Definition:Gradient Operator

Definition
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:

where:
 * $\star_g$ is the Hodge star operator of $\struct {M, g}$
 * $\d_{\d R}$ is de Rham differential.

Real Cartesian Space
The usual context in which the gradient operator is encountered is real Cartesian space:

Also known as
The gradient of $U$ is usually vocalised grad $U$.