Elliptic cylindrical coordinates ($x=a\cosh u\cos v,y=a\sinh u\sin v$)

For $a>0$, $u\ge 0$, $0\le v<2\pi$ and real $z$, these orthogonal coordinates have coordinate tangents $r_u=(a\sinh u\cos v,a\cosh u\sin v,0)$, $r_v=(-a\cosh u\sin v,a\sinh u\cos v,0)$ and $r_z=(0,0,1)$. Their scale factors of orthogonal coordinates are $h_u=h_v=a\sqrt{{\sinh }^{2}u+{\sin }^{2}v}$ and $h_z=1$; the ordered Jacobian determinant is positive away from the two focal lines. Positive constant $u$ gives confocal elliptical cylinders, while generic constant $v$ gives a hyperbolic-cylinder half-branch selected by the signs of $x$ and $y$. The limiting $u=0$ surface is the strip $y=0$, $|x|\le a$, with arbitrary $z$. Constant $z$ gives horizontal planes.

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.