2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216537The classical Schläfli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas which are vector-valued rather than scalar. Some consequences follow, for instance constraints on where cone singularities can appear when a constant curvature manifold is deformed among cone-manifolds.26 pages, no figure. v2: added 2 refs, improved intro. v3: added section on low dim examplesDifferential GeometryGeometric Topologytext