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Partitions of unity: support of a function

  1. Oct 30, 2008 #1
    in my readings, spivak or elsewhere, i've come across this several times but i don't have the formal training (maturity) to know how to use it. intuitively: by the atlas maps on the manifold, we can chop up a manifold into patchs. for each patch, by smoothness or something, there is a smooth function whose (this is where i give up) support in nonzero. since these functions are nonzero on each patch and add up to unity, we can sum up these up over the entire manifold.

    is 'support of a function' an integrability contition that allows for nonzero terms in the sum like the characteristic function on a riemann integral?
     
  2. jcsd
  3. Oct 30, 2008 #2

    quasar987

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    The support of a function is just the closure of the set where it does not vanish. Or said differently, it is the smallest closed set outside of which the function is identically 0.

    A partition of unity is a bunch of functions such that
    (1) around each point, there is a nbhd on which only finitely many functions are non zero.
    (2) the functions add up to 1.

    Because of condition (1), the sum of the functions at each point is actually a finite sum.
     
  4. Oct 31, 2008 #3

    mathwonk

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    its a very useful trick for letting you make constructions locally where they are easier, and then add up the results globally. there is also a version in aklgebraic geometry, where a partition of unity is just an expression like 1 = ax + by+....+cz, of the identity, as an element of an ideal generated by the functions x,y...z. The book Basic algebraic geometry, by shafarevich contains many nice arguments using this trick.
     
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