Definition:Gradient Operator/Riemannian Manifold/Definition 2

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Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.

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

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