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 formula_1 on formula_2
such that for every formula_5, formula_6 is an element of formula_7.
For every formula_8 such that the set
is nonempty, formula_10 is a vector field in the usual sense defined on the open set formula_11.
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 formula_14, formula_15 is an element of the domain of definition of "X" and
Equivalence with time-independent vector fields.
A time-independent vector field formula_17 on formula_18 can be thought of as a vector field formula_19 on formula_20 where formula_21 does not depend on formula_22
Conversely, associated with a time-dependent vector field formula_17 on formula_18 is a time-independent one formula_19
on formula_27 In coordinates,
The system of autonomous differential equations for formula_19 is equivalent to that of non-autonomous ones for formula_30 and formula_31 is a bijection between the sets of integral curves of formula_17 and formula_33 respectively.
Flow.
The flow of a time dependent vector field "X", is the unique differentiable map
such that for every formula_35,
is the integral curve formula_37 of "X" that satisfies formula_38.
Properties.
We define formula_39 as formula_40
Applications.
Let "X" and "Y" be smooth time dependent vector fields and formula_47 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 formula_49 is a smooth time dependent tensor field:
This last identity is useful to prove the Darboux theorem.