[
P
x
P
y
]
=
[
c
θ
−
s
θ
s
θ
c
θ
]
[
P
r
R
θ
]
[PxPy]
所以 P 2 x = ( P r c θ − R θ s θ ) 2 = P 2 r c 2 θ − P r R θ ( 2 s θ c θ ) + R r s 2 θ + R θ ( 2 s θ c θ ) + R 2 θ s 2 θ P^2x=(Prc\theta-R\theta s\theta)^2=P^2rc^2\theta-PrR\theta(2s\theta c\theta)+Rrs^2\theta +\mathbb{R}\theta (2s\theta c\theta)+R^2\theta s^2\theta P2x=(Prcθ−Rθsθ)2=P2rc2θ−PrRθ(2sθcθ)+Rrs2θ+Rθ(2sθcθ)+R2θs2θ
重调和方程就是 ( P 2 r + R r + R 2 θ ) 2 ϕ = 0 (P^2r+Rr+R^2\theta)^2\phi=0 (P2r+Rr+R2θ)2ϕ=0
拉普拉阿斯算子 ∇ 2 = P 2 r + R r + R 2 θ \nabla^2=P^2r+Rr+R^2\theta ∇2=P2r+Rr+R2θ