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 $…

I'll make the codimension $1$ example I'm alluding to explicit: Say $M\subset\mathbb{R}^n$ is a codimension $1$ submanifold with unit normal field $\nu$ and $A\subset M$ is measurable. Consider $A_r=\{p+t\nu(p)\colon p\in A,-r<t<r\}$ (that is, $A$ thickened up in normal direction). I'm not gonna pay attention to the technical details in the forthcoming btw. Assume you have a local parametrization $\psi\colon U\rightarrow V\cap M$, whose image contains $A$ and consider $B=\psi^{-1}(A)$. We form the map $\tau\colon B\times\mathbb{R}\rightarrow\mathbb{R}^n,\,(x,t)\mapsto\psi(x)+t\nu(\psi(x))$.…