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.
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.
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
Relationship with vector fields in the usual sense
A vector field in the usual sense can be thought of as a time dependent vector field defined on even though its value on a point does not depend on the component .
Conversely, given a time dependent vector field X defined on , we can associate to it a vector field in the usual sense on such that the autonomous differential equation associated to is essentially equivalent to the nonautonomous differential equation associated to X. It suffices to impose:
for each , where we identify with . We can also write it as:
To each integral curve of X, we can associate one integral curve of , and vice versa.
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 verifies .
We define as
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.
- Lee, John M., Introduction to Smooth Manifolds, Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.