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We need these definitions for the theorem: enter image description here enter image description here

Theorem 10.20

(a)If $\omega, \lambda $ are $k-$ and $m-$ forms, respectively of class $C^{1}$ in $E$, then $$d(\omega \land \lambda)=(d\omega)\land \lambda + (-1)^{k}\omega \land d\lambda$$

(b) If $\omega$ is of class $C^{1}$, then $d^{2} \omega=0$

enter image description here

Hence it's said that ( read the proof on the photo) $d(\omega \land \lambda)$ = ($df \land dx_I$)$\land$ ($g dx_J$) + (${-1}^k$)($fdx_I$)$\land$($dg \land dx_J$) . ( mark this equality by ($\star$))

I couldn't understand how do we get the ($\star$).

I would be grateful for any kind of help.

JohnNash
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  • There is a lot of text there. I don't see the star. – John Douma Dec 31 '21 at 17:00
  • @JohnDouma what do you mean? – JohnNash Dec 31 '21 at 17:04
  • I did't see your starred equation in the text. I finally found it four lines above the bottom of the text of the book but this is difficult to read. Also, the author is referencing $(59)$ and $(60)$ which is probably necessary to understand this. – John Douma Dec 31 '21 at 17:18
  • @JohnDouma i've added these things. now can you answer? – JohnNash Dec 31 '21 at 18:48
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    You can find this precise question many times on this site. For example, here and here. – Ted Shifrin Dec 31 '21 at 18:56
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    baby Rudin isn't the best place to be learning differential forms. In fact I'd stop with baby Rudin after Ch9. For manifolds and differential forms I'd use Tu (https://link.springer.com/book/10.1007/978-1-4419-7400-6). For Lebesgue integration I'd use Axler (https://link.springer.com/book/10.1007/978-3-030-33143-6). – g.s Dec 31 '21 at 20:06

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