I have trouble understanding this problem of Arnold's ODE book. It goes as follows:
Prove that a diffeomorphism taking the vector field $v$ to the vector field $w$ takes the phase curves of the field $v$ to the phase curves of the field $w$. Is the converse true?
I'am confused about what type of diffeomorphism is supposed to be taken. Either a diffeomorphism between vector fields (for which I don't know if it is possible to define such mapping) or the diffeomorphism between the domains of the vector fields. In the latter case I think the answer is yes, in fact it is a corollarly of the theorem proved just below that problem. But he gives a negative answer by giving two vector fields, namely $v=x\frac{\partial}{\partial x}$ and $w=2x\frac{\partial}{\partial x}$. Hope you can help me to understand this.