Simplex Calculations for Stokes' Theorem

 Posted on November 3, 2018  (Last modified on February 17, 2021)  |  3 minutes  |    |   0 comment

Oriented affine kk-simplex σ=[p0,p1,…,pk]\sigma = [{\bf p}_0,{\bf p}_1,\dots,{\bf p}_k]
A kk-surface given by the affine function
σ(∑i=1kaiei):=p0+∑i=1kai(pi−p0),(1) \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 pi∈Rn{\bf p}_i \in \R^n for all i∈{1,…,k}i \in \{1,\dots,k\}.
In particular, σ(0)=p0\sigma({\bf 0})={\bf p}_0 and for each i∈{1,…,k}i\in\{1,\dots,k\}, σ(ei)=pi\sigma({\bf e}_i)={\bf p}_i.

Standard simplex Qk:=[0,e1,…,ek]Q^k := [{\bf 0}, {\bf e}_1, \dots, {\bf e}_k]
A particular type of oriented affine kk-simplex with the standard basis {e1,…,ek}\{{\bf e}_1, \dots, {\bf e}_k\} of Rk\R^k.
Qk:={∑i=1kaiei∣∀i∈{1,…,k},ai≥0,∑i=1kai=1} 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 kk-simplex σ\sigma has parameter domain QkQ^k.

[Read More]
simplexes