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Corollary 8: Integration in 'Polar Coordinates'

  1. Jan 3, 2015 #1
    I am reading Spivak's Differential Geometry Vol. 1. I am stuck for some days in chapter 8 about integrating forms on manifolds. Maybe someone can clear my doubt.

    First, I will 'type' what the corollary says:

    Screen_Shot_2015_01_03_at_11_48_24_AM.jpg

    My doubt is regarding this affirmation:

    Screen_Shot_2015_01_03_at_11_48_37_AM.jpg

    The book it says is easy to see. Well I think the (-1)n-1, does not exist. I worked in the n=2 case just to see where I was doing an error, but still I have that negative sign does not exist.

    I will post my working out:

    http://postimg.org/image/ghfrg9jq1/

    I believe Fubini is justified. A good person that was trying to help me, said that instead of Fubini, I can only interchange the integrals adding the (-1)n-1 term, but I don't know why.

    Any help is appreciated,

    thanks

    Sergio
     
  2. jcsd
  3. Jan 8, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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