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We can express every k-form in the form $ \omega(x) = \sum_Id_Idx_I $ where $ I$ is k-tuple and $ d_I$ is just some scalar function of x. That's entirely understandable for me. But while reading Munkres "Analysis on manifolds" I stumbled upon something quite confusing. In the discussion of the pullback function $ \alpha^*$, there is the following part:

The form $ \alpha_*(dy_I)$ is a k-form defined on an open set of $ \mathbb{R}^k$, so it has the form $ h dx_1\wedge dx_2 \wedge ... \wedge dx_k$ for some scalar function $ h $.

I completely don't see where that comes from. Can somebody clear this up for me?

Here are 2 pages of the book, puttin the problem in context:

enter image description here

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up vote 3 down vote accepted

The key thing here is that it's specifically a $k$-form on $\Bbb{R}^k$. Since $dx^i \wedge dx^i=0$ for all $i$, this means there's only one possible term in the sum: you need to use all the different coordinates or you won't have enough.

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Oh, that was really simple, shame on me for not noticing :( Thanks a lot! – Ormi Dec 1 '12 at 15:40
Also, any other presentation of $dx^1 \wedge dx^2 \wedge \cdots \wedge dx^k$ can be permuted to go back to the $1,2, \dots , 3$ order due to the algebra of the wedge product. It suffices to consider just scalar function multiples of the cannonical volume form $dx^1 \wedge dx^2 \wedge \cdots \wedge dx^k$ for this reason. – James S. Cook Dec 1 '12 at 15:41

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