Smoooth vector field

Definition

A 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
• it is a rule that associates (smoothly) to every point in the manifold a tangent vector

Related notions

• Tensor field
• Differential form
• Tensor of mixed type