Definition:Gradient Operator/Riemannian Manifold/Definition 1

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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.