Pullback of a Differential Form by a $C^1$ Map

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

$$(F^{*}\omega)_x(v_1,\dots,v_k)=\omega_{F(x)}\bigl(J_F(x)v_1,\dots,J_F(x)v_k\bigr)$$

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 the definition of matrix-vector multiplication, applied to the Jacobian matrix appearing in the differentiability definition.