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 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 indexed by some set , subject to two conditions. First: the collection of supports is a locally finite cover of , which takes a bit to unpack.
The support 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 .
To say that a collection of sets is a locally finite cover means that every point is contained in at least one of them, and that has some neighborhood which intersects only finitely many of them. For instance, the collection of all intervals centered at integers is a locally finite cover of . Every real number is within 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
That is, if we add up all these functions we get the function with constant value . 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
and we know that the supports of form a locally finite cover! That is, there is some neighborhood of which intersects at most finitely many of the . For all of them doesn’t intersect, we are absolutely certain that , and so our big sum really only involves at most finitely many terms at each point!
As an example, consider the function defined by
This is a differentiable — though not smooth — function supported on the interval . We can slide this over to define , getting a differentiable function supported on . From here, it’s an exercise to verify that this is a partition of unity. We must check that on the interval we have .