Editing Partition of unity (section)

==Applications==
A partition of unity can be used to define the integral (with respect to a [[volume form]]) of a function defined over a manifold: one first defines the integral of a function whose support is contained in a single coordinate patch of the manifold; then one uses a partition of unity to define the integral of an arbitrary function; finally one shows that the definition is independent of the chosen partition of unity.

A partition of unity can be used to show the existence of a [[Riemannian metric]] on an arbitrary manifold.

[[Method of steepest descent#The case of multiple non-degenerate saddle points|Method of steepest descent]] employs a partition of unity to construct asymptotics of integrals.

[[Linkwitz–Riley filter]] is an example of practical implementation of partition of unity to separate input signal into two output signals containing only high- or low-frequency components.

The [[Bernstein polynomial]]s of a fixed degree ''m'' are a family of ''m''+1 linearly independent single-variable polynomials that are a partition of unity for the unit interval <math>[0,1]</math>.

The weak [[Hilbert's Nullstellensatz|Hilbert Nullstellensatz]] asserts that if <math>f_1,\ldots, f_r\in \C[x_1,\ldots,x_n]</math> are polynomials with no common vanishing points in <math>\C^n</math>, then there are polynomials <math>a_1, \ldots, a_r</math> with <math>a_1f_1+\cdots+a_r f_r = 1</math>. That is, <math>\rho_i = a_i f_i</math> form a polynomial partition of unity subordinate to the [[Zariski topology|Zariski-open]] cover <math>U_i = \{x\in \C^n \mid f_i(x)\neq 0\}</math>. 

Partitions of unity are used to establish global smooth approximations for [[Sobolev space|Sobolev]] functions in bounded domains.<ref>{{Citation|last=Evans|first=Lawrence|chapter=Sobolev spaces|date=2010-03-02|pages=253–309|publisher=American Mathematical Society|isbn=9780821849743|doi=10.1090/gsm/019/05|title=Partial Differential Equations|volume=19|series=Graduate Studies in Mathematics}}</ref>