Volume computing of the continuous geometry with the boundary described by multiple variables

     In my study on the controllable abundance of linear continuous-time systems, the volume of a special continuous geometry with the boundary described by multiple variables is computed again and again. Hence, the following volume computing method for that geometry is proposed and proven.

     [Theorem] Let $$ $R_{c}$ is a continuous geometry that contains the origin in the $n$ -dimension space. If the any point $z$ in the boundary $\partial R_{c}$ of the geometry $R_{c}$ can be described as the following equation with the multiple variables $z_{i}(i=1,2,\ldots,m)$

$\textrm{d}z=c_{1}(z)\textrm{d}z_{1}+c_{2}(z)\textrm{d}z_{2}+\cdots+c_{m}(z)\textrm{d}z_{m}$

where $m>n-1$ , and parameter vector $\left[c_{1}(z),c_{2}(z),\cdots,c_{m}(z)\right]$ satisfies

    $\mathrm{rank}[z,c_{1}(z),c_{2}(z),\cdots,c_{m}(z)]=n$

    $\mathrm{rank}[c_{1}(z),c_{2}(z),\cdots,c_{m}(z)]=n-1$

that is, $\textrm{d}z$ is a ( $n-1$ )-dimensions parallel polyhedron independent on the vector $z$ .

     For the geometry defined as the above, its volume can be computed as

$\mathrm{Vol}(R_{c})=\frac{1}{n}\sum_{(k_{1},k_{2},\ldots,k_{n-1})\in\varOmega_{1,m}^{n-1}}\int_{z\in\partial R_{c}}\left|\det\left(\left[z,c_{k_{1}}(z),c_{k_{2}}(z),\cdots,c_{k_{n-1}}(z)\right]\right)\right|\textrm{d}z_{k_{1}}\textrm{d}z_{k_{2}}\cdots\textrm{d}z_{k_{n-1}}$

where where $\Omega_{1,m}^{n-1}$ is constituted by the all possible multi-tuple $(k_{1},k_{2},\cdots,k_{n-1})$ which elements are picked from the natural number set $\{1,2,\cdots,m\}$ and sorted by the values.