Given a pair of fields $X,Y$ in $\mathbb R^2$, how can one deduce that exists a system of coordinates $(u,v)$ defined in a open subset of $\mathbb R^2$ such that $$X = \frac{\partial}{\partial u} \text{ and }Y = \frac{\partial}{\partial v}? \quad (I)$$
I'm not familiar with this kind of argument. Take of example the following fields:
- $X = y \frac{\partial}{\partial z} + \frac{\partial}{\partial y}$ and $Y = \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$.
- $X = \frac{\partial}{\partial z}$ and $Y = \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$.
I don't know if the Lie brackets make any difference to conclude, but in the first case $[X,Y] = -x \frac{\partial}{\partial z}$ and in the second one the corresponding Lie Bracket is zero. How can I see if there exists such a system satisfying $(I)$?
Hint Suppose we have coordinates $(u, v)$ in which the coordinate representations of $X, Y$ are $X = \partial_u$ and $Y = \partial_v$; what is $[X, Y]$?