A system of curvilinear coordinates. There are several different conventions for the orientation and designation of these coordinates.
Arfken (1970) defines coordinates 
such that
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
In this work, following Morse and Feshbach (1953), the coordinates 
are used instead. In this convention, the traces of
the coordinate surfaces of the 
-plane are confocal parabolas
with a common axis. The 
curves open into the negative x-axis; the 
curves open into the positive x-axis. The 
and 
curves intersect along the
y-axis.
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
where 
,
,
and 
.
The scale factors are
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |
(10)
|
The Helmholtz differential equation is separable in parabolic cylindrical coordinates.
, 
, 
)." §2.8 in 
."
Table 1.04 in