Definition:Gradient Operator/Riemannian Manifold/Definition 2
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Definition
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 see
- Results about the gradient operator can be found here.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Basic Constructions on Riemannian Manifolds