Simplex Calculations for Stokes' Theorem

Oriented affine $k$-simplex $\sigma = [{\bf p}_0,{\bf p}_1,\dots,{\bf p}_k]$ : A $k$-surface given by the affine function \[ \sigma\left(\sum_{i=1}^k a_i {\bf e}_i \right) := {\bf p}_0 + \sum_{i=1}^k a_i ({\bf p}_i - {\bf p}_0) \tag{1}, \] where ${\bf p}_i \in \R^n$ for all $i \in {1,\dots,k}$. In particular, $\sigma({\bf 0})={\bf p}_0$ and for each $i\in{1,\dots,k}$, $\sigma({\bf e}_i)={\bf p}_i$. Standard simplex $Q^k := [{\bf 0}, {\bf e}_1, \dots, {\bf e}_k]$ : A particular type of oriented affine $k$-simplex with the standard basis ${{\bf e}_1, \dots, {\bf e}_k}$ of $\R^k$. [Read More]