- Time dependent vector field
mathematics, a time dependent vector field is a construction in vector calculuswhich 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 spaceor 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
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
which is called nonautonomous by definition.
integral curveof the equation above (also called an integral curve of "X") is a map
such that , is an element of the
domain of definitionof "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 viceversa.
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
#If and then
#, is a
diffeomorphismwith 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
* Lee, John M., "Introduction to Topological Manifolds", Springer-Verlag, New York (2000), ISBN 0-387-98759-2. "Introduction to Smooth Manifolds", Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.
Wikimedia Foundation. 2010.