極座標のホッジラプラシアン

以下の数式が正しいことをホッジラプラシアンを展開することにより確かめます。

`Delta = d delta + delta d`
`u_(x x) + u_(y y) = u_(r r) + (1 / r) u_r + (1 / r^2) u_(theta theta)` -- Parameters and metrics def N := 2 def x := [|r, θ|] def g_i_j := [| [| 1, 0 |], [| 0, r^2 |] |]_i_j def g~i~j := [| [| 1, 0 |], [| 0, 1 / r^2 |] |]~i~j -- Hodge Laplacian def d %A := !(flip ∂/∂) x A def hodge %A := let k := dfOrder A in withSymbols [i, j] (sqrt (abs (M.det g_#_#))) * (foldl (.) ((ε' N k)_(i_1)..._(i_N) . A..._(j_1)..._(j_k)) (map 1#g~(i_%1)~(j_%1) [1..k])) def δ %A := let k := dfOrder A in -1^(N * (k + 1) + 1) * (hodge (d (hodge A))) def Δ %A := match (dfOrder A) as integer with | #0 -> δ (d A) | #N -> d (δ A) | _ -> d (δ A) + δ (d A) def f := function (r, θ) assertEqual "exterior derivative" (d f) [| ∂/∂ f r, ∂/∂ f θ |] assertEqual "hodge operator" (hodge (d f)) [| (-1 * ∂/∂ f θ) / r, r * (∂/∂ f r) |] assertEqual "Laplacian" (Δ f) ((-1 / r^2) * ((∂/∂ (∂/∂ f θ) θ) + r * (∂/∂ f r) + (r^2 * (∂/∂ (∂/∂ f r) r))))

リンク

Egison 数学ノート目次に戻る