A partition of unity subordinate to an open cover 
of a smooth
manifold 
is a family of smooth functions 
that are nonnegative
and whose supports form a locally
finite family, such that the support of 
is contained in 
and 
everywhere. Often one requires that the 
have compact
closure, meaning that their closures are
compact. If the open cover 
is a locally
finite family, any point 
has only finitely many 
with 
.
A partition of unity can be used to patch together objects defined locally. For vector fields 
on coordinate charts 
, the locally finite sum 
, with each summand extended
by zero, defines a smooth
global vector field. This sum
may vanish identically through cancellation.
To obtain a global vector field which is not identically
zero on a positive-dimensional smooth manifold,
multiply a nonvanishing coordinate vector
field by a nonzero bump
function with compact support inside one coordinate chart, and extend it by zero.
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.
For these patching arguments, normalization to unity may be omitted, but the nonnegative weights must have a strictly positive pointwise sum. Dividing each weight by that sum gives a partition of unity. Normalization also expresses each patched value as a convex combination of local values.
