Definition:Gradient Operator/Real Cartesian Space

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Definition

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

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

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ 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:

\(\displaystyle \grad f\) \(:=\) \(\displaystyle \nabla f\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf u}\) Definition of Del Operator
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k\)


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

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

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


On a Region

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 operator 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:

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


Also see

  • Results about gradient can be found here.


Sources