Smoooth vector field

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This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds


A smooth vector field on a differential manifold can be defined in any of the following equivalent ways:

  • It is a derivation from the algebra of C^\infty functions on the manifold, to itself
  • It is a section of the tangent bundle, which is a smooth map
  • it is a rule that associates (smoothly) to every point in the manifold a tangent vector

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