Definition:Gradient Operator

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

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


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 $f \in \map {\CC^\infty} M : M \to \R$ 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

  • Results about gradient 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