A time dependent vector field on a manifold M is a map from an open subset on
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such that for every , is an element of .
For every such that the set
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is nonempty, is a vector field in the usual sense defined on the open set .
Associated differential equation
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Given a time dependent vector field X on a manifold M, we can associate to it the following differential equation:
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which is called nonautonomous by definition.
An integral curve of the equation above (also called an integral curve of X) is a map
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such that , is an element of the domain of definition of X and
- .
Equivalence with time-independent vector fields
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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
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on In coordinates,
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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
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such that for every ,
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is the integral curve of X that satisfies .
We define as
- If and then
- , 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:
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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:
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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.