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
is an element of
such that the set is nonempty,
is a vector field in the usual sense defined on the open set
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
, α (
is an element of the domain of definition of X and A time dependent vector field
can be thought of as a vector field
Conversely, associated with a time-dependent vector field
is a time-independent one
In coordinates, The system of autonomous differential equations for
is equivalent to that of non-autonomous ones for
is a bijection between the sets of integral curves of
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
We define
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
η
is a smooth time dependent tensor field: This last identity is useful to prove the Darboux theorem.