Spivak's Calculus on Manifolds: Theorem 5-3

[exclamationmarkX10]

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

Now he says that d\omega(x) defined this way is independent of the choice of coordinate system around x. Any hints on how this can be shown?

 

[Orodruin]

Take another coordinate system and show that the definition results in the same p-form.

 

[exclamationmarkX10]

Take another coordinate system and show that the definition results in the same p-form.

Taking the suggested path, I reduced the problem to showing for all j, 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) where \omega_{i_1, \ldots, i_p} are components of \omega, g: V \rightarrow \mathbb{R}^n is another coordinate system around x = g(b), and w_1^{\prime}, \ldots, w_{p+1}^{\prime} \in \mathbb{R}_b^k are the unique vectors such that g_{\ast}(w_i^{\prime}) = v_i.

Now this would follow immediately if we could use chain rule but we can't since the components are defined only on the manifold.

 

[Orodruin]

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.

 

[exclamationmarkX10]

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.

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.