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I have checked Rudin's proof about Poincaré Lemma (Principles of Mathematical Analysis) and it seems to have a mistake. Through Google, I found another guy who has noted such error. More details here. And I look through the other books about differential forms, but they just talked about $C^\infty$.

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  • $\begingroup$ the sum of continuous forms is continuous as the coefficients add in the natural manner. If I read his proof correctly, by that point his induction hypothesis indicates the continuity of each piece which builds $\gamma$ hence $\gamma$ is continuous. $\endgroup$ – James S. Cook Nov 13 '13 at 10:27
  • $\begingroup$ I don't think he says $\gamma$ is smooth, he says there in $Y_{p-1}$ which as I read it says they're once continuously differentiable. ($p-1$ refers to form degree) $\endgroup$ – James S. Cook Nov 13 '13 at 10:42
  • $\begingroup$ But why DjFi is continuously differentiable? $\endgroup$ – gilliatt Nov 13 '13 at 11:30
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    $\begingroup$ If $F_i$ was $C^2$ then $D_jF_i$ would be $C^1$. However, I see you say he only assumes $C^1$ which also leaves me puzzled. $\endgroup$ – James S. Cook Nov 14 '13 at 3:03
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The ambiguity goes away when coefficients are taken in the basis of $\Omega^k(M)$ given by wedges $dx_{i_1}\wedge\cdots\wedge dx_{i_k}$ of ordered tuples of $1$-forms $(dx_{i_1},\ldots,dx_{i_k})$ where $i_1<\cdots<i_k$.

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  • $\begingroup$ Could you please elaborate? And I want know why d(dw)=0,when w belongs to class C^1 $\endgroup$ – gilliatt Nov 13 '13 at 10:24

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