Given a smooth manifold with an open cover , a partition of unity subject to the cover is a collection of smooth, nonnegative functions , such that the support of is contained in and everywhere. Often one requires that the have compact closure, which can be interpreted as finite, or bounded, open sets. In the case that the is a locally finite cover, any point has only finitely many with .
A partition of unity can be used to patch together objects defined locally. For instance, there always exist smooth global vector fields, possibly vanishing somewhere, but not identically zero. Cover with coordinate charts such that only finitely many overlap at any point. On each coordinate chart , there are the local vector fields . Label these and, for each chart, pick the vector field . Then is a global vector field. The sum converges because at any , only finitely many .
Other applications require the objects to be interpreted as functions, or a generalization of functions called bundle sections, such as a Riemannian metric. By viewing such a metric as a section of a bundle, it is easy to show the existence of a smooth metric on any smooth manifold. The proof uses a partition of unity and is similar to the one used above.
Strictly speaking, the sum doesn't have to be identically unity for the arguments to work. It goes with the name, because at every point the functions partition the value 1. Also, it is convenient when considered from the point of view of convexity.