For a video presentation of the contents of this page, visit the Khan Academy.