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1: 1.5 Calculus of Two or More Variables
§1.5(ii) Coordinate Systems
Polar Coordinates
Cylindrical Coordinates
Spherical Coordinates
For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. …
2: 36.13 Kelvin’s Ship-Wave Pattern
In a reference frame where the ship is at rest we use polar coordinates r and ϕ with ϕ = 0 in the direction of the velocity of the water relative to the ship. …
3: 18.39 Applications in the Physical Sciences
where x is a spatial coordinate, m the mass of the particle with potential energy V ( x ) , = h / ( 2 π ) is the reduced Planck’s constant, and ( a , b ) a finite or infinite interval. … Now use spherical coordinates (1.5.16) with r instead of ρ , and assume the potential V to be radial. …By (1.5.17) the first term in (18.39.21), which is the quantum kinetic energy operator T e , can be written in spherical coordinates r , θ , ϕ as … Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …
4: 22.18 Mathematical Applications
In polar coordinates, x = r cos ϕ , y = r sin ϕ , the lemniscate is given by r 2 = cos ( 2 ϕ ) , 0 ϕ 2 π . …
5: 36.7 Zeros
There are also three sets of zero lines in the plane z = 0 related by 2 π / 3 rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates ( x = r cos θ , y = r sin θ ) is given by …
6: 36.5 Stokes Sets
For z 0 , the Stokes set is expressed in terms of scaled coordinates
36.5.7 X = 9 20 + 20 u 4 Y 2 20 u 2 + 6 u 2 sign ( z ) ,
36.5.10 160 u 6 + 40 u 4 = Y 2 .
With coordinates
36.5.17 Y S ( X ) = Y ( u , | X | ) ,
7: 20 Theta Functions
Chapter 20 Theta Functions
8: 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(ii) Ellipsoidal Coordinates
The wave equation (29.18.1), when transformed to ellipsoidal coordinates α , β , γ : …
9: 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. … …
10: 9.16 Physical Applications
The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). …