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