# First Integral of a Vector Field.

In do Carmo's Differential Geometry of Curves and Surfaces, In the Section about Vector Fields, first Lemma, he proves that for every differentiable vector field, there exists a function that is constant along the trajectories of the field and such that it's differential is not zero. He defines it as the inverse of the local flow restricted to the points with $x=0$ followed by the projection to the $y$-axis. I don't understand why this is the function that works. In particular I don't know how to prove that it is constant alongthe trajetories.

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It looks likes that this projection is unnecessary! –  Tomás Oct 24 '12 at 16:16

Note that $\tilde{\alpha}(0,y,t)$ is the point obtained by "walking" in the trajectory of $(0,y)$ an time $t$.
On the other hand, $\tilde{\alpha}^{-1}(x,y)$ are the points of the form $(0,y',t)$ for some $y'$ and some $t\in I$. By the first paragraph, if you take $(x,y)$ and $(x_1,y_1)$ in the same trajectory, you must have $\tilde{\alpha}^{-1}(x,y)=\tilde{\alpha}^{-1}(x_1,y_1)$, so the function $\tilde{\alpha}^{-1}(x,y)$ is contanst on trajectories.