Definition:Gradient Operator/Riemannian Manifold/Definition 1
Jump to navigation
Jump to search
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:
\(\ds \grad f\) | \(:=\) | \(\ds \nabla f\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds g^{-1} \d_{\d R} f\) |
where $\d_{\d R}$ is de Rham differential.