TITLE: Use of partitions of unity
QUESTION [6 upvotes]: So, I'm working through introduction to smooth manifolds, and I've seen the fact that partitions of unity exist. They seem like extremely useful theoretical tools, but I haven't seen them used yet.
I'd like some nice examples as to where they can be used, and why they're "special" in some sense (I don't have a good intuition as to why you need a smooth manifold to exhibit this partition). 
What's the intuition with partitions of unity? asks the intuition of this object. I am after uses.
Non-Theoretical Applications of Partitions of Unity asks about, well, applied uses.
I want something along the lines of "what is the most dazzling thing you can do with this tool" - Something like the awesome uses of orthogonalization, or the spectral theorem you would see in linear algebra.

REPLY [3 votes]: On Riemannian manifold, if there is an locally finite open cover $U_i$ and $X_i$ is a vector field on $U_i$, then by partition of unity $f_i$, we have a global vector field $X =\sum_i\ f_iX_i$.