Simplexes, Orientations, Boundaries, and Simplicial Subdivisions

Abstract

This chapter contains an introduction to a number of topics in the geometry of n-dimensional Euclidean space. It treats barycentric coordinates, convex sets, simplexes, the orientation of simplexes, chains of simplexes, boundaries of simplexes and chains, three identities and an inequality for determinants, affine and barycentric transformations, and simplicial subdivisions of cubes and simplexes in ℝⁿ. The results in this chapter are developed in preparation for their use in the proof of the fundamental theorem of the integral calculus in ℝⁿ and of other theorems. This introduction begins the treatment of these subjects by reviewing some of the basic facts about (n — l)-dimensional planes in ℝⁿ, convex sets, and linear independence and dependence of vectors inℝⁿ.