Computation of the Volume of the n-dimensions Geometry $R$

$\mathrm{Volume} (R)=\int _{s\in R}\mathrm{d}s =\int _{z\in \partial R} z\otimes \mathrm{d}z$

where $\partial R$ is the boundary of the n-dimensions geometry $R$ , $\mathrm{d}z=c_1\mathrm{d}z_1+c_2\mathrm{d}z_2+...+c_{n-1}\mathrm{d}z_{n-1}$ ,

$\mathrm{rank}[z,c_1 \mathrm{d}z_1,c_2\mathrm{d}z_2,...,c_{n-1}\mathrm{d}z_{n-1}]=n$ , $z \otimes \mathrm{d}z$ is the volume of the n-dimensions geometry that is constituted by the edge $z$ and the ( $n$ -1)-dimensions geometry $\mathrm{d}z$ as follows

$z \otimes \mathrm{d}z=\frac{1}{n}\left | \mathrm {det} [z,c_1 \mathrm{d}z_1,c_2\mathrm{d}z_2,...,c_{n-1}\mathrm{d}z_{n-1}] \right |$

then, we have

$\mathrm{Volume} (R)= \cfrac {1} {n} \int _{z\in \partial R} \left | \mathrm {det} [z,c_1 ,c_2,...,c_{n-1}] \right | \mathrm{d}z_1 \mathrm{d}z_2 ... \mathrm{d}z_{n-1}$