Let $U\subset\mathbb R^n$ be an open set. A differential $k$-form on $U$ is a map to the $k$th exterior product of the dual space of $\mathbb R^n$: \(\phi:U\to\wedge^k((\mathbb R^n)^*).\)
A $0$-form is just a function. Wedging a $0$-form with something amounts to multiplication by the function.
Let $U\subset\mathbb R^n$ be an open set. Then we write $\mathcal A^k(U)$ for the vector space of k-forms on $U$.
Wikidata ID: Q1047080