Partitions of Unity

And, finally, one to go up today!

A partition of unity is a useful, though technical, tool that helps us work in local coordinates. This can be a tricky matter when we’re doing things all over our manifold, since it’s almost never the case that the entire manifold fits into a single coordinate patch. A (smooth) partition of unity is a way of breaking the function with the constant value $1$ up into a bunch of (smooth) pieces that will be easier to work with.

More specifically, a partition of unity is a collection of nonnegative smooth functions $\phi_\alpha:M\to\mathbb{R}$ indexed by some set $\alpha\in A$ , subject to two conditions. First: the collection of supports $\{\mathrm{supp}(\phi_\alpha)\}_{\alpha\in A}$ is a locally finite cover of $M$ , which takes a bit to unpack.

The support $\mathrm{supp}(f)$ of a real-valued (or vector-valued) function is the closure of the set on which it takes nonzero values. In other words, the complement of the support is the largest open set on which $f(p)=0$ .

To say that a collection of sets is a locally finite cover means that every point $p\in M$ is contained in at least one of them, and that $p$ has some neighborhood which intersects only finitely many of them. For instance, the collection of all intervals $[n-1,n+1]$ centered at integers $n$ is a locally finite cover of $\mathbb{R}$ . Every real number is within $1$ of some integer, and around each real number we can draw a small neighborhood that meets at most three of these intervals (why three?).

The other condition is that the sum

$\displaystyle\sum\limits_{\alpha\in A}\phi_\alpha=1$

That is, if we add up all these functions we get the function with constant value $1$ . But we made no restriction on the index set, so how do we know that this sum remotely makes sense? Because we evaluate it at each point

$\displaystyle\sum\limits_{\alpha\in A}\phi_\alpha(p)$

and we know that the supports of $\phi_\alpha$ form a locally finite cover! That is, there is some neighborhood $N$ of $p$ which intersects at most finitely many of the $\mathrm{supp}(\phi_\alpha)$ . For all of them $N$ doesn’t intersect, we are absolutely certain that $\phi_\alpha(p)=0$ , and so our big sum really only involves at most finitely many terms at each point!

As an example, consider the function $\phi_0$ defined by

$\displaystyle\phi_0(x)=\left\{\begin{array}{cc}0& x\leq-1\\\cos(\frac{\pi}{2}x)^2&-1<x<1\\{0}&1\leq x\end{array}\right.$

This is a differentiable — though not smooth — function supported on the interval $[-1,1]$ . We can slide this over to define $\phi_n=\phi_0(x-n)$ , getting a differentiable function supported on $[n-1,n+1]$ . From here, it’s an exercise to verify that this is a partition of unity. We must check that on the interval $[n,n+1]$ we have $\phi_n(x)+\phi_{n+1}(x)=1$ .

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March 7, 2011 - Posted by John Armstrong | Differential Topology, Topology

8 Comments »

[...] know what a partition of unity is, but not all partitions of unity are very useful. For instance, the single function defined by [...]

Pingback by Partitions of Unity Subordinate to a Cover « The Unapologetic Mathematician | March 8, 2011 | Reply
John, it would be useful if you could put a second pair of previous/next buttons at the top of the first page too. Thank you.
- Charlie C

Comment by Charlie C | March 10, 2011 | Reply
I agree, Charlie, but I don’t have control over this theme. I may have to change themes entirely at some point, but so far I’ve kept putting it off.

Comment by John Armstrong | March 10, 2011 | Reply
- Not a problem, John. The current layout works just fine.
  
  Comment by Charlie C | March 11, 2011 | Reply
You know, I had never noticed those “previous/next” buttons before! They’re quite handy, so thanks!

Comment by Landau | March 10, 2011 | Reply
The theme is useful for being able to stretch to accommodate long typeset display equations, but I’ve been becoming dissatisfied with it.

Unfortunately, none of the WP.com-available themes are particularly great for everything I want, and I just don’t have the time to maintain the look/feel/installation myself, either here or on my own host with WP.org; nor do I have the money to pay someone else to do it for me. We can’t all be John Baez, with Jacques Distler at our beck and call.

Comment by John Armstrong | March 10, 2011 | Reply
[...] we come to the heart of our partitions of unity: the bump functions. These are like smooth analogues of characteristic functions. A characteristic [...]

Pingback by Bump Functions, part 1 « The Unapologetic Mathematician | March 12, 2011 | Reply
[...] we’ve asserted: given any open cover of a smooth manifold we can find a countable smooth partition of unity subordinate to [...]

Pingback by Partitions of Unity (proof) « The Unapologetic Mathematician | March 14, 2011 | Reply

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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