Usefulness of partition of unity?

[JonnyG]

I am going through Spivak's Calculus on manifolds. I am on the chapter now regarding partitions of unity. I understand the construction of it, but why exactly is a partition of unity useful? Why do we care about it?

 

[ToBePhysicist]

Greetings,
few Words.
Hope I have provided the right answer in my answer...[pssst: I don't know what is this!I just know how to ultra google and tera- research!]

 

[JonnyG]

I read that already, but it didn't give me the clearest picture.

 

[Bacle2]

Basically, P.O.U's allow you to patch locally-defined objects, like Riemannian metrics in the manifold into globally defined objects.

 

[ToBePhysicist]

Basically, P.O.U's allow you to patch locally-defined objects, like Riemannian metrics in the manifold into globally defined objects.

I already said that.
Where is @JonnyG anyways? (Huh, oops...That will show a new alert on his page )

 

[JonnyG]

@Bacle2 It seems as if a partition of unity will allow us to break up a map into smaller maps. For example, if there was a set $A \subset \mathbb{R}^n$ and $\{ \phi_i \}$ is a partition of unity on $A$ and $f$ was a map on $A$ then given any $x \in A$ we can write $f(x) = \sum_{i = 1}^{\infty} \phi_i (x) f(x)$. Each $\phi_i (x) f(x)$ is smaller than $f(x)$ because the $\phi_i(x)f(x)$ will vanish outside of some open set about $x$. Thus we are concentrating the map just into that open set. Thus we can prove something or construct something regarding $f$ just in that small open set, then use a partition of unity to "glue it all together". I get that now. My question is - what advantages does this provide? I mean, why not just break up the set into smaller pieces rather than breaking up the map?

 

[mathwonk]

the domains of the maps overlap, so there is no disjoint partition of the set. all you have to do is keep reading spivak and see how he uses it. it is also used in algebraic geometry, in a different version where the key fact is that when elements f1,...,fn generate the unit ideal then there exiust g1,...,gn such that sum fjgj = 1.

 

[ToBePhysicist]

@Bacle2 It seems as if a partition of unity will allow us to break up a map into smaller maps. For example, if there was a set $A \subset \mathbb{R}^n$ and $\{ \phi_i \}$ is a partition of unity on $A$ and $f$ was a map on $A$ then given any $x \in A$ we can write $f(x) = \sum_{i = 1}^{\infty} \phi_i (x) f(x)$. Each $\phi_i (x) f(x)$ is smaller than $f(x)$ because the $\phi_i(x)f(x)$ will vanish outside of some open set about $x$. Thus we are concentrating the map just into that open set. Thus we can prove something or construct something regarding $f$ just in that small open set, then use a partition of unity to "glue it all together". I get that now. My question is - what advantages does this provide? I mean, why not just break up the set into smaller pieces rather than breaking up the map?

Noticing you have not taken advantage of the link I provided...All I can say is that you didn't read the whole Spivak's Calculus.

 

[JonnyG]

I have read the link provided...anyway, after working on some of the problems in Spivak's book and Munkres' book, I am starting to get a more clear picture of how a P.O.U. should be used. As I continue on, I am sure it will become even more clear and apparent. Thanks for the help everybody.