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1: 29.18 Mathematical Applications
§29.18(i) Sphero-Conal Coordinates
when transformed to sphero-conal coordinates r , β , γ : …
29.18.4 u ( r , β , γ ) = u 1 ( r ) u 2 ( β ) u 3 ( γ ) ,
29.18.5 d d r ( r 2 d u 1 d r ) + ( ω 2 r 2 - ν ( ν + 1 ) ) u 1 = 0 ,
29.18.6 d 2 u 2 d β 2 + ( h - ν ( ν + 1 ) k 2 sn 2 ( β , k ) ) u 2 = 0 ,
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 ξ , η , ζ , 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
§14.31(i) Toroidal Functions
§14.31(ii) Conical Functions
The conical functions P - 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates14.19(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
§30.14(i) Oblate Spheroidal Coordinates
Oblate spheroidal coordinates ξ , η , ϕ are related to Cartesian coordinates x , y , z by …
§30.14(ii) Metric Coefficients
§30.14(iii) Laplacian
6: 13.28 Physical Applications
§13.28(i) Exact Solutions of the Wave Equation
The reduced wave equation 2 w = k 2 w in paraboloidal coordinates, x = 2 ξ η cos ϕ , y = 2 ξ η sin ϕ , z = ξ - η , can be solved via separation of variables w = f 1 ( ξ ) f 2 ( η ) e i p ϕ , where …
7: 30.13 Wave Equation in Prolate Spheroidal Coordinates
§30.13 Wave Equation in Prolate Spheroidal Coordinates
§30.13(i) Prolate Spheroidal Coordinates
§30.13(ii) Metric Coefficients
§30.13(iii) Laplacian
8: 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 …
9: 23.21 Physical Applications
§23.21(iii) Ellipsoidal Coordinates
Ellipsoidal coordinates ( ξ , η , ζ ) may be defined as the three roots ρ of the equation …where x , y , z are the corresponding Cartesian coordinates and e 1 , e 2 , e 3 are constants. The Laplacian operator 2 1.5(ii)) is given by
23.21.2 ( η - ζ ) ( ζ - ξ ) ( ξ - η ) 2 = ( ζ - η ) f ( ξ ) f ( ξ ) ξ + ( ξ - ζ ) f ( η ) f ( η ) η + ( η - ξ ) f ( ζ ) f ( ζ ) ζ ,
10: 32.6 Hamiltonian Structure
P I P VI  can be written as a Hamiltonian system …
32.6.3 q = p ,
32.6.4 p = 6 q 2 + z .
32.6.5 σ = H I ( q , p , z ) ,
32.6.7 q = - σ ,