chapter  2
52 Pages

Two–Dimensional Sphere Ω

We start with the general settings for the sphere. Much of the notation is adapted to the Euclidean concept in Chapter 1.

A spherical cap with radius ρ ∈ (0, 2] and center ξ ∈ Ω is a subregion of the unit sphere Ω defined by

Γρ(ξ) = {η ∈ Ω : 1− ξ · η < ρ}. (2.1) For the choice ρ = 2, the spherical cap coincides with the punctured sphere Ω \ {ξ}. If the underlying spherical cap is meant with respect to a sphere of radius R > 0 and center x ∈ ΩR, we still write Γρ(x) = {y ∈ ΩR : R2 − x · y < ρ}, where ρ can be chosen from the interval (0, 2R2] (the fact x ∈ ΩR already indicates that Γρ(x) is meant with respect to ΩR). While BR0,R1(y) = {x ∈ R3 : R0 < |x− y| < R1}, 0 ≤ R0 < R1, denotes a spherical shell around the center y ∈ R3, the set

CR0,R1(Γ) = { x ∈ R3 : x|x| ∈ Γ, R0 < |x| < R1

} (2.2)

is called a conical shell if Γ is a subregion of the sphere Ω. It is always meant with the origin as its center. As already known, any vector x ∈ R3 \ {0} can be represented by x = rξ, with r = |x| and ξ = x|x| ∈ Ω (generally, Greek letters ξ, η, ζ denote vectors of the unit sphere). Using spherical coordinates, we write

x(r, ϕ, t) = rξ = r

⎛ ⎝

√ 1− t2 cosϕ√ 1− t2 sinϕ

t

⎞ ⎠ , r > 0, ϕ ∈ [0, 2π), t ∈ [−1, 1],

(2.3)

where ϕ denotes longitude, θ ∈ [0, π] colatitude, and t = cos(θ) polar distance. As mentioned before, the vectors ε1, ε2, ε3 denote the canonical Cartesian basis vectors. Beside this basis, the typical orthonormal basis in a spherical framework is given by the moving triad

εr(ϕ, t) =

⎛ ⎝

√ 1− t2 cosϕ√ 1− t2 sinϕ

t

⎞ ⎠ , (2.4)

εϕ(ϕ, t) =

⎛ ⎝ − sinϕcosϕ

⎞ ⎠ , εt(ϕ, t) =

⎛ ⎝ −t cosϕ−t sinϕ√

1− t2

⎞ ⎠ .