nLab vector field

Idea

A vector field is a section of a vector bundle.

More specifically, a vector field is a tangent vector field which is a section of a tangent bundle. Hence this is a function which picks a tangent vector at each point of a manifold, such that this assignment is suitably differentiable.

Equivalently this is a rank $(1,0)$-tensor field on $X$

Properties

As derivations of smooth functions

Vector fields may be identified with derivations on the algebra of smooth functions. See the article derivations of smooth functions are vector fields.

Related concepts

• flow of a vector field

• pushforward of vector fields

• derivations of smooth functions are vector fields

• differential form

• frame field

• Euler vector field

• Hamiltonian vector field

• Reeb vector field

• evolutionary vector field

• field (physics)

References

For instance section 1.3 and 1.4 of

• Theodore Frankel, The Geometry of Physics - An Introduction