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Proof of unique derivative

  1. Dec 23, 2011 #1
    Hi--

    I am trying to work through Spivak's Calculus on Manifolds over the holidays, and I am a little stuck on his proof of the unique derivative (on p. 16 as well as below).

    Specifically,
    (i) Why does the ≤ inequality hold, and
    (ii) Why does the last equality of the second-to-last-line hold?

    I would very much appreciate any help!
     

    Attached Files:

  2. jcsd
  3. Dec 23, 2011 #2

    micromass

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    (i) The triangle inequality [tex]|x+y|\leq |x|+|y|[/tex]

    (ii) Linearity of [itex]\lambda[/itex] and [itex]\mu[/itex]:

    [tex]\frac{|\lambda(tx)-\mu(tx)|}{|tx|} = \frac{|t\lambda(x)-t\mu(x)|}{|tx|} = \frac{|t||\lambda(x)-\mu(x)|}{|t||x|} = \frac{|\lambda(x)-\mu(x)|}{|x|}[/tex]
     
  4. Dec 24, 2011 #3
    Thanks so much!
     
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