The Unapologetic Mathematician

Mathematics for the interested outsider

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.

About these ads

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

8 Comments »

  1. [...] 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

  2. 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

  3. 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

  4. You know, I had never noticed those “previous/next” buttons before! They’re quite handy, so thanks!

    Comment by Landau | March 10, 2011 | Reply

  5. 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

  6. [...] 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

  7. [...] 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


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 388 other followers

%d bloggers like this: