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

for every $x\in U$ and every vectors $v_1,\dots,v_k\in\mathbb{R}^n$, where $J_F(x)v_r$ denotes the matrix-vector product from \reftext{def:matrix-vector-product-2026a}{the definition of matrix-vector multiplication}, applied to the Jacobian matrix appearing in \reftext{def:differentiable-map-at-point-euclidean-2026a}{the differentiability definition}.