z12a.nba 整理:VxP(xy少=~1 hh h3 auau aus 对标量函数(x1,x2,x3)=(1,a2,l3) V·(Vy)=V v·[via+V2l2+l3a3l 1[a(1h2h3)a(2hh).d(V3h1h2) h, hr h3I au h, h, h3 a Q柱坐标系和球坐标系中的 laplacian V 坐标 a( ap 182pa24 e apap) p2lad2)a32 Laplacian[φ[p,φ,z],(p,φ,z},"cy1 indica1"] FullSimplify [% 0(0,0,2)[p,φ,z]+-(φ10,2;0)[,中,z]+pφ(1;0,0[p,中,z])+φ2,0,0[p,中,z] 球坐标: x=rsin 0 cos d 故:{h2 rsin e a0) sin 0 a 82整理： ∇ V(x, y, z) = 1 h1 h2 h3 h1 u  1 h2 u  2 h3 u  3 ∂ ∂ u1 ∂ ∂ u2 ∂ ∂ u3 h1 V1 h2 V2 h3 V3 Laplacian ∇2： 对标量函数 φ(x1, x2, x3) = φ(u1, u2, u3)， ∇2 φ ≡ ∇ ·(∇φ) = ∇ · 1 h1 ∂ φ ∂ u1 V1 u  1 + 1 h2 ∂ φ ∂ u2 V3 u  2 + 1 h3 ∂ φ ∂ u3 V3 u  3 = ∇ ·[V1 u  1 + V2 u  2 + V3 u  3] = 1 h1 h2 h3 ∂ (V1 h2 h3) ∂ u1 + ∂ (V2 h3 h1) ∂ u2 + ∂ (V3 h1 h2) ∂ u3 = 1 h1 h2 h3 ∂ ∂ u1 h2 h3 h1 ∂ φ ∂ u1 + ∂ ∂ u2 h3 h1 h2 ∂ φ ∂ u2 + ∂ ∂ u3 h1 h2 h3 ∂ φ ∂ u3  柱坐标系和球坐标系中的 Laplacian ∇2 柱坐标： x = ρ cos ϕ y = ρ sin ϕ z = z 故： h1 = ∂ x ∂ ρ 2 + ∂ y ∂ ρ 2 + ∂ z ∂ ρ 2 1/2 = 1 h2 = ∂ x ∂ ϕ 2 + ∂ y ∂ ϕ 2 + ∂ z ∂ ϕ 2 1/2 = ρ h3 = ∂ x ∂ z 2 + ∂ y ∂ z 2 + ∂ z ∂ z 2 1/2 = 1 ∇2 φ = 1 ρ ∂ ∂ ρ ρ ∂ φ ∂ ρ + 1 ρ2 ∂2 φ ∂ ϕ2 + ∂2 φ ∂ z2 Laplacian[φ[ρ, ϕ, z], {ρ, ϕ, z}, "Cylindrical"]; FullSimplify[%] φ(0,0,2)[ρ, ϕ, z] + 1 ρ2 (φ(0,2,0)[ρ, ϕ, z] + ρ φ(1,0,0)[ρ, ϕ, z]) + φ(2,0,0)[ρ, ϕ, z] 球坐标： x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ 故： h1 = ∂ x ∂ r 2 + ∂ y ∂ r 2 + ∂ z ∂ r 2 1/2 = 1 h2 = ∂ x ∂ θ 2 + ∂ y ∂ θ 2 + ∂ z ∂ θ 2 1/2 = r h3 = ∂ x ∂ ϕ 2 + ∂ y ∂ ϕ 2 + ∂ z ∂ ϕ 2 1/2 = rsin θ ∇2 φ = 1 r2 sin θ sin θ ∂ ∂ r r2 ∂ φ ∂ r + ∂ ∂ θ sin θ ∂ φ ∂ θ + 1 sin θ ∂2 φ ∂ ϕ2 z12a.nb 7