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{{Calculus}}
In [[Mathematics|mathematics]], a '''time dependent vector field''' is a construction in [[Vector calculus|vector calculus]] which generalizes the concept of [[Vector field|vector field]]s. 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|manifold]].

==Definition==
A '''time dependent vector field''' on a manifold ''M'' is a map from an open subset <math>\Omega \subset \mathbb{R} \times M</math> on <math>TM</math>

:<math>\begin{align}
X: \Omega \subset \mathbb{R} \times M &\longrightarrow TM \\
(t,x) &\longmapsto X(t,x) = X_t(x) \in T_xM
\end{align}</math>

such that for every <math>(t,x) \in \Omega</math>, <math>X_t(x)</math> is an element of <math>T_xM</math>.

For every <math>t \in \mathbb{R}</math> such that the set

:<math>\Omega_t=\{x \in M \mid (t,x) \in \Omega \} \subset M</math>

is nonempty, <math>X_t</math> is a vector field in the usual sense defined on the open set <math>\Omega_t \subset M</math>.

==Associated differential equation==
Given a time dependent vector field ''X'' on a manifold ''M'', we can associate to it the following [[Differential equation|differential equation]]:

:<math>\frac{dx}{dt}=X(t,x)</math>

which is called [[Autonomous system (mathematics)|nonautonomous]] by definition.

==Integral curve==

An [[Integral curve|integral curve]] of the equation above (also called an integral curve of ''X'') is a map

:<math>\alpha : I \subset \mathbb{R} \longrightarrow M</math>

such that <math>\forall t_0 \in I</math>, <math>(t_0,\alpha (t_0))</math> is an element of the domain of definition of ''X'' and

:<math>\frac{d \alpha}{dt} \left.{\!\!\frac{}{}}\right|_{t=t_0} =X(t_0,\alpha (t_0))</math>.

==Equivalence with time-independent vector fields==
A time dependent vector field <math>X</math> on <math>M</math> can be thought of as a vector field <math>\tilde{X}</math> on <math>\mathbb{R} \times M,</math> where <math>\tilde{X}(t,p) \in T_{(t,p)}(\mathbb{R} \times M)</math> does not depend on <math> t. </math>

Conversely, associated with a time-dependent vector field <math>X</math> on <math>M</math> is a time-independent one <math>\tilde{X}</math>

:<math>\mathbb{R} \times M \ni (t,p) \mapsto \dfrac{\partial}{\partial t}\Biggl|_t + X(p) \in T_{(t,p)}(\mathbb{R} \times M)</math>

on <math>\mathbb{R} \times M.</math> In coordinates,

:<math>\tilde{X}(t,x)=(1,X(t,x)).</math>

The system of autonomous differential equations for <math>\tilde{X}</math> is equivalent to that of non-autonomous ones for <math>X,</math> and <math>x_t \leftrightarrow (t,x_t)</math> is a bijection between the sets of integral curves of <math>X</math> and <math>\tilde{X},</math> respectively.

==Flow==
The [[Flow (mathematics)|flow]] of a time dependent vector field ''X'', is the unique differentiable map

:<math>F:D(X) \subset \mathbb{R} \times \Omega \longrightarrow M</math>

such that for every <math>(t_0,x) \in \Omega</math>,

:<math>t \longrightarrow F(t,t_0,x)</math>

is the integral curve <math>\alpha</math> of ''X'' that satisfies <math>\alpha (t_0) = x</math>.

===Properties===
We define <math>F_{t,s}</math> as <math>F_{t,s}(p)=F(t,s,p)</math>

#If <math>(t_1,t_0,p) \in D(X)</math> and <math>(t_2,t_1,F_{t_1,t_0}(p)) \in D(X)</math> then <math>F_{t_2,t_1} \circ F_{t_1,t_0}(p)=F_{t_2,t_0}(p)</math>
#<math>\forall t,s</math>, <math>F_{t,s}</math> is a [[Diffeomorphism|diffeomorphism]] with [[Inverse function|inverse]] <math>F_{s,t}</math>.

==Applications==
Let ''X'' and ''Y'' be smooth time dependent vector fields and <math>F</math> the flow of ''X''. The following identity can be proved:

:<math>\frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} (F^*_{t,t_0} Y_t)_p = \left( F^*_{t_1,t_0} \left( [X_{t_1},Y_{t_1}] + \frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} Y_t \right) \right)_p</math>

Also, we can define time dependent tensor fields in an analogous way, and prove this similar identity,  assuming that <math>\eta</math> is a smooth time dependent tensor field:

:<math>\frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} (F^*_{t,t_0} \eta_t)_p = \left( F^*_{t_1,t_0} \left( \mathcal{L}_{X_{t_1}}\eta_{t_1} + \frac{d}{dt} \left .{\!\!\frac{}{}}\right|_{t=t_1} \eta_t \right) \right)_p</math>

This last identity is useful to prove the Darboux theorem.

==References==
* Lee, John M., ''Introduction to Smooth Manifolds'', Springer-Verlag, New York (2003) {{isbn|0-387-95495-3}}. Graduate-level textbook on smooth manifolds.

[[Category:Differential geometry]]
[[Category:Vector calculus]]

{{Sourceattribution|Time dependent vector field}}