We give an easy extension of the Poincaré Index Formula from the disc to any surface with boundary. In particular we show that the sum of the indices of the vector field at the critical points depends only on the Euler characteristic of the surface and on the behaviour of its trajectories in the boundary. Our theorem improves previous results on the same formula.
"A Poincaré index formula for surfaces with boundary." Differential Integral Equations 11 (1) 191 - 199, 1998. https://doi.org/10.57262/die/1367414143