Spezielle Funktionen 4 Skalare Kugelfunktionen M. Gutting 28. Mai 2015
4.1 Scalar Spherical Harmonics We start by introducing some basic spherical notation. S 2 = {ξ R 3 ξ = 1} is the unit sphere in R 3, we use Greek letters for elements of S 2. Polar coordinate representation of x R 3 : x(r, ϕ, t) = r 1 t 2 cos(ϕ) r 1 t 2 sin(ϕ) rt where r = x R 0 is the distance to the origin, ϕ [0, 2π) the longitude and t = cos(θ) [ 1, 1] the polar distance and θ [0, π] the latitude. M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 2 / 11
The canonical basis in R 3 is denoted by ε 1, ε 2, ε 3 and another orthonormal basis is given by the moving frame consisting of the following three vectors (depending on the spherical coordinates ϕ and t): ε r (ϕ, t) = ε t (ϕ, t) = 1 t 2 cos(ϕ) 1 t 2 sin(ϕ) t t cos(ϕ) t sin(ϕ) 1 t 2., ε ϕ (ϕ, t) = sin(ϕ) cos(ϕ) 0, ε ϕ and ε t are tangential vectors. Note the vector product ε r ε ϕ = ε t. M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 3 / 11
The gradient in R 3 can be composed into a radial and an angular part, i.e. where = ε r r + 1 r = ε ϕ 1 1 t 2 ϕ + εt 1 t 2 t and the tangential operator is called surface gradient. Another tangential operator is the surface curl gradient L which is defined by L ξ F (ξ) = ξ ξ F (ξ) for F C (1) (S 2 ), ξ S 2, i.e. in local coordinates: L = ε ϕ 1 t 2 t + 1 εt 1 t 2 ϕ. M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 4 / 11
We canonically define the surface divergence and the surface curl ξ f (ξ) = L ξ f (ξ) = 3 ξ F i(ξ) ε i i=1 3 L ξ F i(ξ) ε i, i=1 where f = (F 1, F 2, F 3 ) T c (1) (S 2 ). Note that we use lower-case letters for vector fields and capital letters for scalar fields. The same convention applies to the corresponding function spaces such as c (1) (S 2 ) for the space of continuously differentiable vector fields and C (1) (S 2 ) for the space of continuously differentiable scalar fields on the sphere. The surface curl represents a scalar-valued function on the unit sphere: L ξ f (ξ) = ξ (f (ξ) ξ). M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 5 / 11
Finally, the Laplace operator can be decomposed into where = 2 r 2 + 2 r r + 1 r 2 ξ = t (1 t2 ) t + 1 1 t 2 2 ϕ 2 and denotes the Beltrami operator. It holds that = L L =. M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 6 / 11
Definition 4.1 (Regular Region on the Sphere) A bounded region S 2 is called regular, if its boundary is an orientable piecewise smooth Lipschitzian manifold of dimension 1. An example is a spherical cap of radius r around ξ S 2, i.e., C(ξ, r) = {η S 2 : 1 r ξ η 1}. Theorem 4.2 (Surface Theorems of Gauß and Stokes) Suppose that S 2 is a regular region with continuously differentiable boundary curve. Let f be a tangential vector field of class c (1) tan(), i.e., f (ξ) ξ = 0 for all ξ. Then, ξ f (ξ)ds(ξ) = ν ξ f (ξ)dσ(ξ), L ξ f (ξ)ds(ξ) = τ ξ f (ξ)dσ(ξ), where dσ is the arc element. M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 7 / 11
It is important to point out the assumption of the vector field f being tangential in Theorem 4.2. This causes an additional term in Green s formulas involving, but it does not affect those for L, which is due to the fact that ξ = 2, but L ξ = 0 for ξ S 2. The normal derivative is given by for F C (1) (). F ν (ξ) = ν ξ ξ F (ξ) = τ ξ L ξ F (ξ) M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 8 / 11
Lemma 4.3 Let S 2 be a regular region with continuously differentiable boundary. Suppose that F, G are of class C (1) (). Then, G(η) ηf (η)ds(η) + F (η) ηg(η)ds(η) = ν η (F (η)g(η))dσ(η) + 2 η(f (η)g(η))dσ(η), G(η)L ηf (η)ds(η) + F (η)l ηg(η)ds(η) = τ η (F (η)g(η))dσ(η). Green s surface identities (Theorem 4.4) are an immediate consequence of the surface Gauß theorem (Theorem 4.2). M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 9 / 11
Theorem 4.4 Scalar Spherical Harmonics Let G C (2) (), S 2 be a regular region with a continuously differentiable boundary and a unit outward normal vector field ν. Then, we have 1 Green s first surface identity for F C (1) (), i.e., ( ξ G(ξ) ) ( ξ F (ξ)) ds(ξ) + F (ξ) ξ G(ξ)dS(ξ) = F (ξ) ν G(ξ)dσ(ξ), 2 Green s second surface identity for F C (2) (), i.e., F (ξ) ξ G(ξ) G(ξ) ξ F (ξ)ds(ξ) = F (ξ) G(ξ) G(ξ) ν ν F (ξ)dσ(ξ). M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 10 / 11
The aforementioned statements (Lemma 4.3 and Theorem 4.4) hold as well for the entire sphere S 2 instead of a subregion, thereby observing that the occurring boundary integrals vanish. For functions F C (1) (S 2 ) and tangential vector fields f c tan(s (1) 2 ), this implies the following identities: f (η) ηf (η)ds(η) = F (η) η f (η)ds(η), S 2 S 2 f (η) L ηf (η)ds(η) = F (η)l η f (η)ds(η), S 2 S 2 η f (η)ds(η) = L η f (η)ds(η) = 0. S 2 S 2 M. Gutting (Uni Siegen) Spezielle Funktionen 28. Mai 2015 11 / 11