Time dependent vector field

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In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.



A time dependent vector field on a manifold M is a map from an open subset on

such that for every , is an element of .

For every such that the set

is nonempty, is a vector field in the usual sense defined on the open set .

Associated differential equation

Given a time dependent vector field X on a manifold M, we can associate to it the following differential equation:

which is called nonautonomous by definition.

Integral curve

An integral curve of the equation above (also called an integral curve of X) is a map

such that , is an element of the domain of definition of X and


Equivalence with time-independent vector fields

A time dependent vector field on can be thought of as a vector field on where does not depend on

Conversely, associated with a time-dependent vector field on is a time-independent one

on In coordinates,

The system of autonomous differential equations for is equivalent to that of non-autonomous ones for and is a bijection between the sets of integral curves of and respectively.


The flow of a time dependent vector field X, is the unique differentiable map

such that for every ,

is the integral curve of X that satisfies .


We define as

  1. If and then
  2. , is a diffeomorphism with inverse .


Let X and Y be smooth time dependent vector fields and the flow of X. The following identity can be proved:

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity, assuming that is a smooth time dependent tensor field:

This last identity is useful to prove the Darboux theorem.

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