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.
