Definition:Gradient Operator

Definition

Geometrical Representation

Let $R$ be a region of space.

Let $F$ be a scalar field acting over $R$.

The gradient of $F$ at a point $A$ in $R$ is defined as:

$\grad F = \dfrac {\partial F} {\partial n} \mathbf {\hat n}$

where:

$\mathbf {\hat n}$ denotes the unit normal to the equal surface $S$ of $F$ at $A$

$n$ is the magnitude of the normal vector to $S$ at $A$.

Real Cartesian Space

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

\(\ds \)

\(=\)

\(\ds \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k\)

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 known as

The gradient of a scalar field $U$ is usually vocalised grad $U$.

Also see

• Definition:Gradient

• Divergence Operator
• Curl Operator

• Definition:Hessian

• Results about the gradient operator can be found here.

Historical Note

During the course of development of vector analysis, various notations for the gradient operator were introduced, as follows:

Symbol	Used by
$\nabla$	Josiah Willard Gibbs and Edwin Bidwell Wilson Oliver Heaviside Max Abraham Vladimir Sergeyevitch Ignatowski
$\grad$	Hendrik Antoon Lorentz Cesare Burali-Forti and Roberto Marcolongo

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): gradient: 3.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): gradient: 3.