Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Partitions of unity

  1. Apr 22, 2007 #1
    what are they exactly?
     
  2. jcsd
  3. Apr 24, 2007 #2

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    smooth functions which are each zero off a small interval, but which add up to the constant fucntion 1 on the whole line or whole mANIFOLD. hence they "partition" the constant fucntion 1, i.e. unity.

    they aRE USED TO PATCH TOGETHER THIngS WHICH ARE only CONSTRUCTED LOCALLY.

    i.e. given another function f which we want to integrate over a whole manifold M, WE COVER M by small nbhds and take a pof1 subordinate to tht cover. then multiplying f by one of our pof1 functions makes the product non zero only in SMLL NBHD AND WE INTEGRATE THERE USING LOCAL COORDINATES.

    doing this over all nbhds we then add the results.


    in affine algebraic geometry, a similar technique is uised when we haVE GENERATORS f1,...fm FOR the unit IDEAL R. i.e. this emans there exist multipliers g1,...,gm such that the sum of the products figi equals 1.

    then we can make a local construction using the gi, and modify it with the fi to get a global construction.

    this technique for example can be used to prove the first cohomology of the structure sheaf O on an affine variety is zero.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Partitions of unity
Loading...