Parabolic cylindrical coordinates

File:Parabolic coords.svg

The parabolic cylindrical coordinates {{math|(σ, τ, z)}} are defined in terms of the Cartesian coordinates {{math|(x, y, z)}} by:

:$\backslash begin\{align\}$

x &= \sigma \tau \\

y &= \frac{1}{2} \left( \tau^2 - \sigma^2 \right) \\

z &= z

\end{align}

The surfaces of constant {{math|σ}} form confocal parabolic cylinders

:$$

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

that open towards {{math|+y}}, whereas the surfaces of constant {{math|τ}} form confocal parabolic cylinders

:$$

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

that open in the opposite direction, i.e., towards {{math|−y}}. The foci of all these parabolic cylinders are located along the line defined by {{math|x {{=}} y {{=}} 0}}. The radius {{math|r}} has a simple formula as well

:$$

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

that proves useful in solving the Hamilton–Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics; for further details, see the Laplace–Runge–Lenz vector article.

The scale factors for the parabolic cylindrical coordinates {{math|σ}} and {{math|τ}} are:

:$\backslash begin\{align\}$

h_\sigma &= h_\tau = \sqrt{\sigma^2 + \tau^2} \\

h_z &= 1

\end{align}

The infinitesimal element of volume is

:$dV\; =\; h\_\backslash sigma\; h\_\backslash tau\; h\_z\; d\backslash sigma\; d\backslash tau\; dz\; =\; (\; \backslash sigma^2\; +\; \backslash tau^2\; )\; d\backslash sigma\; \backslash ,\; d\backslash tau\; \backslash ,\; dz$

The differential displacement is given by:

:$d\backslash mathbf\{l\}\; =\; \backslash sqrt\{\backslash sigma^2\; +\; \backslash tau^2\}\; \backslash ,\; d\backslash sigma\; \backslash ,\; \backslash boldsymbol\{\backslash hat\{\backslash sigma\}\}\; +\; \backslash sqrt\{\backslash sigma^2\; +\; \backslash tau^2\}\; \backslash ,\; d\backslash tau\; \backslash ,\; \backslash boldsymbol\{\backslash hat\{\backslash tau\}\}\; +\; dz\; \backslash ,\; \backslash mathbf\{\backslash hat\{z\}\}$

The differential normal area is given by:

:$d\backslash mathbf\{S\}\; =\; \backslash sqrt\{\backslash sigma^2\; +\; \backslash tau^2\}\; \backslash ,\; d\backslash tau\; \backslash ,\; dz\; \backslash boldsymbol\{\backslash hat\{\backslash sigma\}\}\; +\; \backslash sqrt\{\backslash sigma^2\; +\; \backslash tau^2\}\; \backslash ,\; d\backslash sigma\; \backslash ,\; dz\; \backslash boldsymbol\{\backslash hat\{\backslash tau\}\}\; +\; \backslash left(\backslash sigma^2\; +\; \backslash tau^2\backslash right)\; \backslash ,\; d\backslash sigma\; \backslash ,\; d\backslash tau\; \backslash mathbf\{\backslash hat\{z\}\}$

Let {{math|f}} be a scalar field. The gradient is given by

:$\backslash nabla\; f\; =\; \backslash frac\{1\}\{\backslash sqrt\{\backslash sigma^\{2\}\; +\; \backslash tau^\{2\}\}\}\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; \backslash sigma\}\backslash boldsymbol\{\backslash hat\{\backslash sigma\}\}\; +\; \backslash frac\{1\}\{\backslash sqrt\{\backslash sigma^\{2\}\; +\; \backslash tau^\{2\}\}\}\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; \backslash tau\}\backslash boldsymbol\{\backslash hat\{\backslash tau\}\}\; +\; \{\backslash partial\; f\; \backslash over\; \backslash partial\; z\}\backslash mathbf\{\backslash hat\{z\}\}$

The Laplacian is given by

:$\backslash nabla^2\; f\; =\; \backslash frac\{1\}\{\backslash sigma^\{2\}\; +\; \backslash tau^\{2\}\}$

\left(\frac{\partial^{2} f}{\partial \sigma^{2}} +

\frac{\partial^{2} f}{\partial \tau^{2}} \right) +

\frac{\partial^{2} f}{\partial z^{2}}

Let {{math|A}} be a vector field of the form:

:$\backslash mathbf\; A\; =\; A\_\backslash sigma\; \backslash boldsymbol\{\backslash hat\{\backslash sigma\}\}\; +\; A\_\backslash tau\; \backslash boldsymbol\{\backslash hat\{\backslash tau\}\}\; +\; A\_z\; \backslash mathbf\{\backslash hat\{z\}\}$

The divergence is given by

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

The curl is given by

:$\backslash nabla\; \backslash times\; \backslash mathbf\; A\; =$

\left(

\frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \tau}

- \frac{\partial A_\tau}{\partial z}

\right) \boldsymbol{\hat{\sigma}}

- \left(

\frac{1}{\sqrt{\sigma^2 + \tau^2}} \frac{\partial A_z}{\partial \sigma}

- \frac{\partial A_\sigma}{\partial z}

\right) \boldsymbol{\hat{\tau}}

+ \frac{1}{\sigma^2 + \tau^2} \left(

\frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\tau \right)}{\partial \sigma}

- \frac{\partial \left(\sqrt{\sigma^2 + \tau^2} A_\sigma\right)}{\partial \tau}

\right) \mathbf{\hat{z}}

Other differential operators can be expressed in the coordinates {{math|(σ, τ)}} by substituting the scale factors into the general formulae found in orthogonal coordinates.

Relationship to cylindrical coordinates {{math|(ρ, φ, z)}}:

:$\backslash begin\{align\}$

\rho\cos\varphi &= \sigma \tau\\

\rho\sin\varphi &= \frac{1}{2} \left( \tau^2 - \sigma^2 \right) \\

z &= z \end{align}

Parabolic unit vectors expressed in terms of Cartesian unit vectors:

:$\backslash begin\{align\}$

\boldsymbol{\hat{\sigma}} &= \frac{\tau \hat{\mathbf x} - \sigma \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\

\boldsymbol{\hat{\tau}} &= \frac{\sigma \hat{\mathbf x} + \tau \hat{\mathbf y}}{\sqrt{\tau^2+\sigma^2}} \\

\mathbf{\hat{z}} &= \mathbf{\hat{z}}

\end{align}

Since all of the surfaces of constant {{math|σ}}, {{math|τ}} and {{math|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:

:$V\; =\; S(\backslash sigma)\; T(\backslash tau)\; Z(z)$

and Laplace's equation, divided by {{math|V}}, is written:

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

Since the {{math|Z}} equation is separate from the rest, we may write

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

where {{math|m}} is constant. {{math|Z(z)}} has the solution:

:$Z\_m(z)=A\_1\backslash ,e^\{imz\}+A\_2\backslash ,e^\{-imz\}$

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

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

We may now separate the {{math|S}} and {{math|T}} functions and introduce another constant {{math|n²}} to obtain:

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

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

The solutions to these equations are the parabolic cylinder functions

:$S\_\{mn\}(\backslash sigma)\; =\; A\_3\; y\_1(n^2\; /\; 2m,\; \backslash sigma\; \backslash sqrt\{2m\})\; +\; A\_4\; y\_2(n^2\; /\; 2m,\; \backslash sigma\; \backslash sqrt\{2m\})$

:$T\_\{mn\}(\backslash tau)\; =\; A\_5\; y\_1(n^2\; /\; 2m,\; i\; \backslash tau\; \backslash sqrt\{2m\})\; +\; A\_6\; y\_2(n^2\; /\; 2m,\; i\; \backslash tau\; \backslash sqrt\{2m\})$

The parabolic cylinder harmonics for {{math|(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:

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

The classic applications of parabolic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which such coordinates allow a separation of variables. A typical example would be the electric field surrounding a flat semi-infinite conducting plate.

• Parabolic coordinates
• Orthogonal coordinate system
• Curvilinear coordinates

• {{cite book | author = Morse PM, Feshbach H | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X|lccn=52011515 | page = 658}}
• {{cite book | author = Margenau H, Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | url = https://archive.org/details/mathematicsofphy0002marg| url-access = registration| publisher = D. van Nostrand | location = New York | pages = [https://archive.org/details/mathematicsofphy0002marg/page/186 186]–187 | lccn = 55010911 }}
• {{cite book | author = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | page = 181 | lccn = 59014456}}
• {{cite book | author = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 96 | lccn = 67025285}}
• {{cite book | author = Zwillinger D | year = 1992 | title = Handbook of Integration | publisher = Jones and Bartlett | location = Boston, MA | isbn = 0-86720-293-9 | page = 114}} Same as Morse & Feshbach (1953), substituting u_k for ξ_k.
• {{cite book | author = Moon P, Spencer DE | year = 1988 | chapter = Parabolic-Cylinder Coordinates (μ, ν, z) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print | publisher = Springer-Verlag | location = New York | pages = 21–24 (Table 1.04) | isbn = 978-0-387-18430-2}}

• [http://mathworld.wolfram.com/ParabolicCylindricalCoordinates.html MathWorld description of parabolic cylindrical coordinates]

{{Orthogonal coordinate systems}}

Category:Three-dimensional coordinate systems

Category:Orthogonal coordinate systems