toroidal coordinates

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1: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

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§14.19(i) Introduction

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates

(\eta,\theta,\phi)

, which are related to Cartesian coordinates

(x,y,z)

by … ►

§14.19(iv) Sums

… ►

§14.19(v) Whipple’s Formula for Toroidal Functions

…

2: 14.31 Other Applications

… ►

§14.31(i) Toroidal Functions

►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). ►

§14.31(ii) Conical Functions

►The conical functions

\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)

appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …

3: 29.18 Mathematical Applications

… ►

§29.18(i) Sphero-Conal Coordinates

… ►when transformed to sphero-conal coordinates

r,\beta,\gamma

: … ►

29.18.4

u(r,\beta,\gamma)=u_{1}(r)u_{2}(\beta)u_{3}(\gamma),

ⓘ

Symbols:: $u$ : solution, $r$ : sphero-conal coordinate, $\beta$ : sphero-conal coordinate, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Permalink:: http://dlmf.nist.gov/29.18.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §29.18(i), §29.18 and Ch.29

… ►

§29.18(ii) Ellipsoidal Coordinates

►The wave equation (29.18.1), when transformed to ellipsoidal coordinates

\alpha,\beta,\gamma

: …

4: 12.17 Physical Applications

§12.17 Physical Applications

… ►in Cartesian coordinates

x, y, z

of three-dimensional space (§1.5(ii)). By using instead coordinates of the parabolic cylinder

\xi,\eta,\zeta

, defined by … ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … …

5: 28.27 Addition Theorems

… ►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

6: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

►Oblate spheroidal coordinates

\xi,\eta,\phi

are related to Cartesian coordinates

x, y, z

by … ►

§30.14(ii) Metric Coefficients

… ►

§30.14(iii) Laplacian

…

7: 13.28 Physical Applications

… ►

§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation

\nabla^{2}w=k^{2}w

in paraboloidal coordinates,

x=2\sqrt{\xi\eta}\cos\phi

y=2\sqrt{\xi\eta}\sin\phi

z=\xi-\eta

, can be solved via separation of variables

w=f_{1}(\xi)f_{2}(\eta)e^{\mathrm{i}p\phi}

, where …

8: 23.21 Physical Applications

… ►

§23.21(iii) Ellipsoidal Coordinates

►Ellipsoidal coordinates

(\xi,\eta,\zeta)

may be defined as the three roots

\rho

of the equation …where

x, y, z

are the corresponding Cartesian coordinates and

e_{1}

e_{2}

e_{3}

are constants. The Laplacian operator

\nabla^{2}

(§1.5(ii)) is given by ►

23.21.2

(\eta-\zeta)(\zeta-\xi)(\xi-\eta)\nabla^{2}=(\zeta-\eta)f(\xi)f^{\prime}(\xi)% \frac{\partial}{\partial\xi}+(\xi-\zeta)f(\eta)f^{\prime}(\eta)\frac{\partial}% {\partial\eta}+(\eta-\xi)f(\zeta)f^{\prime}(\zeta)\frac{\partial}{\partial% \zeta},

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\xi$ : ellipsoidal coordinate, $\eta$ : ellipsoidal coordinate, $\zeta$ : ellipsoidal coordinate and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

…

9: 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.13 Wave Equation in Prolate Spheroidal Coordinates

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§30.13(i) Prolate Spheroidal Coordinates

… ►

§30.13(ii) Metric Coefficients

… ►

§30.13(iii) Laplacian

… ►

10: 28.32 Mathematical Applications

… ►

§28.32(i) Elliptical Coordinates and an Integral Relationship

►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … ► ►

§28.32(ii) Paraboloidal Coordinates

►The general paraboloidal coordinate system is linked with Cartesian coordinates via …