Could you please help me solve it? Let X and Y be smooth vector fields on $\mathbb{R}^n$. Suppose that $[X,Y]=0$ (Lie bracket). Show that around each point there exists local diffeomorphism f such that $f_*(X)=∂/∂x$ and $f_*(Y)= ∂/∂y$.
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1Have you looked at John M. Lee's book? (http://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/0387954481) – voldemort Jan 21 '14 at 06:27
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go with the flow. – James S. Cook Jan 21 '14 at 06:48
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2This is the Frobenius theorem and you can check it on wiki. – gaoxinge Jan 22 '14 at 08:24