Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions ψ_i, such that the support of ψ_i is contained in U_i and sum_i ψ_i = 1 everywhere. Often one requires that the U_i have compact closure, which can be interpreted as finite, or bounded, open sets. In the case that the U_i is a locally finite cover, any point x element M has only finitely many i with ψ_i(x)!=0.

convex set | open cover | Riemannian metric | section | smooth manifold | vector field