Definition
If $x_0,...,x_n$ are $(k+1)$ affinely indipendent point of $\Bbb R^n$ (which means that the vectors $(x_1-x_0),...,(x_k-x_0)$ are linearly independent) then simplex determined by them is the set $$ S:=\Biggl\{x\in\Bbb R^n: x=\alpha^i\vec v_i, \sum_{i=1}^n\alpha_i\le1\,\,\,\text{and}\,\,\,\alpha_i\ge0\,\,\,\text{for all}\,i\Biggl\} $$ where $v_i:=(x_i-x_0)$ for each $i=1,\dots, k$.
So using the change of variables theorem (see here for details) it is possible to prove that the volume of a $n$-symplex $S$ in $\Bbb R^n$ is given by the formula $$ \operatorname{vol}(S)=\Big|\frac{1}{n!}\det\big[(x_1-x_0),...,(x_n-x_0)\big]\Big| $$ for each $n\in\Bbb N$. Moreover we know that any $k$-simplex is a compact convex smooth orientable $k$-manifold with corners (see theese three link -I, II and III - for more details) so that I ask if the same formula holds for any $k$-simplex in $\Bbb R^n$. So as proved in the link II any coordinate patch of $\mathcal S$ is equal to the composition of an affine map $f:\Bbb R^k\rightarrow\Bbb R^n$ with a coordinate patch $\alpha:U\rightarrow V$ of the standard $k$-simplex $\mathcal E_k$ of $\Bbb R^k$ so that if $\{\phi_i:i=1,\dots,l\}$ is a partition of unity for $\mathcal S$ then $\{\phi_i\circ f:i=1,\dots,l\}$ is a partition of unity for $\mathcal E_k$; moreover the derivative of $\alpha$ is a square matrix so that by the Leibniz rule and by the Binet formula it follows that $$ \operatorname{vol}\big(D(f\circ\alpha)\big):=\sqrt{\det\Big(\big(D(f\circ\alpha)\big)^T\cdot\big(D(f\circ\alpha)\big)\Big)}=\\ \sqrt{\det\Big((D\alpha)^T\cdot\big((Df)^T\cdot(Df)\big)\cdot(D\alpha)}=\\ \sqrt{\det\big(D\alpha)^T\cdot\det\big((Df)^T\cdot(Df)\big)\cdot\det(D\alpha)}=\\ \sqrt{\det\big((Df)^T\cdot(Df)\big)}\cdot\sqrt{\det\big(D\alpha)^T\cdot\det(D\alpha)}=\\ \sqrt{\det\big((Df)^T\cdot(Df)\big)}\cdot\sqrt{\det\big((D\alpha)^T\cdot(D\alpha)\big)}=\\ \operatorname{vol}\big(Df\big)\cdot\operatorname{vol}\big(D\alpha\big)=\operatorname{vol}(\vec v_1,\dots,\vec v_k)\cdot\operatorname{vol}\big(D\alpha\big) $$ and thus $$ \operatorname{vol}(\mathcal S)=\int_\mathcal S1=\sum_{i=1}^l\int_\mathcal S\phi_i=\sum_{i=1}^l\int_{\operatorname{int}U_i}\big(\phi_i\circ(f\circ\alpha_i)\big)\cdot\operatorname{vol}\big(D(f\circ\alpha_i)\big)=\\\operatorname{vol}(\vec v_1,\dots,\vec v_k)\cdot\sum_{i=1}^l\int_{\operatorname{int} U_i}\big((\phi_i\circ f)\circ\alpha_i\big)\cdot\operatorname{vol}\big(D\alpha_i\big)=\operatorname{vol}(\vec v_1,\dots,\vec v_k)\cdot\sum_{i=1}^l\int_{\mathcal E_k}(\phi_i\circ f)=\\ \operatorname{vol}(\vec v_1,\dots,\vec v_k)\cdot\int_{\mathcal E_k}1=\operatorname{vol}(\vec v_1,\dots,\vec v_k)\cdot\operatorname{vol}\big(\mathcal E_k\big)=\frac 1{k!}\operatorname{vol}(\vec v_1,\dots,\vec v_k) $$ so that the statement holds. So I ask if my arguments are correct. Could someone help me, please?
DETAILS ABOUT INTEGRATION ON MANIFOLDS
If you do not understand completely the meaning of the symbols see here for explanations about the volume of a (compact) manifold.
