I'm having trouble trying to start this. Here is the problem statement:

Show that if $w : S \to \mathbb{R^3}$ is a differentiable vector field on a regular surface $S \subset \mathbb{R}^3$, and $w(p) \neq 0$ for some $p \in S$, then it is possible to parametrize a neihghborhood of $p$ by $x(u,v)$ in such a way that $x_u = w$.

How do I start this? What do I need to show to prove this?

  • 1
    $\begingroup$ It would be helpful to give more context. What, specifically, do you not understand in the question? Where are you stuck? $\endgroup$ – Joshua Mundinger Mar 23 at 14:49
  • $\begingroup$ @JoshuaMundinger I'm not sure what steps would be necessary in order to prove this. "it is possible to parametrize... in a way such that $x_u=w$ " makes me think that this is a question of existence, but if that is the case, I don't know how to show it exists. $\endgroup$ – JB071098 Mar 23 at 14:55
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    $\begingroup$ The main theorem of Section 3.4 in do Carmo's Differential Geometry of Curves and Surfaces is exactly a theorem of existence of some type. What is the difference between the text's results and your problem? This may help you get started so that we can give more specific help. $\endgroup$ – Joshua Mundinger Mar 24 at 1:36
  • $\begingroup$ Is there any progression? $\endgroup$ – izimath May 7 at 2:37
  • $\begingroup$ @izimath Still cannot formulate a formal proof. $\endgroup$ – JB071098 May 20 at 20:38

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