## 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

$\nabla$ Josiah Willard Gibbs and Edwin Bidwell Wilson
$\grad$ Hendrik Antoon Lorentz