Let $n,m\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$} and $k\in\mathbb{N}\cup\{0\}$. Let $U\subseteq\mathbb{R}^n$ and $V\subseteq\mathbb{R}^m$ be \reftext{def:open-subset-euclidean-space-2026a}{open} subsets of \reftext{def:euclidean-space-rn-2026a}{Euclidean space}, let $F:U\to V$ be a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map}, and let $\omega$ be a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{$C^1$ differential $k$-form} on $V$. Then the \reftext{def:pullback-differential-form-c1-euclidean-2026a}{pullback} $F^{*}\omega$ is a $C^1$ differential $k$-form on $U$, and

where on the left $d$ denotes the \reftext{def:exterior-derivative-c1-differential-form-euclidean-open-set-2026b}{exterior derivative} on $U$ applied to $F^{*}\omega$, and on the right $d\omega$ is the exterior derivative of $\omega$ on $V$, whose pullback under $F$ is again taken in the sense of \ref{def:pullback-differential-form-c1-euclidean-2026a}.