2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157116We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros.7 pages, no figures; errors corrected and a section on an alternative formulation addedGeometric TopologyAlgebraic Topology57R25; 55M25text