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

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


Cartesian $3$-Space

In $3$ dimensions this is usually rendered as follows:


Let $R$ be a region of Cartesian $3$-space $\R^3$.

Let $\map F {x, y, z}$ be a scalar field acting over $R$.

Let $\tuple {i, j, k}$ be the standard ordered basis on $\R^3$.


The gradient of $F$ is defined as:


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


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 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:

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


Riemannian Manifold

Definition 1

Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.

Let $f \in \map {\CC^\infty} M$ be a smooth mapping on $M$.


The gradient of $f$ is defined as:

\(\ds \grad f\) \(:=\) \(\ds \nabla f\)
\(\ds \) \(=\) \(\ds g^{-1} \d_{\d R} f\)

where $\d_{\d R}$ is de Rham differential.


Definition 2

Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.

Let $f \in \map {C^\infty} M : M \to \R$ be a smooth mapping on $M$.


The gradient of $f$ is the vector field obtained from the differential $\rd f$ obtained by raising an index:

$\grad f := \paren {\rd f}^\sharp$


Also see

  • Results about the gradient operator can be found here.


Sources

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