Part of a series of articles about |
Calculus |
---|
![{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766) |
- Rolle's theorem
- Mean value theorem
- Inverse function theorem
|
Differential Definitions |
---|
| Concepts |
---|
- Differentiation notation
- Second derivative
- Implicit differentiation
- Logarithmic differentiation
- Related rates
- Taylor's theorem
| Rules and identities |
---|
|
|
| Definitions |
---|
| Integration by |
---|
|
|
|
|
|
|
|
|
|
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
on
![{\displaystyle {\begin{aligned}X:\Omega \subset \mathbb {R} \times M&\longrightarrow TM\\(t,x)&\longmapsto X(t,x)=X_{t}(x)\in T_{x}M\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094f1639114a63d8424bfb4334f3aee060bc4545)
such that for every
,
is an element of
.
For every
such that the set
![{\displaystyle \Omega _{t}=\{x\in M\mid (t,x)\in \Omega \}\subset M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b184d39d2d297d704ba72a8ee4b39c95a4231d45)
is nonempty,
is a vector field in the usual sense defined on the open set
.
Associated differential equation
Given a time dependent vector field X on a manifold M, we can associate to it the following differential equation:
![{\displaystyle {\frac {dx}{dt}}=X(t,x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e58676c199faf97f92a2d0b29f30c7c9ecf5fe)
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
![{\displaystyle \alpha :I\subset \mathbb {R} \longrightarrow M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8dcd9fa9ed69993b028c695c57c25fa57e416f8)
such that
,
is an element of the domain of definition of X and
.
Equivalence with time-independent vector fields
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
![{\displaystyle \mathbb {R} \times M\ni (t,p)\mapsto {\dfrac {\partial }{\partial t}}{\Biggl |}_{t}+X(p)\in T_{(t,p)}(\mathbb {R} \times M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f082d52bcb5349753d2893761d498981300e16b)
on
In coordinates,
![{\displaystyle {\tilde {X}}(t,x)=(1,X(t,x)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc0473d05c08810716bb1a7f268f9a88986b39d7)
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.
Flow
The flow of a time dependent vector field X, is the unique differentiable map
![{\displaystyle F:D(X)\subset \mathbb {R} \times \Omega \longrightarrow M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/964d706186ed35afe5b9d606a91658be6b890534)
such that for every
,
![{\displaystyle t\longrightarrow F(t,t_{0},x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e865fc4f30fa7c7356c19d788b614ab206044fb5)
is the integral curve
of X that satisfies
.
Properties
We define
as
- If
and
then ![{\displaystyle F_{t_{2},t_{1}}\circ F_{t_{1},t_{0}}(p)=F_{t_{2},t_{0}}(p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd1d7d694c55f2a191d74db5de4aae586b5c30f)
,
is a diffeomorphism with inverse
.
Applications
Let X and Y be smooth time dependent vector fields and
the flow of X. The following identity can be proved:
![{\displaystyle {\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc669b795d3f438a4bb215cb104f523dbc0191de)
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:
![{\displaystyle {\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f32ca46ed2579e62a78898411db0f34f7c54d145)
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.