Let $m,n\in\mathbb N$, let $\Omega\subset \mathbb R^n$ be bounded and open, let $1\le r\le \min\{m,n\}$, and let $p\in[1,\infty)$ satisfy $p\ge r$. Suppose that $(u_k)_{k=1}^{\infty}$ and $u$ belong to $W^{1,p}(\Omega;\mathbb R^m)$ and that $u_k \rightharpoonup u$ in $W^{1,p}(\Omega;\mathbb R^m)$. Write $u_k=(u_{k,1},\dots,u_{k,m})$ and $u=(u_1,\dots,u_m)$. Let $Ju_k:\Omega\to\mathbb R^{m\times n}$ and $Ju:\Omega\to\mathbb R^{m\times n}$ denote the weak Jacobian matrices, viewed as a.e.-defined maps whose entries are $(Ju_k)_{ab}=\partial_{x_b}u_{k,a}$ and $(Ju)_{ab}=\partial_{x_b}u_a$. For every pair of strictly increasing index maps $\alpha:\{1,\dots,r\}\to\{1,\dots,m\}$ and $\beta:\{1,\dots,r\}\to\{1,\dots,n\}$, define

\begin{align*} M_{\alpha,\beta}(Ju_k)=\det\big(\partial_{x_{\beta_b}}u_{k,\alpha_a}\big)_{a,b=1}^{r} \end{align*}

and

\begin{align*} M_{\alpha,\beta}(Ju)=\det\big(\partial_{x_{\beta_b}}u_{\alpha_a}\big)_{a,b=1}^{r}. \end{align*}

Then $M_{\alpha,\beta}(Ju_k)\to M_{\alpha,\beta}(Ju)$ in the sense of distributions on $\Omega$. Equivalently, for every $\varphi\in C_c^\infty(\Omega)$,

\begin{align*} \lim_{k\to\infty}\int_\Omega \varphi\,M_{\alpha,\beta}(Ju_k)\,d\mathcal L^n=\int_\Omega \varphi\,M_{\alpha,\beta}(Ju)\,d\mathcal L^n. \end{align*}

If $p>r$, then the same convergence holds weakly in $L^q(\Omega)$, where

\begin{align*} q=\frac{p}{r}. \end{align*}