Vector calculus in elliptic cylindrical coordinates ($h=a\sqrt{{\sinh }^{2}u+{\sin }^{2}v}$)

In elliptic cylindrical coordinates, write a vector field as $A=A_u{e}_u+A_v{e}_v+A_z{e}_z$ using the local orthonormal unit vectors. The scale factors of orthogonal coordinates are $(h,h,1)$, so the divergence and curl areApplying the divergence to the gradient of a scalar function gives $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/h^2+{\varphi }_{zz}$. These formulas hold where $h>0$. They use physical components along unit vectors, rather than components along the unnormalized coordinate tangent vectors.