Rectification of Vector fields I am trying to understand how to rectify vector fields on manifolds. Namely if there is a manifold $M$ and a given vector field $X_p$ where for a $p_0$ we have $X(p_0)\neq0$ then there exists a neighborhood of $p_0$ and some other coordinate systme $y$ where $X=\frac{\partial}{\partial y_1}$. I would like some refference to study this phenomenon, mainly excersices or examples. I have read the proof in Spivak's book but I have a hard time solving actual excersises. Also I found Arnold's book on Ordinary Differential Equations lacking of any worked examples (although I may be wrong here). For example, I would appreciate if someone could give me some on direction on rectifying the following $\mathbb{R^2}$ vector field, given on cartesian coordinates, around the point $p=(1,2)$: $$V=x_1\frac{\partial}{\partial x_1} -2x_2\frac{\partial}{\partial x_2}.$$ I would be perticularly grateful if you could illuminate the procedure using flows.

I) Generally stratification/rectification of a non-vanishing vector field is guaranteed locally by the local existence theorem for ODEs.

II) In OP's example, it is easiest to stratify/rectify the vector field $V$ in steps:

1. Choose first $$(y_1,y_2)~=~(\ln |x_1|,\ln|x_2| ).$$ Then the vector field becomes $$V ~=~ \frac{\partial}{\partial y_1} - 2\frac{\partial}{\partial y_2}$$ by the chain rule.

2. Next choose $$(y_1,y_2)~=~(z_1+f(z_2),-2z_1+g(z_2)),$$ so that the Jacobian is non-vanishing. Then the vector field $$V ~=~ \frac{\partial}{\partial z_1}$$ is stratified.