Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 479Note that S 1 = S 1 and B 1 = B 1 . And also note that the definition ofS t and B t can be formulated asS t = H t (S 1 ) , B t = H t (B 1 ),where H t : R n → R n is the homothetic transformation u ↦→ t · u, and wherewe are using the covariant functoriality of distributions of compact support.For low dimensions, we shall describe the distributions S t , B t , S t andB t explicitly:Dimension 1:Dimension 2:Dimension 3:< S t , ψ >=< B t , ψ >=< S t , ψ >=< B t , ψ >=< S t , ψ >= ψ(−t) + ψ(t), < B t , ψ >=< S t , ψ >= ψ(−t) + ψ(t), < B t , ψ >=< S t , ψ >=< B t , ψ >=< S t , ψ >=< B t , ψ >=∫ π ∫ 2π0 0∫ t ∫ π ∫ 2π00∫ π ∫ 2π∫ 2π0∫ t ∫ 2π0 0∫ 2π0∫ 1 ∫ 2π0ψ(t cos θ, t sin θ) t dθ,∫ t−t∫ 1−1ψ(s cos θ, s sin θ) s dθ ds,ψ(t cos θ, t sin θ) dθ,0ψ(s) ds,ψ(t · s) ds.ψ(ts cos θ, t s sin θ) s dθ ds.ψ(t cos θ sin φ, t sin θ sin φ, t cos φ)t 2 sin φ dθ dφ,00 0∫ 1 ∫ π ∫ 2π00ψ(s cos θ sin φ, s sin θ sin φ, s cos φ) s 2 sin φ dθ dφ ds,ψ(t cos θ sin φ, t sin θ sin φ, t cos φ) sin φ dθ dφ,0ψ(ts cos θ sin φ, ts sin θ sin φ, ts cos φ) s 2 sin φ dθ dφ ds.These formulas make sense for all t, whereas set–theoretically S t and B t(as point sets) only make good sense for t > 0.