chapter  1
18 Pages

Three-Dimensional Euclidean Space R3

The letters N, N0, Z, R, and C denote the sets of positive integers, non-negative integers, integers, real numbers, and complex numbers, respectively.

Let us use x, y, . . . to represent the elements of Euclidean space R3. For all x ∈ R3, x = (x1, x2, x3)T, different from the origin, we have

x = rξ, r = |x| = √ x21 + x

2 2 + x

2 3, (1.1)

where ξ = x|x| is the uniquely determined directional unit vector of x ∈ R3. BR(x) designates the (open) ball in R3 with center x and radius R:

BR(x) = {y ∈ R3 : |x− y| < R}. (1.2)

The sphere of radius R around x (i.e., the boundary ∂BR(x) of the ball BR(x)) is denoted by ΩR(x):

ΩR(x) = ∂BR(x) = {y ∈ R3 : |x− y| = R}. (1.3)

Throughout this work, the unit sphere Ω1(0) around the origin is denoted simply by Ω, while ΩR denotes the sphere with radius R around the origin.