Parabolic cylindrical coordinates
Parabolic cylindrical coordinates extend parabolic coordinates to 3D,
by describing a point in space using the variables .
The -axis is the same as in the Cartesian system, (hence the name cylindrical),
while the coordinate lines of and are confocal parabolas.
and this system are related by:
Conversely, a point given in can be converted
to using these formulae:
Parabolic cylindrical coordinates form
an orthogonal curvilinear system,
whose scale factors , and we need.
To get those, we calculate the unnormalized local basis:
By normalizing the local basis vectors
, and ,
we arrive at these expressions,
where we have defined the abbreviation for convenience:
Thanks to these scale factors, we can easily convert calculus from the Cartesian system
using the standard formulae for orthogonal curvilinear coordinates.
For line integrals,
the tangent vector element for a curve is as follows:
For surface integrals,
the normal vector element for a surface is given by:
And for volume integrals,
the infinitesimal volume takes the following form:
The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
Uncommon operations include:
the gradient of a divergence ,
the gradient of a vector ,
the advection of a vector with respect to ,
the Laplacian of a vector ,
and the divergence of a 2nd-order tensor :
- M.L. Boas,
Mathematical methods in the physical sciences, 2nd edition,