Time dependent vector field


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.

Definition

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 Autonomous [system (mathematics)|nonautonomous] by definition.

Integral curve

An integral curve of the equation above 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.

Flow

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.

Properties

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

Applications

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.