Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 483consisting of f(0) and ˙ f(0) the initial state of f. We can now, for each ofthe cases n = 1, n = 3, and n = 2 describe fundamental solutions to thewave equations. By fundamental solutions, we mean solutions whose initialstate is either a constant times (δ(0, 0)), or a constant times (0, δ(0).In dimension 1 : The function R → D ′ c(R) given byt ↦→ S t (= S t )is a solution of the WE; its initial state is 2(δ(0), 0).The function R → D ′ c(R) given byt ↦→ B tis a solution of the WE with initial state 2(0, δ(0)).In dimension 3: The function R → D ′ c(R 3 ) given byt ↦→ S t + t 2 ∆(B t )is a solution of the WE with initial state 4πδ(0), 0). The function R →D ′ c(R 3 ) given byt ↦→ t · S tis a solution of the WE with initial state 4π(0, δ(0)).In dimension 2: The function R → D ′ c(R 2 ) given byt ↦→ p ∗ (S t + t 2 ∆(B t ))is a fundamental solution of the WE in dimension 2; its initial state is4π(δ(0), 0). The function R → D ′ c(R 2 ) given byt ↦→ p ∗ (t · S t )is a fundamental solution of the WE in dimension 2; its initial state is4π(0, δ(0)).