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Spivak Inverse Function Theorem Proof

  1. Nov 27, 2009 #1
    On p. 36 of "Calculus on Manifolds" Spivak writes:

    "If the theorem is true for ([tex]\lambda[/tex][tex]^{-1}[/tex])[tex]\circ[/tex]f , it is clearly true for f."

    This far I understand. However, he next says:

    "Therefore we may assume at the outset that [tex]\lambda[/tex] is the identity."

    I don't understand how this follows, since he previously defined [tex]\lambda[/tex] = Df(x).

    I would appreciate someone adding a bit more explanation here.

    Thank you.
     
  2. jcsd
  3. Nov 28, 2009 #2

    HallsofIvy

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    What he is saying is that since, whenever some theorem is true for [itex]\lambda^{-1}\circle f[/itex], it is true for f, we can work with f rather than [itex]\lambda^{-1}\circle f[/itex]- that is that we can take [itex]\lambda= 1[/itex]. What you need to do is look at why "if the theorem is true for [itex]\lambda^{-1}\circle f[/itex], it is clearly true for f.[/itex]
     
  4. Nov 29, 2009 #3
    I can understand why

    a)if this particular theorem is true for [tex]\lambda^{-1}\circ[/tex]f it is true for f, but

    b) is it true as your posting suggests that any theorem true for [tex]\lambda^{-1}\circ[/tex]f is true for f?

    and

    c) how does his proof depend upon (a)? I mean, how does the subsequent argument depend upon (a)? I understand that [tex]\lambda[/tex] is a linear transformation with non-zero determinant. Doesn't that already imply that f(x+a) -f(a) <> 0 in some neighborhood of a?

    I hate to look a gift horse in the mouth but I'm studying this stuff on my own with no one to talk to - not in a class.

    Thank you all.
     
  5. Dec 29, 2011 #4
    I know this thread is old, but I wanted to put in my two cents since I had trouble with this at first as well.
    Suppose the theorem is true for any function with derivative at a equal to the identity function and suppose we have a function f with λ = Df(a) not necessarily the identity. As the author points out, [itex]\lambda^{-1} \circ f[/itex] is a continuously differentiable function with derivative at a equal to the identity, so the theorem is true for [itex]\lambda^{-1} \circ f[/itex] by the assumption above. Hence, the theorem is true for f.
     
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