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Given a vector field $X$ of the form $f_1(x_1,x_2) \partial/ \partial_{x_1} + f_2(x_1,x_2) \partial/ \partial_{x_2}$ on a smooth 2-dimensional manifold $M \subset \mathbb{R}^2$ with a fixed point at the origin. Let $N \subset M$ be a smooth 1-dimensional submanifold. Suppose that $N$ intersects the $x_1$-axes perpendicularly at the origin. Can I then obtain any nice information on the local normal form of $X|_N$ at the origin?

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Is $M$ a 2-dimensional manifold? If so, it's just an open subset of $\mathbb R^2$. I also don't understand how $X$ and $N$ are related and what the axes have to do with all this. –  user31373 Jun 27 '12 at 22:56
Thanks for the comment, I just edited it. –  Novo Jun 27 '12 at 23:48
What do you mean by "local normal form"? –  Jason DeVito Jun 28 '12 at 2:20
Ah yes, I of course need to add a fixed point. This should be the final edit. –  Novo Jun 28 '12 at 8:26

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