Simplex Calculations for Stokes' Theorem

Sat, 03 Nov 2018 02:05:58 +0100

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 $\R^k$.
$$ Q^k := \left\{ \sum_{i=1}^k a_i {\bf e}_i \Biggm| \forall i \in \{1,\dots,k\}, a_i \ge 0, \sum_{i=1}^k a_i = 1 \right\} $$

Note that an oriented affine $k$-simplex $\sigma$ has parameter domain $Q^k$.

Simplexes on Solarized Sublime Sekai