partition of unity (original) (raw)

A partition of unity is locally finite if each x in X is contained in an open set on which only a finite number of εi are non-zero. That is, if the cover {εi-1⁢((0,1])} is locally finite.

A partition of unity is subordinate to an open cover {Ui} of X if each εi is zero on the complement of Ui.

Example 1 (Circle)

A partition of unity for S1 is given by{sin2⁡(θ/2),cos2⁡(θ/2)}subordinate to the covering{(0,2⁢π),(-π,π)}.

Application to integration

Let M be an orientable manifold with volume form ωand a partition of unity {εi⁢(x)}. Then, the integral of a function f⁢(x) over M is given by

It is of the choice of partition of unity.