= Parabolic cylindrical coordinates
{title2=$x=u^2-v^2,\quad y=2uv,\quad z=w$}
These <orthogonal coordinates> extend two confocal families of <parabolas> parallel to an axis. In the displayed scaling their <scale factors of orthogonal coordinates> are $h_u=h_v=2\sqrt{u^2+v^2}$ and $h_w=1$, so the <Laplacian> is $\Delta\phi=(\phi_{uu}+\phi_{vv})/[4(u^2+v^2)]+\phi_{ww}$ away from $u=v=0$. With $u\geq0$, signed $v$ distinguishes the upper and lower half-planes; the two signs at $u=0$ describe the same negative $x$ half-axis. Fixed positive $u^2$ or $v^2$ describes a <parabolic cylinder>. This differs from the rotational three-dimensional <parabolic coordinates>, whose coordinate surfaces are <elliptic paraboloids>.
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