What to follow is a reference from the text Differential forms by Victor Guillemin and Peter Haine
So clearly the theorems $2.2.4$ and $2.2.5$ follow directely by the Cauchy theorem for ordinary differential equation provided that the vector field if locally lipschitz or of class $C^r$ for $r\ge 1$ since any function of calss $C^1$ is locally lipschitz. However how prove the last theorem? In particular I know that if the vector field $\vec v$ is of class $C^r$ for $r\ge 1$ then the solution of the correspondent system of differential equations is of class $C^r$ too but unfortunately this did not help me to prove the theorem. So could someone idicates where I can find the proof of the last theorem, please? I point out I did NOT study lebesgue integration and measure theory so that I ask courteously to not use them, thanks.
MY PROOF ATTEMPT
So by the regularity theorem we know that the function $\gamma_p$ is of class $C^\infty$ so that if $A_p$ is an open neighborhood of any $p\in V$ contained in $V$ then the set $J:=\gamma^{-1}_p[A_p]$ is open an open interval containing $a\in I$ and so if $\pi$ is the projection of $V\times I$ onto $I$ the statement follows proving that
$$
h(q,t)=[\gamma_p\circ\pi](q,t)
$$
for any $(q,t)\in A_p\times J$ because the set $A_p\times J$ is open and the function $\gamma_p\circ\pi$ is of class $C^\infty$ being a composition of such functions. So how prove the last equality? Is effectively it hold?
Best Answer
It is possible to find it at the $12$-th chapter of the text Introduction to Smooth Manifolds written by John M. Lee