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Therefore, the exterior derivative $d\omega\in\Omega^k(M)$ is uniquely defined by the local definition~\eqref{eq:localdw}.
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Therefore, the exterior derivative $d\omega\in\Omega^{k+1}(M)$ is uniquely defined by the local definition~\eqref{eq:localdw}.
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\end{corollary}
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\begin{proof}
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Follows from Theorem~\ref{thm:differentialpushforward} applied with $F = \varphi_1\circ\varphi_2^{-1} : \varphi_2(U) \to\varphi_1(U)$, where $U=U_1\cap U_2$.
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