# Smoooth vector field

From Diffgeom

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 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