Let $ M = \mathbb{S}^2 $ be a manifold with the coordinate patch $U$parameterized using spherical coordinates with $(\phi, \theta) \in (0,2\pi)\times (0,\pi)$. Let $Y = \frac{\partial}{\partial \phi}$ be vector fields over $U$. Show $Y$ cannot be extended continously to a vector field in $\chi(M)$.
I am aware of the idea that I have to show that if I consider a patch that covers the points $ M -{U}$ then the vector field $Y$ won't be defined at least on the intersection of the patches. However, I am unsure how to achieve this. Moreover, is it correct to suggest that the information about the spherical coordinate parametrization is redundant since they have already provided me the $Y$ vector field?