# partition of unity

## Primary tabs

Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

54D20 Noncompact covering properties (paracompact, LindelÃ¶f, etc.)
58A05 Differentiable manifolds, foundations

### partition of unity

Could you tell me why there always exists such functions to form partition of unity ?

### Re: partition of unity

It's quite simple. First, we note that the function

f(x) = exp (-1/x) |x| < 1
0 x >= 1

is infinitely differentiable and never negative. Likewise, in n dimensions, the function

g (x_1, ... x_n) = f ( sqrt{x_1^2 + ... sqrt x_n^2)

is infinitely differentiable and differs from zero only when the point lies in the unit ball. Now, if M is a manifold then, by the paracompactness property, we can cover M with a (possibly infinite) number of balls such that any point is contained in a finite number of these balls. To each ball B, we associate a function like g_B as above (which is stricltly positive inside the ball and zero outside). Sum these functions. Because of our paracompactness property, this sum converges trivially for any point p of M, and define h(p) to be the sum of g_B (p) ofer all balls B in the cover. Note that h(p) is strictly positive for all p. Thus g_B / h exists. The collection of all g_B / h's for all B is your partition of unity.

### Definition insufficient

Surely we can do better than a circular, one-word semantic definition here....try Kelley, (even Wikepedia!)....
To quote someone: 'incomplete truth is better than no information'.

Best wishes.

### Re: Definition insufficient

It looked like only one word because the article needed rerendering (which I've now done).

Aaron Krowne said some time back that this problem needs to be fixed urgently. I agree, but I don't think it's going to happen.

### Re: Definition insufficient

Thanks....................that looks much better!