I'm trying to apply the Poincare-Hopf theorem for a vector field over a closed disk. The vector fields sometimes have zeros on the boundary (if number of zeros is infinite, then it's zero over the whole boundary). Furthermore, the fields are undefined at the center, so I guess it's a vector field over a closed disk with a hole. In all other ways the vector field satisfies the conditions for the Poincare-Hopf theorem.
In a different scenario, the vector field is defined over the whole 2D plane except the origin but is not differentiable at the boundary of the unit disk. The field tends to zero at infinity.
How do I apply the Poincare-Hopf Theorem (or an extension of it) in these two scenarios?
What if I take away the condition that the vector field is perpendicular to the boundary for scenario 1?
The Poincaré-Hopf theorem does not hold for either manifolds with boundary or for noncompact manifolds; it is specific to compact manifolds without boundary.
As an example of the first situation, you can write down nonvanishing vector fields on $[0, 1]$ despite the fact that it has Euler characteristic $1$. As an example of the second situation, you can write down nonvanishing vector fields on $\mathbb{R}^2$ despite the fact that it has Euler characteristic $1$. Both of these examples also have compactly supported Euler characteristic $1$, so switching to that doesn't help either.