Definition:Gradient Operator/Real Cartesian Space

At a Point
Let:


 * $f: \R^n \to \R, \mathbf x \mapsto f\left({\mathbf x}\right)$ be a real-valued function where:


 * $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$

is a vector in $\R^n$.

Let the partial derivative of $f$ with respect to $x_i$ exist for all $x_i$.

The gradient of $f$ (at $\mathbf x$) is defined as the column matrix:

On a Region
Let $S \subseteq \R^n$.

Let $\left[{X \to Y}\right]$ be the space of functions from $X$ to $Y$.

Suppose that for all $\mathbf x \in S$, $\nabla f\left({\mathbf x}\right)$ exists.

The gradient can then be defined as an operator acting on $f$:


 * $\nabla: \mathbf F \to \left[{S \to \R^n}\right]$
 * $\left({f: \mathbf x \mapsto f\left({\mathbf x}\right)}\right) \mapsto \left({\nabla f: \mathbf x \mapsto \nabla f\left({\mathbf x}\right)}\right)$

where:


 * $\mathbf F = \left \{ f \in \left[{S \to \R}\right] : \nabla f \text{ is defined} \right\}$.

That is:

Also see

 * Definition:Directional Derivative