Implicit Functions

Abstract

In one spatial dimension, suppose we divide the real line into three distinct pieces using the points x = -1 and x = 1. That is, we define (-∞, -1), (-1, 1), and (1, ∞) as three separate subdomains of interest, although we regard the first and third as two disjoint pieces of the same region. We refer to Ω- = (-1, 1) as the inside portion of the domain and Ω+ = (-∞, -1) ∪ (1, ∞) as the outside portion of the domain. The border between the inside and the outside consists of the two points ∂Ω = {-1, 1} and is called the interface. In one spatial dimension, the inside and outside regions are one-dimensional objects, while the interface is less than one-dimensional. In fact, the points making up the interface are zero-dimensional. More generally, in ℜn, subdomains are n-dimensional, while the interface has dimension n - 1. We say that the interface has codimension one.