Paraboloidal coordinates

The Cartesian coordinates $(x,\; y,\; z)$ can be produced from the ellipsoidal coordinates

$(\backslash mu,\; \backslash nu,\; \backslash lambda)$ by the equations{{Citation

| last1 = Yoon

| first1 = LCLY

| last2 = M

| first2 = Willatzen

| title = Separable Boundary-Value Problems in Physics

| publisher = Wiley-VCH

| year = 2011

| isbn = 978-3-527-63492-7

| page = 217

}}

:$$

x^{2} = \frac{4}{b-c}(\mu - b)(b - \nu)(b - \lambda)

:$$

y^{2} = \frac{4}{b-c}(\mu - c)(c - \nu)(\lambda - c)

:$$

z = \mu + \nu + \lambda - b - c

with

:$$

\mu > b > \lambda > c > \nu > 0

Consequently, surfaces of constant $\backslash mu$ are downward opening elliptic paraboloids:

:$$

\frac{x^2}{\mu - b} + \frac{y^2}{\mu - c} = -4 (z - \mu)

Similarly, surfaces of constant $\backslash nu$ are upward opening elliptic paraboloids,

:$$

\frac{x^2}{b - \nu} + \frac{y^2}{c - \nu} = 4 (z - \nu)

whereas surfaces of constant $\backslash lambda$ are hyperbolic paraboloids:

:$$

\frac{x^2}{b - \lambda} - \frac{y^2}{\lambda - c} = 4 (z - \lambda)

The scale factors for the paraboloidal coordinates $(\backslash mu,\; \backslash nu,\; \backslash lambda)$ areWillatzen and Yoon (2011), p. 219

:$$

h_{\mu} = \left[ \frac{\left(\mu - \nu \right) \left(\mu - \lambda \right)}{ \left(\mu - b \right) \left(\mu - c \right)} \right]^{1/2}

:$$

h_{\nu} = \left[\frac{\left(\mu - \nu \right) \left( \lambda - \nu \right)}{ \left(b - \nu \right) \left(c - \nu \right)} \right]^{1/2}

:$$

h_{\lambda} = \left[\frac{\left(\lambda - \nu \right) \left(\mu - \lambda \right)}{ \left(b - \lambda \right) \left(\lambda - c \right)} \right]^{1/2}

Hence, the infinitesimal volume element is

:$$

dV = \frac{(\mu - \nu)(\mu - \lambda)(\lambda - \nu)}{\left[(\mu - b)(\mu - c)(b - \nu)(c - \nu)(b - \lambda)(\lambda - c) \right]^{1/2}} \ d\lambda d\mu d\nu

Common differential operators can be expressed in the coordinates $(\backslash mu,\; \backslash nu,\; \backslash lambda)$ by substituting the scale factors into the general formulas for these operators, which are applicable to any three-dimensional orthogonal coordinates. For instance, the gradient operator is

:$$

\nabla = \left[\frac{ \left(\mu - b \right) \left(\mu - c \right)}{\left(\mu - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \mathbf{e}_{\mu} \frac{\partial}{\partial \mu} + \left[\frac{\left(b - \nu \right) \left(c - \nu \right)}{\left(\mu - \nu \right) \left( \lambda - \nu \right)} \right]^{1/2} \mathbf{e}_{\nu} \frac{\partial}{\partial \nu} + \left[\frac{\left(b - \lambda \right) \left(\lambda - c \right)}{\left(\lambda - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \mathbf{e}_{\lambda} \frac{\partial}{\partial \lambda}

and the Laplacian is

:$$

\begin{align}

\nabla^2 = &\left[\frac{ \left(\mu - b \right) \left(\mu - c \right)}{\left(\mu - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \frac{\partial}{\partial \mu} \left[(\mu - b)^{1/2}(\mu - c)^{1/2} \frac{\partial}{\partial \mu} \right] \\

&+ \left[\frac{\left(b - \nu \right) \left(c - \nu \right)}{\left(\mu - \nu \right) \left( \lambda - \nu \right)} \right]^{1/2} \frac{\partial}{\partial \nu} \left[ (b - \nu)^{1/2}(c - \nu)^{1/2}\frac{\partial}{\partial \nu} \right] \\

&+ \left[\frac{\left(b - \lambda \right) \left(\lambda - c \right)}{\left(\lambda - \nu \right) \left(\mu - \lambda \right)} \right]^{1/2} \frac{\partial}{\partial \lambda} \left[(b - \lambda)^{1/2}(\lambda -c)^{1/2} \frac{\partial}{\partial \lambda} \right]

\end{align}

Paraboloidal coordinates can be useful for solving certain partial differential equations. For instance, the Laplace equation and Helmholtz equation are both separable in paraboloidal coordinates. Hence, the coordinates can be used to solve these equations in geometries with paraboloidal symmetry, i.e. with boundary conditions specified on sections of paraboloids.

The Helmholtz equation is $(\backslash nabla^2\; +\; k^2)\; \backslash psi\; =\; 0$. Taking $\backslash psi\; =\; M(\backslash mu)N(\backslash nu)\backslash Lambda(\backslash lambda)$, the separated equations areWillatzen and Yoon (2011), p. 227

:$$

\begin{align}

&(\mu - b)(\mu - c) \frac{d^2M}{d\mu^2} + \frac{1}{2}\left[2 \mu - (b + c) \right] \frac{dM}{d\mu} + \left[k^2 \mu^2 + \alpha_3 \mu - \alpha_2\right] M = 0 \\

&(b - \nu)(c - \nu) \frac{d^2N}{d\nu^2} + \frac{1}{2}\left[2 \nu - (b + c) \right] \frac{dN}{d\nu} + \left[k^2 \nu^2 + \alpha_3 \nu - \alpha_2\right] N = 0 \\

&(b - \lambda)(\lambda - c) \frac{d^2\Lambda}{d\lambda^2} - \frac{1}{2}\left[2 \lambda - (b + c) \right] \frac{d\Lambda}{d\lambda} - \left[k^2 \lambda^2 + \alpha_3 \lambda - \alpha_2\right] \Lambda = 0 \\

\end{align}

where $\backslash alpha\_2$ and $\backslash alpha\_3$ are the two separation constants. Similarly, the separated equations for the Laplace equation can be obtained by setting $k\; =\; 0$ in the above.

Each of the separated equations can be cast in the form of the Baer equation. Direct solution of the equations is difficult, however, in part because the separation constants $\backslash alpha\_2$ and $\backslash alpha\_3$ appear simultaneously in all three equations.

Following the above approach, paraboloidal coordinates have been used to solve for the electric field surrounding a conducting paraboloid.{{Citation

| last1 = Duggen

| first1 = L

| last2 = Willatzen

| first2 = M

| last3 = Voon

| first3 = L C Lew Yan

| year = 2012

| title = Laplace boundary-value problem in paraboloidal coordinates

| journal =European Journal of Physics

| volume = 33

| issue = 3

| pages = 689–696

| doi = 10.1088/0143-0807/33/3/689

| bibcode = 2012EJPh...33..689D

}}

{{reflist}}

• {{cite book | vauthors = Lew Yan Voon LC, Willatzen M | year = 2011| title = Separable Boundary-Value Problems in Physics | publisher = Wiley-VCH| isbn = 978-3-527-41020-0}}
• {{cite book | vauthors = Morse PM, Feshbach H | author-link = Philip M. Morse | author2-link = Herman Feshbach | year = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | isbn = 0-07-043316-X|lccn=52011515 | page = 664}}
• {{cite book | vauthors = Margenau H, Murphy GM | author-link = Henry Margenau | 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/184 184]–185 | lccn = 55010911 }}
• {{cite book | vauthors = Korn GA, Korn TM |year = 1961 | title = Mathematical Handbook for Scientists and Engineers | url = https://archive.org/details/mathematicalhand0000korn | url-access = registration | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | page = [https://archive.org/details/mathematicalhand0000korn/page/180 180] | lccn = 59014456}}
• {{cite book | author = Arfken G | year = 1970 | title = Mathematical Methods for Physicists | edition = 2nd | publisher = Academic Press | location = Orlando, FL | pages = 119–120}}
• {{cite book | vauthors = Sauer R, Szabó I | year = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York | page = 98 | 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 | vauthors = Moon P, Spencer DE | year = 1988 | chapter = Paraboloidal Coordinates (μ, ν, λ) | 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 = 44–48 (Table 1.11) | isbn = 978-0-387-18430-2}}

• [http://mathworld.wolfram.com/ConfocalParaboloidalCoordinates.html MathWorld description of confocal paraboloidal coordinates]

{{Orthogonal coordinate systems}}

Category:Three-dimensional coordinate systems

Category:Orthogonal coordinate systems