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I am trying to understand the following proof about how two vector fields determine a paramaterisation of a surface. What I do not understand is why the inverse of the constructed $\phi$ is the parametrisation we are looking for. I tried to differentiate the parameter curve $\phi^{-1}(t,p_{2})$, but don't see why the resulting vector is parallel/linear dependend to/of $w_{1}$.

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The proof is from DoCarmos book Differential Geometry of Curves and Surfaces on page 185.

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  • $\begingroup$ $\varphi^{-1}(t,p_2)$ gives points $q$ with $f_2(q)=p_2$, and by construction the level curves of $f_2$ are tangent to $w_1$. You have to go back and look at what the first integral of a vector field is. For an alternative proof, see pp. 119-120 of my differential geometry text. $\endgroup$ – Ted Shifrin Oct 12 '18 at 19:11

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