Let $X$ be a non-vanishing vector field on the unit square $I^2$ in $\mathbb{R}^2$. I would like to show that every integral curve to $X$ exits the unit square in finite time.
This fact is used in a paper I am reading, in which the author says "(assume there is an integral curve that does not exit the unit square,) it would approach asymptotically some simple closed curve in $I^2$. In the interior of this curve the vector field would have to have a singularity." This does seem reasonable, since integral curves cannot cross themselves (unless they are simply closed curves, but as the author has noted this would result in a contradiction in the interior of the closed curve) so they should have no place to go except wrapping around. However I cannot make this idea rigorous at all.
Any help is appreciated!
Poincare-Hopf Theorem. The turning number on the boundary of the disk is ...what?