The parabolic cylindrical coordinates (σ, τ, z) are defined in terms of the Cartesian coordinates(x, y, z) by:
The surfaces of constant σ form confocal parabolic cylinders
that open towards +y, whereas the surfaces of constant τ form confocal parabolic cylinders
that open in the opposite direction, i.e., towards −y. The foci of all these parabolic cylinders are located along the line defined by x = y = 0. The radius r has a simple formula as well
Other differential operators can be expressed in the coordinates (σ, τ) by substituting the scale factors into the general formulae found in orthogonal coordinates.
Parabolic unit vectors expressed in terms of Cartesian unit vectors:
Parabolic cylinder harmonicsedit
Since all of the surfaces of constant σ, τ and z are conicoids, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written:
and Laplace's equation, divided by V, is written:
Since the Z equation is separate from the rest, we may write
where m is constant. Z(z) has the solution:
Substituting −m2 for , Laplace's equation may now be written:
We may now separate the S and T functions and introduce another constant n2 to obtain:
The parabolic cylinder harmonics for (m, n) are now the product of the solutions. The combination will reduce the number of constants and the general solution to Laplace's equation may be written:
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 658. ISBN 0-07-043316-X. LCCN 52011515.
Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 186–187. LCCN 55010911.
Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 181. LCCN 59014456. ASIN B0000CKZX7.
Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. p. 96. LCCN 67025285.
Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting uk for ξk.
Moon P, Spencer DE (1988). "Parabolic-Cylinder Coordinates (μ, ν, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 21–24 (Table 1.04). ISBN 978-0-387-18430-2.
External linksedit
MathWorld description of parabolic cylindrical coordinates