本文探讨了在球坐标系统下如何应用拉普拉斯算子（Δ）。通过分解算子并分别对径向、纬度和经度方向进行偏导数运算，展示了求解球坐标中函数梯度和拉普拉斯方程的方法。
球坐标下的Laplace
∇=r^∂r+θ^1r∂θ+ϕ^1rsinθ∂ϕΔφ=∇⋅∇φ=∇⋅(r^∂rφ+θ^r∂θφ+ϕ^rsinθ∂ϕφ)=∇(r2sinθ∂rφ)⋅r^r2sinθ+∇(sinθ∂θφ)⋅θ^rsinθ+∇(1sinθ∂ϕφ)⋅ϕ^r=1r2∂r(r2∂rφ)+1r2sinθ∂θ(sinθ∂θφ)+1r2sin2θ∂ϕ∂ϕφ
\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
∇=r^∂r+θ^r1∂θ+ϕ^rsinθ1∂ϕΔφ=∇⋅∇φ=∇⋅(r^∂rφ+rθ^∂θφ+rsinθϕ^∂ϕφ)=∇(r2sinθ∂rφ)⋅r2sinθr^+∇(sinθ∂θφ)⋅rsinθθ^+∇(sinθ1∂ϕφ)⋅rϕ^=r21∂r(r2∂rφ)+r2sinθ1∂θ(sinθ∂θφ)+r2sin2θ1∂ϕ∂ϕφ
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