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Homework Help: Help with smooth manifolds

  1. Dec 4, 2003 #1
    i am trying to solve this problem:

    Give the paraboloid [tex] y_{3}=(y_{1})^2+(y_{2})^2 [/tex] the structure of a smooth manifold.

    But i am unsure what it means by structure. Can anyone give me some help here?
     
    Last edited: Dec 5, 2003
  2. jcsd
  3. Dec 5, 2003 #2
    i kept trying to solve it on my own, but all i was able to get was:

    [tex]
    x^1(y_{1},y_{2},y_{3})=\pm\sqrt{y_{3}-(y_{2})^2}
    [/tex]

    [tex]
    x^2(y_{1},y_{2},y_{3})=\pm\sqrt{y_{3}-(y_{1})^2}
    [/tex]

    [tex]
    x^3(y_{1},y_{2},y_{3})=(y_{1})^2 + (y_{2})^2
    [/tex]

    is that the answer? i'm feeling very lost here.

    edit:

    Ok i think i figured it out...

    i set the local coordinates [tex] x^1,x^2 [/tex] to lie in the[tex] y^1;y^3,y^2;y^3 [/tex] planes respectively. thus from the above I get:

    [tex]
    x^1(y_{1},0,y_{3})=\pm\sqrt{y_{3}}
    [/tex]

    [tex]
    x^2(0,y_{2},y_{3})=\pm\sqrt{y_{3}}
    [/tex]

    I also realised that the manifold is 2-dimensional (being embedded in [tex] E_{3} [/tex]) so there is no [tex] x^3 [/tex] coordinate.


    If any one can give any confirmation on this answer i would greatly appreciate it.
     
    Last edited: Dec 5, 2003
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