円筒座標

円筒座標$(r,\theta,z)$では $(h_1,h_2,h_3)=(1,r,1)$だから

$$\bigtriangleup = \frac 1r \{ \partial_r(r\partial_r) +\partial_\theta(\frac 1r\partial_\theta) +\partial_z(r\partial_z) \}$$ (2.14)

$$= \frac 1r\frac\partial{\partial r}r\frac\partial{\partial r} +\frac 1{r^2} \frac{\partial^2}{\partial\theta^2} +\frac{\partial^2}{\partial z^2}$$ (2.15)