Definition:Gradient Operator/Real Cartesian Space

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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 $\mathbf u = u_1 \mathbf e_1 + u_2 \mathbf e_2 + \cdots + u_n \mathbf e_n = \displaystyle \sum_{k \mathop = 1}^n u_n \mathbf e_n$ 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:

\(\displaystyle \operatorname {grad} f\) \(:=\) \(\displaystyle \nabla f\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} }\right) f \left({\mathbf u}\right)\) $\quad$ Definition of Del Operator $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \dfrac {\partial f \left({\mathbf u}\right)} {\partial x_k} \mathbf e_k\) $\quad$ $\quad$


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

\(\displaystyle \operatorname {grad} f\) \(:=\) \(\displaystyle \nabla f\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} }\right) f\) $\quad$ Definition of Del Operator $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\partial f} {\partial x} \mathbf i + \dfrac {\partial f} {\partial y} \mathbf j + \dfrac {\partial f} {\partial z} \mathbf k\) $\quad$ $\quad$

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:

\(\displaystyle \nabla f\) \(=\) \(\displaystyle \begin{bmatrix} \frac{\partial f}{\partial x_1} \\ \frac{\partial f}{\partial x_2} \\ \vdots \\ \frac{\partial f}{\partial x_n} \end{bmatrix}\) $\quad$ $\quad$


Also see

  • Results about gradient can be found here.


Sources