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:
Related: flow of a non-autonomous vector field

Given a vector field, we have a local group of transformations ${\varphi }_t$ associated to it. The existence is guaranteed by the following theorem:

Flow theorem

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

See Jacobian of the flow: first the Jacobian matrix $d({\varphi }_t{)}_x$, then the formula for its determinant.