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In the proof of Poincare's lemma, it's said every differential $(k+1)-form $ ${\omega} \in\Omega^{k+1}(U\times\mathbb{R})$ ($U$ is an open set in $\mathbb{R}^n$) could be uniquely written as $$\omega=\sum f_I(x,t)dx_I+\sum g_J(x,t)dt\wedge dx_J .\quad (*)$$ Here $I=(i_1,\cdots,i_{k+1}),J=(j_1,\cdots,j_k),\ 1\le i_1<i_2<\cdots<i_{k+1}\le n, 1\le j_1<j_2<\cdots <j_k\le n.$

Question: I wonder where the second part in $(*)$ comes from, and what's the meaning of symbol $dt$ here. Any help would be appreciated.

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In this notation, we are writing the coordinates in $U\subseteq\Bbb R^n$ as $x_1,\ldots,x_n$ and in $\Bbb R$ as $t$ so that the coordinates on $U\times \Bbb R$ are $x_1,\ldots,x_n,t$. So $dx_1,\ldots,dx_n,dt$ are the basic $1$-forms.

A typical $3$-form might then be say $$t\,dx_1\wedge dx_3-x_1^2dx_2\wedge dx_3+dt\wedge(\sin t\,dx_2-e^{x_2}\,dx_3).$$

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