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Properties of Differentials, Smooth Manifolds.

  1. Feb 24, 2013 #1
    I'm reading the second edition of John M. Lee's Introduction to Smooth Manifolds and he has a proposition that I'd like to understand better

    Let M, N, and P be smooth manifolds with or without boundary, let F:M→N and G:N→P be smooth maps and let p[itex]\in[/itex]M

    Proposition: TpF : TpM → TF(p) is linear

    ok I know that v[itex]\in[/itex]TpM means that

    v:C(M)→ℝ is a derivation and that TpM is a vector space.

    Does this mean that the image of (av+bw) under TpF where v,w [itex]\in[/itex] TpM and a,b [itex]\in[/itex] ℝ

    is aTpF(v) + bTpF(w) which means TpF is linear?
  2. jcsd
  3. Feb 24, 2013 #2
    Yes, that's what it means.
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