Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Spivak's Calculus on Manifolds: Theorem 5-3

  1. Aug 2, 2015 #1
    I am trying to finish the last chapter of Spivak's Calculus on Manifolds book. I am stuck in trying to understand something that seems like it's supposed to be trivial but I can't figure it out.

    Suppose [itex]M[/itex] is a manifold and [itex]\omega[/itex] is a p-form on [itex]M[/itex]. If [itex]f: W \rightarrow \mathbb{R}^n[/itex] is a coordinate system around [itex]x = f(a)[/itex] and [itex] v_1, \ldots, v_{p+1} \in M_x[/itex], there are unique [itex] w_1, \ldots, w_{p+1} \in \mathbb{R}_a^k[/itex] such that [itex]f_{\ast}(w_i) = v_i[/itex]. Define [itex]d\omega(x)(v_1, \ldots, v_{p+1}) = d(f^{\ast}\omega)(a)(w_1, \ldots, w_{p+1})[/itex].

    Now he says that [itex]d\omega(x)[/itex] defined this way is independent of the choice of coordinate system around [itex]x[/itex]. Any hints on how this can be shown?
     
  2. jcsd
  3. Aug 3, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Take another coordinate system and show that the definition results in the same p-form.
     
  4. Aug 3, 2015 #3
    Taking the suggested path, I reduced the problem to showing for all [itex]j[/itex], [itex]d(\omega_{i_1, \ldots, i_p} \circ g)(b)(w_j^{\prime}) = d(\omega_{i_1, \ldots, i_p} \circ f)(a)(w_j)[/itex] where [itex]\omega_{i_1, \ldots, i_p}[/itex] are components of [itex]\omega[/itex], [itex]g: V \rightarrow \mathbb{R}^n[/itex] is another coordinate system around [itex]x = g(b)[/itex], and [itex]w_1^{\prime}, \ldots, w_{p+1}^{\prime} \in \mathbb{R}_b^k[/itex] are the unique vectors such that [itex]g_{\ast}(w_i^{\prime}) = v_i[/itex].

    Now this would follow immediately if we could use chain rule but we can't since the components are defined only on the manifold.
     
    Last edited: Aug 3, 2015
  5. Aug 3, 2015 #4

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    You need to use the fact that the coordinate transformation between the two systems is smooth and that you can apply the chain rule for it.
     
  6. Aug 3, 2015 #5
    Maybe that is what Spivak meant when he says "Precisely the same considerations hold for forms" on page 116.

    I see it now, thanks for all your help Orodruin.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Spivak's Calculus on Manifolds: Theorem 5-3
Loading...