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| {{Calculus}}
| | Myrtle Benny is how I'm called and I really feel comfy when people use the complete name. To do aerobics is a factor that I'm totally addicted to. My working day job is a librarian. Years in the past we moved to North Dakota.<br><br>My blog; at home std test - [http://musical.sehan.ac.kr/?document_srl=2110508 Related Web Page] - |
| In [[mathematics]], a '''time dependent vector field''' is a construction in [[vector calculus]] which generalizes the concept of [[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 (geometric)|vector]] to every point in a [[Euclidean space]] or in a [[manifold]].
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| ==Definition==
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| A '''time dependent vector field''' on a manifold ''M'' is a map from an open subset <math>\Omega \subset \Bbb{R} \times M</math> on <math>TM</math>
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| :<math>X: \Omega \subset \Bbb{R} \times M \longrightarrow TM</math>
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| ::::<math>(t,x) \longmapsto X(t,x)=X_t(x) \in T_xM</math>
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| such that for every <math>(t,x) \in \Omega</math>, <math>X_t(x)</math> is an element of <math>T_xM</math>.
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| For every <math>t \in \Bbb{R}</math> such that the set
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| :<math>\Omega_t=\{x \in M | (t,x) \in \Omega \} \subset M</math>
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| 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>.
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| ==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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| :<math>\frac{dx}{dt}=X(t,x)</math>
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| which is called [[Autonomous system (mathematics)|nonautonomous]] by definition.
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| ==Integral curve==
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| An [[integral curve]] of the equation above (also called an integral curve of ''X'') is a map
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| :<math>\alpha : I \subset \Bbb{R} \longrightarrow M</math>
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| 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
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| :<math>\frac{d \alpha}{dt} \left.{\!\!\frac{}{}}\right|_{t=t_0} =X(t_0,\alpha (t_0))</math>.
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| ==Relationship with vector fields in the usual sense==
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| A vector field in the usual sense can be thought of as a time dependent vector field defined on <math>\Bbb{R} \times M</math> even though its value on a point <math>(t,x)</math> does not depend on the component <math>t \in \Bbb{R}</math>.
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| Conversely, given a time dependent vector field ''X'' defined on <math>\Omega \subset \Bbb{R} \times M</math>, we can associate to it a vector field in the usual sense <math>\tilde{X}</math> on <math>\Omega</math> such that the autonomous differential equation associated to <math>\tilde{X}</math> is essentially equivalent to the nonautonomous differential equation associated to ''X''. It suffices to impose:
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| :<math>\tilde{X}(t,x)=(1,X(t,x))</math>
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| for each <math>(t,x) \in \Omega</math>, where we identify <math>T_{(t,x)}(\Bbb{R}\times M)</math> with <math>\Bbb{R}\times T_x M</math>. We can also write it as:
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| :<math> \tilde{X}=\frac{\partial{}}{\partial{t}}+X</math>.
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| To each integral curve of ''X'', we can associate one integral curve of <math>\tilde{X}</math>, and vice versa.
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| ==Flow==
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| The [[Flow (mathematics)|flow]] of a time dependent vector field ''X'', is the unique differentiable map
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| :<math>F:D(X) \subset \Bbb{R} \times \Omega \longrightarrow M</math> | |
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| such that for every <math>(t_0,x) \in \Omega</math>,
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| :<math>t \longrightarrow F(t,t_0,x)</math>
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| is the integral curve of ''X'' <math>\alpha</math> that verifies <math>\alpha (t_0) = x</math>.
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| ===Properties===
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| We define <math>F_{t,s}</math> as <math>F_{t,s}(p)=F(t,s,p)</math>
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| #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>
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| #<math>\forall t,s</math>, <math>F_{t,s}</math> is a [[diffeomorphism]] with [[Inverse function|inverse]] <math>F_{s,t}</math>.
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| ==Applications==
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| Let ''X'' and ''Y'' be smooth time dependent vector fields and <math>F</math> the flow of ''X''. The following identity can be proved:
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| :<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>
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| 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:
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| :<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>
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| This last identity is useful to prove the [[Darboux theorem]].
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| ==References==
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| * Lee, John M., ''Introduction to Smooth Manifolds'', Springer-Verlag, New York (2003) ISBN 0-387-95495-3. Graduate-level textbook on smooth manifolds.
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| [[Category:Differential geometry]]
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| [[Category:Vector calculus]]
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Myrtle Benny is how I'm called and I really feel comfy when people use the complete name. To do aerobics is a factor that I'm totally addicted to. My working day job is a librarian. Years in the past we moved to North Dakota.
My blog; at home std test - Related Web Page -