Definition:Gradient Operator/Real Cartesian Space

Definition
Let $\R^n$ denote the real Cartesian space of $n$ dimensions.

Let $f \left({x_1, x_2, \ldots, x_n}\right)$ denote a real-valued function on $\R^n$.

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\R^n$.

Let $\mathbf u = u_1 \mathbf e_1 + u_2 \mathbf e_2 + \cdots + u_n \mathbf e_n = \displaystyle \sum_{k \mathop = 1}^n u_k \mathbf e_k$ be a vector in $\R^n$.

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

The gradient of $f$ (at $\mathbf u$) is defined as:

In $3$ dimensions with the standard ordered basis $\left({\mathbf i, \mathbf j, \mathbf k}\right)$, this is usually rendered:

for a vector $\mathbf u = x \mathbf i + y \mathbf j + z \mathbf k$.

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