= Uniqueness of the exterior derivative from its axioms
{title2=$d\alpha=\sum_{I,j}\partial_ja_I\,dx^j\wedge dx^I$}
An $\mathbb R$-linear degree-one <graded derivation> of the <exterior algebra> of smooth <differential forms> that equals the differential on functions and squares to zero is unique. A <smooth bump function> and the <graded Leibniz rule> first establish locality. Then $d(dx^i)=0$, and applying the <graded Leibniz rule> to $\alpha=\sum_Ia_Idx^I$ gives the displayed coordinate formula. The <chain rule> makes that formula coordinate-independent.
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