Laplace operator (ラプラス演算子)

Δf=∂2f∂x2+∂2f∂y2+∂2f∂z2=1ρ∂∂ρ(ρ∂f∂ρ)+1ρ2∂2f∂θ2+∂2f∂z2=1r2∂∂r(r2∂f∂r)+1r2sinθ∂∂θ(sinθ∂f∂θ)+1r2sin2θ∂2f∂ϕ2\begin{aligned} \Delta f & = & \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \\ & = & {1 \over \rho} {\partial \over \partial \rho} \left( \rho {\partial f \over \partial \rho} \right) + {1 \over \rho^2} {\partial^2 f \over \partial \theta^2} + {\partial^2 f \over \partial z^2 } \\ & = & {1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial f \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \phi^2} \end{aligned}