I'm trying to prove that "Given a unit vector field $V$, it can always be uniquely determined a differentiable function $f$ that satisfies $\nabla f = V$."
To provide you more information, the unit vector field $V$ is actually determined from $V=\frac{\nabla \phi}{\| \nabla \phi \|}$, where $\phi$ is a level-set function. That is, I want to prove the existence and uniqueness of a function $f$ satisfying $\nabla f = \frac{\nabla \phi}{\| \nabla \phi \|}$.
Can any of you give me a proof or at least some advice? Any kind of help will be greatly appreciated.
Thank you all in advance. :)