= Vector calculus in elliptic cylindrical coordinates
{title2=$h=a\sqrt{\sinh^2u+\sin^2v}$}
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> are
$$
\nabla\cdot A=\frac{\partial_u(hA_u)+\partial_v(hA_v)+\partial_z(h^2A_z)}{h^2},
$$
$$
\nabla\times A=\frac{\partial_vA_z-\partial_z(hA_v)}h e_u+\frac{\partial_z(hA_u)-\partial_uA_z}h e_v+\frac{\partial_u(hA_v)-\partial_v(hA_u)}{h^2}e_z.
$$
Applying the <divergence> to the <gradient> of a scalar <function> gives $\Delta\phi=(\phi_{uu}+\phi_{vv})/h^2+\phi_{zz}$. These formulas hold where $h>0$. They use physical components along <unit vectors>, rather than components along the unnormalized coordinate <tangent vectors>.
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