Smoooth vector field

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

Definition

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 $C^{\mathrm{\infty }}$ 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

Related notions

• Continuous vector field
• Tensor field
• Differential form
• Tensor of mixed type