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.