The basic idea should go something like this: When you have $k$ ($k\le n$) vectors $v_1,...,v_k$ in $\mathbb{R}^n$, you can arrange them into an $n\times k$-matrix $V$. They span a parallelepiped, which has $k$-dimensional volume $\sqrt{\det(V^tV)}$. Assume you have a map $f\colon U\rightarrow\mathbb{R}^n$, where $U\subseteq\mathbb{R}^k$ is open that is sufficiently differentiable (GMT likes to work with Lipschitz maps, which implies differentiable a.e. and that will suffice). Then you can form the Gramian determinant $J_kf=\sqrt{\det(J_f^tJ_f)}$ pointwise. If $M$ is a smooth manifold and $…