Fundamental theorem on vector fields. Flow of a vector field

There is a better approach in @lee2013smooth page 314.
(I have to fix the notation)
What follows is from @warner prop 1.48:

Proposition
Let $X$ be a vector field on a manifold $M$. For each $p\in M$, there exist $a(p),b(p)\in \mathbb{R}\cup \{\pm \mathrm{\infty }\}$ and a differentiable curve

such that:

1. $0\in (a(p),b(p))$ and ${\gamma }_p(0)=p$
2. ${\gamma }_p$ is an integral curve of $X$ (i.e. $d({\gamma }_p{)}_s(\frac{d}{dt})=X({\gamma }_p(s))$)
3. Uniqueness: If $\mu :(c,d)⟶M$ is another curve satisfying the above conditions, then $(c,d)\subseteq (a(p),b(p))$ and ${\gamma }_p{|}_{(c,d)}=\mu$.
4. For each $t\in \mathbb{R}$, we define a transformation $X_t$ with domain $D_t=\{q\in M:t\in (a(q),b(q))\}$ by the expression $X_t(q)={\gamma }_q(t)$, so that for each $p\in M$, there exists an open neighborhood $V$ and an $ϵ>0$ such that the map

is defined on $(ϵ,ϵ)\times V$. This transformation is called the flow of $X$ and is usually denoted by ${\varphi }_X(t,q).$
5. $D_t$ is open for each $t$.
6. $\bigcup _{t>0}{D}_t=M$.
7. $X_t:D_t⟶D_{-t}$ is a diffeomorphism with inverse $X_{-t}$.
8. Let $s,t\in \mathbb{R}$. The domain of $X_s\circ X_t$ is contained in $D_{t+s}$ (but in general, it is not equal to $D_{t+s}$). It is equal to $D_{t+s}$ if $s$ and $t$ have the same sign. Moreover, in the domain of $X_s\circ X_t$, we have $X_s\circ X_t=X_{t+s}$. Therefore, we have a local group of transformations.

Important facts

Flow of the sum

See flow of the sum of two vector fields

Flow of the Lie bracket

See Lie bracket#Useful formula.

Re-scaling of the flow

Proposition
Let $X$ be a vector field, and $k$ a constant. Let's denote ${\varphi }_X$ and ${\varphi }_{kX}$the flow of $X$ and $kX$ respectively. It is satisfied that:

Proof
We only have to check that $\hat{\varphi }(t,q)={\varphi }_X(kt,q)$ satisfies 1. and 2. for the vector field $kX$, and then we will have the result. In fact:

1. $\hat{\varphi }(0,q)={\varphi }_X(0,q)=q$.
2. $\hat{\varphi }(t,q)$ are integral curves of $kX$:

$◼$

Jacobian of the flow of a vector field

Proposition
Let $X$ be a vector field and ${\varphi }_t(p)$ its flow. For every $t$ the map ${\varphi }_t(-)$ is a diffeomorphism, so we can consider its Jacobian

It is satisfied that

where $\text{div}$ is the divergence.

Moreover

Proof
(See this answer in Mathstackexchange)
It is satisfied that

and then, taking derivatives with respect to the coordinates $x_1,\dots ,x_n$ we obtain the matrix equation

We can consider the linear system given by

and then $d({\varphi }_t{)}_x$ is a matrix whose columns are solutions to it. We now apply Liouville's formula to this linear system, obtaining

and deriving with respect to $t$:

$◼$

Geometric intuition: population growth with varying rate. The ODE $\frac{d}{dt}J=J\cdot \mathrm{div}X({\varphi }_t(x))$ has exactly the structure of a population growth equation $\frac{dP}{dt}=r(t)\cdot P$, but with a non-constant per-capita rate $r(t)=\mathrm{div}X({\varphi }_t(x))$ that changes at each instant because the orbit ${\varphi }_t(x)$ moves through regions of space where $\mathrm{div}X$ takes different values. Think of a population living in a landscape where fertility depends on location: as the population migrates along the orbit, the birth rate changes according to the terrain it currently occupies. The solution is $J(t,x)=\exp ({\int }_0^t\mathrm{div}X({\varphi }_s(x))ds)$, where the exponent is the Malthusian parameter accumulated along the trajectory — the time-integral of the local rate, not the net new volume.