什么是 differential form？

Let $U \subseteq \R^n$ be open. A differential $k$-form on $U$ is a smooth map $\omega\colon U \to \Lambda^k(\R^n)^*$; in the basis of Proposition 21.4 (with $\dd x_i$ written for $e_i^*$),

with smooth coefficients $a_I \in \mathcal C^\infty(U)$. Their space is $\Omega^k(U)$; $\Omega^0(U) = \mathcal C^\infty(U)$. A $0$-form is a function; a $1$-form is a field of linear forms (e.g. the differential $\dd f$ of a function); an $n$-form is $a\,\dd x_1\wedge\dots\wedge\dd x_n$, the natural integrand of Chapter 11.