What is differential form? — Mathematics

Definition 21.7 University Mathematics — Year 3 · Chapter 21 — Differential Forms and Stokes’ Theorem

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 $e_i^*$ ),

\omega = \sum_{\abs I = k} a_I\,\dd x_I, \qquad \dd x_I = \dd x_{i_1} \wedge \dots \wedge \dd x_{i_k},

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.

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