Inmathematics,parabolic cylindrical coordinates are a three-dimensionalorthogonalcoordinate system that results from projecting the two-dimensionalparabolic coordinate system in theperpendicular${\displaystyle z}$-direction. Hence, thecoordinate surfaces areconfocalparabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., thepotential theory of edges.

The parabolic cylindrical coordinates(σ,τ,z) are defined in terms of theCartesian coordinates(x,y,z) by:

The surfaces of constantσ form confocal parabolic cylinders

${\displaystyle 2y=\frac{x^2}{{\sigma }^2}-{\sigma }^2}$

that open towards+y, whereas the surfaces of constantτ form confocal parabolic cylinders

${\displaystyle 2y=-\frac{x^2}{{\tau }^2}+{\tau }^2}$

that open in the opposite direction, i.e., towards−y. The foci of all these parabolic cylinders are located along the line defined byx =y = 0. The radiusr has a simple formula as well

${\displaystyle r=\sqrt{x^2+y^2}=\frac{1}{2}\left({\sigma }^2+{\tau }^2\right)}$

that proves useful in solving theHamilton–Jacobi equation in parabolic coordinates for theinverse-squarecentral force problem ofmechanics; for further details, see theLaplace–Runge–Lenz vector article.

The scale factors for the parabolic cylindrical coordinatesσ andτ are:

The infinitesimal element of volume is

${\displaystyle dV=h_{\sigma }{h}_{\tau }{h}_{z}d\sigma d\tau dz=({\sigma }^2+{\tau }^2)d\sigma d\tau dz}$

The differential displacement is given by:

${\displaystyle d\mathbf{l}=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \hat{\mathit{\tau }}+dz\hat{\mathbf{z}}}$

The differential normal area is given by:

${\displaystyle d\mathbf{S}=\sqrt{{\sigma }^2+{\tau }^2}d\tau dz\hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\sigma dz\hat{\mathit{\tau }}+\left({\sigma }^2+{\tau }^2\right)d\sigma d\tau \hat{\mathbf{z}}}$

Letf be a scalar field. Thegradient is given by

${\displaystyle \mathrm{\nabla }f=\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\mathrm{\partial }f}{\mathrm{\partial }\sigma }\hat{\mathit{\sigma }}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\mathrm{\partial }f}{\mathrm{\partial }\tau }\hat{\mathit{\tau }}+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\hat{\mathbf{z}}}$

TheLaplacian is given by

${\displaystyle {\mathrm{\nabla }}^{2}f=\frac{1}{{\sigma }^2+{\tau }^2}\left(\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\sigma }^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{\tau }^2}\right)+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2}}$

LetA be a vector field of the form:

${\displaystyle \mathbf{A}=A_{\sigma }\hat{\mathit{\sigma }}+A_{\tau }\hat{\mathit{\tau }}+A_z\hat{\mathbf{z}}}$

Thedivergence is given by

${\displaystyle \mathrm{\nabla }\cdot \mathbf{A}=\frac{1}{{\sigma }^2+{\tau }^2}\left(\frac{\mathrm{\partial }(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma })}{\mathrm{\partial }\sigma }+\frac{\mathrm{\partial }(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau })}{\mathrm{\partial }\tau }\right)+\frac{\mathrm{\partial }{A}_z}{\mathrm{\partial }z}}$

Thecurl is given by

${\displaystyle \mathrm{\nabla }\times \mathbf{A}=\left(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\mathrm{\partial }{A}_z}{\mathrm{\partial }\tau }-\frac{\mathrm{\partial }{A}_{\tau }}{\mathrm{\partial }z}\right)\hat{\mathit{\sigma }}-\left(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\mathrm{\partial }{A}_z}{\mathrm{\partial }\sigma }-\frac{\mathrm{\partial }{A}_{\sigma }}{\mathrm{\partial }z}\right)\hat{\mathit{\tau }}+\frac{1}{{\sigma }^2+{\tau }^2}\left(\frac{\mathrm{\partial }\left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau }\right)}{\mathrm{\partial }\sigma }-\frac{\mathrm{\partial }\left(\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma }\right)}{\mathrm{\partial }\tau }\right)\hat{\mathbf{z}}}$

Other differential operators can be expressed in the coordinates(σ,τ) by substituting the scale factors into the general formulae found inorthogonal coordinates.

Relationship tocylindrical coordinates(ρ,φ,z):

Parabolic unit vectors expressed in terms of Cartesian unit vectors:

Since all of the surfaces of constantσ,τ andz areconicoids, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of theseparation of variables, a separated solution to Laplace's equation may be written:

${\displaystyle V=S(\sigma )T(\tau )Z(z)}$

and Laplace's equation, divided byV, is written:

${\displaystyle \frac{1}{{\sigma }^2+{\tau }^2}\left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right]+\frac{\ddot{Z}}{Z}=0}$

Since theZ equation is separate from the rest, we may write

${\displaystyle \frac{\ddot{Z}}{Z}=-m^2}$

wherem is constant.Z(z) has the solution:

${\displaystyle Z_m(z)=A_1{e}^{imz}+A_2{e}^{-imz}}$

Substituting−m² for${\displaystyle \ddot{Z}/Z}$, Laplace's equation may now be written:

${\displaystyle \left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right]=m^2({\sigma }^2+{\tau }^2)}$

We may now separate theS andT functions and introduce another constantn² to obtain:

${\displaystyle \ddot{S}-(m^2{\sigma }^2+n^2)S=0}$

${\displaystyle \ddot{T}-(m^2{\tau }^2-n^2)T=0}$

The solutions to these equations are theparabolic cylinder functions

${\displaystyle S_{mn}(\sigma )=A_3{y}_1(n^2/2m,\sigma \sqrt{2m})+A_4{y}_2(n^2/2m,\sigma \sqrt{2m})}$

${\displaystyle T_{mn}(\tau )=A_5{y}_1(n^2/2m,i\tau \sqrt{2m})+A_6{y}_2(n^2/2m,i\tau \sqrt{2m})}$

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:

${\displaystyle V(\sigma ,\tau ,z)=\sum _{m,n}{A}_{mn}{S}_{mn}{T}_{mn}{Z}_m}$

The classic applications of parabolic cylindrical coordinates are in solvingpartial differential equations, e.g.,Laplace's equation or theHelmholtz equation, for which such coordinates allow aseparation of variables. A typical example would be theelectric field surrounding a flat semi-infinite conducting plate.

• Parabolic coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

• Morse PM,Feshbach H (1953).Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 658.ISBN 0-07-043316-X.LCCN 52011515.{{cite book}}:ISBN / Date incompatibility (help)
• 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), substitutingu_k for ξ_k.
• 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.

• MathWorld description of parabolic cylindrical coordinates

• Three-dimensional coordinate systems
• Orthogonal coordinate systems

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• Short description matches Wikidata
• CS1 errors: ISBN date