partition of unity
Let X be a topological space.
A partition of unity
is a collection
of continuous functions
{εi:X→[0,1]} such that
∑iεi(x)=1 | (1) |
A partition of unity is locally finite if each in is contained in an open set on which only a finite number of are non-zero.
That is, if the cover is locally finite.
A partition of unity is subordinate to an open cover of if each is zero on the complement of .
Example 1 (Circle)
A partition of unity for is given by subordinate to the covering .
Application to integration
Let be an orientable manifold with volume form
and a partition of unity .
Then, the integral of a function over is given by
It is of the choice of partition of unity.
Title | partition of unity |
---|---|
Canonical name | PartitionOfUnity |
Date of creation | 2013-03-22 13:29:23 |
Last modified on | 2013-03-22 13:29:23 |
Owner | mhale (572) |
Last modified by | mhale (572) |
Numerical id | 10 |
Author | mhale (572) |
Entry type | Definition |
Classification | msc 54D20 |
Classification | msc 58A05 |
Defines | locally finite partition of unity |
Defines | subordinate to an open cover |