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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Spivak's Calculus on Manifolds: Theorem 5-3

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