##  [Partition of Unity](/partition-unity) 

  ##  [Partition of Unity](https://natural.quantumdictionary.io/partition-unity-0) 

  

 [![Natural & Formal Sciences Dictionary](/sites/default/files/styles/large/public/2026-01/Natural%20%26%20Formal%20Sciences.png.webp?itok=2kCDRVQv)](/topic-specific-dictionaries/natural-formal-sciences)



**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A locally finite collection of nonnegative smooth functions on a topological manifold, subordinate to an open cover, whose pointwise sum equals one; used to glue local constructions into global objects.

 

 

 

 

 





 

 



 ##  [Partition of Unity](https://mathlogic.quantumdictionary.io/partition-unity-1) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A family {φ_i} of continuous (often smooth) functions on a topological space such that for each point only finitely many φ_i are nonzero (local finiteness), each φ_i has support contained in a member of a given open cover, and the sum over i of φ_i equals the constant function 1 everywhere on the space.

 

 

 

 

 





 

 



 ##  [Partition of Unity](https://puremath.quantumdictionary.io/partition-unity-2) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A locally finite collection of nonnegative smooth functions on a topological manifold (or domain) subordinate to an open cover whose pointwise sum is identically one; used to localize global constructions by weighting and gluing local data.