Assume that $F(t,x(t)) = a(t,x(t))\frac{dx}{dt} + b(t, x(t))$ which satisfies $\frac{dG}{dt} = F$ for some function G(t, x(t)). I want to show that : $\frac{\partial a}{\partial t} = \frac{\partial b}{\partial x}$. I want to use the chain rule and this properties : $\frac{\partial^2H}{\partial x\partial t} = \frac{\partial^2H}{\partial t\partial x}$.
I'm totally lost, if somebody has an idea or a clue I take it! Thank you in advance
In the given formulation, this is about properties of functions on curves, or only one curve. What is meant is probably that these identities hold for all curves. This is better stated as a property on tangential planes, using total differentials or general differential forms
In the end, after eliminating $F$ and writing everything in total differentials, you get $$ dG(t,x) = a(t,x)\,dx+b(t,x)\,dt. $$ This is an equation for the tangent plane of $G$, which identifies $a=\partial_xG$ and $b=\partial_t G$. This implies the stated identity as a consequence of the Schwarz lemma.