sphero-conal coordinates

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1: 29.18 Mathematical Applications

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§29.18(i) Sphero-Conal Coordinates

… ►when transformed to sphero-conal coordinates $r,\beta ,\gamma$: … ►

29.18.4 $$u(r,\beta ,\gamma )=u_1(r)u_2(\beta )u_3(\gamma ),$$

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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

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29.18.5 $$\frac{d}{dr}\left(r^2\frac{d{u}_1}{dr}\right)+({\omega }^2{r}^2-\nu (\nu +1))u_1=0,$$

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Symbols:

$\frac{df}{dx}$: derivative, $\nu$: real parameter, $\omega$: parameter, $r$: sphero-conal coordinate and $u_j$: solutions

Referenced by:

§29.18(i)

Permalink:

http://dlmf.nist.gov/29.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.18(i), §29.18 and Ch.29

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29.18.6 $$\frac{d^2{u}_2}{{d\beta }^2}+(h-\nu (\nu +1)k^2{\mathrm{sn}}^2(\beta ,k))u_2=0,$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative, $h$: real parameter, $k$: real parameter, $\nu$: real parameter, $\beta$: sphero-conal coordinate and $u_j$: solutions

Referenced by:

§29.18(i)

Permalink:

http://dlmf.nist.gov/29.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.18(i), §29.18 and Ch.29

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2: 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. … …

3: 14.31 Other Applications

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§14.31(i) Toroidal Functions

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§14.31(ii) Conical Functions

►The conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^m\left(x\right)$ 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)). … ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

4: 28.27 Addition Theorems

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

5: 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 $\xi ,\eta ,\varphi$ are related to Cartesian coordinates $x,y,z$ by … ►

§30.14(ii) Metric Coefficients

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§30.14(iii) Laplacian

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6: 13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …

7: 23.21 Physical Applications

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§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:

$\frac{\partial f}{\partial x}$: partial derivative, $\partial x$: partial differential, $\xi$: 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

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8: 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

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§30.13(ii) Metric Coefficients

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§30.13(iii) Laplacian

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9: 28.32 Mathematical Applications

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§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 …

10: 32.6 Hamiltonian Structure

… ► ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ can be written as a Hamiltonian system … ►

32.6.3 $$q^{\prime }=p,$$

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Symbols:

$q$: coordinate and $p$: momentum

Referenced by:

§32.6(ii)

Permalink:

http://dlmf.nist.gov/32.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

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32.6.4 $$p^{\prime }=6q^2+z.$$

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Symbols:

$z$: real, $q$: coordinate and $p$: momentum

Referenced by:

§32.6(ii)

Permalink:

http://dlmf.nist.gov/32.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

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32.6.5 $$\sigma ={\mathrm{H}}_{\text{I}}(q,p,z),$$

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Symbols:

$z$: real, $q$: coordinate, $p$: momentum, $\mathrm{H}(q,p,z)$: Hamiltonian and $\sigma$: function

Permalink:

http://dlmf.nist.gov/32.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

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32.6.7 $$q=-{\sigma }^{\prime },$$

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Symbols:

$q$: coordinate and $\sigma$: function

Permalink:

http://dlmf.nist.gov/32.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(ii), §32.6 and Ch.32

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