# Definition:Gradient Operator/Riemannian Manifold

## Definition

### 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
$\grad$ Hendrik Antoon Lorentz