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球坐标下的Laplace ∇ = r ^ ∂ r + θ ^ 1 r ∂ θ + ϕ ^ 1 r sin θ ∂ ϕ Δ φ = ∇ ⋅ ∇ φ = ∇ ⋅ ( r ^ ∂ r φ + θ ^ r ∂ θ φ + ϕ ^ r sin θ ∂ ϕ φ ) = ∇ ( r 2 sin θ ∂ r φ ) ⋅ r ^ r 2 sin θ + ∇ ( sin θ ∂ θ φ ) ⋅ θ ^ r sin θ + ∇ ( 1 sin θ ∂ ϕ φ ) ⋅ ϕ ^ r = 1 r 2 ∂ r ( r 2 ∂ r φ ) + 1 r 2 sin θ ∂ θ ( sin θ ∂ θ φ ) + 1 r 2 sin 2 θ ∂ ϕ ∂ ϕ φ \nabla=\hat{r}\partial_r+\hat{\theta}\frac{1}{r}\partial_\theta+\hat{\phi}\frac{1}{r\sin\theta}\partial_\phi\\ \Delta\varphi=\nabla\cdot\nabla\varphi\\ =\nabla\cdot(\hat{r}\partial_r\varphi+\frac{\hat{\theta}}{r}\partial_\theta\varphi+\frac{\hat{\phi}}{r\sin\theta}\partial_\phi\varphi)\\ =\nabla(r^2\sin\theta\partial_r\varphi)\cdot\frac{\hat{r}}{r^2\sin\theta}+\nabla(\sin\theta\partial_\theta\varphi)\cdot\frac{\hat{\theta}}{r\sin\theta}+\nabla(\frac{1}{\sin\theta}\partial_\phi\varphi)\cdot\frac{\hat{\phi}}{r}\\ =\frac{1}{r^2}\partial_r(r^2\partial_r\varphi)+\frac{1}{r^2\sin\theta}\partial_\theta(\sin\theta\partial_\theta\varphi)+\frac{1}{r^2\sin^2\theta}\partial_\phi\partial_\phi\varphi ∇=
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