# Definition:Gradient Operator/Riemannian Manifold

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

- $\grad f := \paren {\rd f}^\sharp$

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## 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 |