Is a simplex an orientable manifold with corners?

Question

First of all we remember some elementary definitions and results about manifolds and simplexes.

Definition

A function $f$ defined in a subset $S$ of $\Bbb R^k$ is said of class $C^r$ if it can be extended to a function $\phi$ (said $C^r$-extension) that is of class $C^r$ in a open neighborhood of $S$.

Lemma

If $f$ is a function defined in a subset $S$ of $\Bbb R^n$ such that for any $x\in S$ there exists a function $f_x$ defined in a neighborhood of $x$ that is of class $C^r$ and compatible with $f$ on $U_x\cap S$ then $f$ is of class $C^r$.

Lemma

If $U$ is an open set of $H_n:=[0,+\infty)^n$ for any $k\le n$ then the derivatives of two different extensions $\phi$ and $\varphi$ of a $C^r$-function $f$ agree in $U$.

Definition

A $k$-manifold with corners in $\Bbb R^n$ of class $C^r$ is a subspace $M$ of $\Bbb R^n$ whose points have a neighborhood $V$ in $M$ that is the immage of a homeomorphism $\phi$ of calss $C^r$ defined an open set $U$ of $\Bbb R^k$ or of $H_k$ and whose derivative has rank $k$.

Definition

If $M$ is a $k$-manifold with corners in $\Bbb R^n$ we say that two coordinate patch $\alpha_i:U_i\rightarrow V_i$ for $i=1,2$ of $M$ overlap if $V_1\cap V_2$ is not empty and in this case we say that they overlap positively if the transition function $\alpha^{-1}_2\circ\alpha_1$ is orientations preserving, i.e. its derivative has positive determinant.Finally if $M$ can be covered by a collection of coordinate patches each pair of which overlap positively then $M$ is said to be orientable.

Definition

If $x_0,...,x_k$ are $(k+1)$ affinely indipendent points 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 $$ \mathcal S:=\Biggl\{x\in\Bbb R^n: x=x_0+\alpha^i\vec v_i\,\text{and}\,\sum_{i=1}^k\alpha^i\le1\,\text{and}\,\alpha^i\ge0\,\text{for all}\,i=1,\dots,k\Biggl\} $$ where $\vec v_i:=(x_i-x_0)$ for each $i=1,\dots,k$.

So here I showed that any $k$-simplex is a $k$-manifold with corners and thus now I ask if it is orietable too. So could someone help me, please?

Answer 1

So in the posted link it is shown that 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 $\mathcal E_k$ and thus if $(f\circ\alpha_1)$ and $(f\circ\alpha_2)$ are two coordinate patch that overlap then $$ (f\circ\alpha_2)^{-1}\circ(f\circ\alpha_1)=\alpha_2^{-1}\circ f^{-1}\circ f\circ\alpha_1=\alpha_2^{-1}\circ\alpha_1 $$ so that if we prove that $\mathcal E_k$ is orientable the even $\mathcal S$ will be orientable but $\mathcal E_k$ is trivially orientable because it is a $k$-manifold in $\Bbb R^k$ and any such manifold can be equipped with the natural orientation as here showed.