8z12a.nb Laplacian[φ【r,θ,中],(x,θ,中," Spherica1"] FullSimplify [9] 2(cct]2v00,2[x,O,的+cote]9010[x,,的]+ 0,20)[r,O,]+2r(,00[x,O,])+q20r,e, 取巧的方法: 柱坐标 sim1={x→pCos[中,Y→p8in[中}; sim3={ Arccos[c。s[中]]→φ t1=D[φ[p,中,z]//.sim2,{x,2}] t2=D[φ[p,中,z]//.sim2,{y,2}]; t3=D[φ[p,中,z]//.sim2,{z,2}]; t= Simplify[(tl + t2+t3)/. siml, [p>0, E Reals]] t= Simplify[t//. sim3] / TraditionalForm y(0,2,)(p, =) y(,0,o)(e, =) +p2o)(p,b,) p 七较V2g 1an).1(22 球坐标 sim1={x→rsin[e]cos[中,y→rsin[e]sin[中,z→rc。s[e] sim2: +V2+y2+2, p+Vx+y2, e+Arccos[=]++arccos Il im3=[ ArcCos[cos[e]]→e, ArCCos[cos[中]]→φ}; t1=D[φ[r,日,中]//.sim2,{x,2}]; t2=D[φ[r,θ,中]//.sim2,{Y,2}]; t3=D[φ[r,日,中]//.sim2,{z,2}]; = Simplify[(t1+t2+t3)/.sim1,(r>0,e∈Rea1s,φ∈Rea1s}] t= Simplify[t//.sim3,{x>0,0<θ<丌,0sφs2丌}] t=u11 Simplify[t//.sim3,r>0,0<e<丌,0≤φ≤2x}]// Traditiona1Form 02Vr,B,的)+2r0,B,d)+cot(yup(r,B,d)+csc2()y002r,B,d)+20r,B,d) a0)sin 0 a62Laplacian[φ[r, θ, ϕ], {r, θ, ϕ}, "Spherical"]; FullSimplify[%] 1 r2 Csc[θ]2 φ(0,0,2)[r, θ, ϕ] + Cot[θ] φ(0,1,0)[r, θ, ϕ] + φ(0,2,0)[r, θ, ϕ] + 2 r φ(1,0,0)[r, θ, ϕ] + φ(2,0,0)[r, θ, ϕ] 取巧的方法： 柱坐标： sim1 = {x  ρ Cos[ϕ], y  ρ Sin[ϕ]}; sim2 = ρ  x2 + y2 , ϕ  ArcCos x ρ ; sim3 = {ArcCos[Cos[ϕ]]  ϕ}; t1 = D[φ[ρ, ϕ, z] //. sim2, {x, 2}]; t2 = D[φ[ρ, ϕ, z] //. sim2, {y, 2}]; t3 = D[φ[ρ, ϕ, z] //. sim2, {z, 2}]; t = Simplify[(t1 + t2 + t3) /. sim1, {ρ > 0, ϕ ∈ Reals}]; t = Simplify[t //. sim3] // TraditionalForm φ(0,2,0) (ρ, ϕ, z) ρ2 + φ(0,0,2) (ρ, ϕ, z) + φ(1,0,0) (ρ, ϕ, z) ρ + φ(2,0,0) (ρ, ϕ, z) 比较 ∇2 φ = 1 ρ ∂ ∂ ρ ρ ∂ φ ∂ ρ + 1 ρ2 ∂2 φ ∂ ϕ2 + ∂2 φ ∂ z2 球坐标： sim1 = {x  r Sin[θ] Cos[ϕ], y  r Sin[θ] Sin[ϕ], z  r Cos[θ]}; sim2 = r  x2 + y2 + z2 , ρ  x2 + y2 , θ  ArcCos z r  , ϕ  ArcCos x ρ ; sim3 = {ArcCos[Cos[θ]]  θ, ArcCos[Cos[ϕ]]  ϕ}; t1 = D[φ[r, θ, ϕ] //. sim2, {x, 2}]; t2 = D[φ[r, θ, ϕ] //. sim2, {y, 2}]; t3 = D[φ[r, θ, ϕ] //. sim2, {z, 2}]; t = Simplify[(t1 + t2 + t3) /. sim1, {r > 0, θ ∈ Reals, ϕ ∈ Reals}]; t = Simplify[t //. sim3, {r > 0, 0 < θ < π, 0 ≤ ϕ ≤ 2 π}]; t = FullSimplify[t //. sim3, {r > 0, 0 < θ < π, 0 ≤ ϕ ≤ 2 π}] // TraditionalForm 1 r2 φ(0,2,0) (r, θ, ϕ) + 2 r φ(1,0,0) (r, θ, ϕ) + cot(θ) φ(0,1,0) (r, θ, ϕ) + csc2(θ) φ(0,0,2) (r, θ, ϕ) + φ(2,0,0) (r, θ, ϕ) 比较：∇2 φ = 1 r2 sin θ sin θ ∂ ∂ r r2 ∂ φ ∂ r + ∂ ∂ θ sin θ ∂ φ ∂ θ + 1 sin θ ∂2 φ ∂ ϕ2 8 z12a.nb