2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78479Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector fields on $M$ with coefficients in the space of linear differential operators on $\cal S$. This cohomology space is closely related to the $Vect(M)$-modules, ${\cal D}_λ(M)$, of linear differential operators on the space of tensor densities on $M$ of degree $λ$.14 pages, LaTeXDifferential Geometrytext