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About diffeomorphism

  1. Aug 10, 2007 #1
    q1. The problem statement, all variables and given/known data
    Let f : X ->Y, g : Y->Z be smooth. Show the composite is smooth. If f, g are
    diffeomorphisms, so is the composite.

    q2.Let B= {x : |x|^2 < a^2}. Show that
    x -> ax/[(a^2 − |x|^2)^1/2]
    is a diffeomorphism.
    2. Relevant equations

    3. The attempt at a solution
    For q1 :A map is smooth if smooth functions pull back to smooth functions. If h : Z->R is
    smooth, then by g’s smoothness ,so is hg, then by f’s smoothness so is hgf = h(gf).
    Since this holds for all h, gf is smooth.

    Actually i don't understand the answer and why one needs to come up with the function h.

    For q2 : |f(x)| = a|x|/[(a^2 − |x|^2)^1/2]
    and then rearrange symbols so that only |x| is on the right hand side

    I don't know why I should start with the absolute value of f first ?
  2. jcsd
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