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Table 1
Pairs (q1,q2), function f, and associated area element d2A for the three most popular coordinate systems.
| q1, q2 | f | d2 A | Surface  |
Hyperkernel κq1q2 | |
|
|
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| Cartesian coordinates | x,y | 1 | dxdy | rectangular sheet | κxy, see Eq. (A.4) |
| y,z | 1 | dydz | rectangular sheet | κyz (see κxy) | |
| x,z | 1 | dxdz | rectangular sheet | κxz (see κxy) | |
| Cylindrical coordinates | θ,z | R | Rdθdz | piece of hollow cylinder | – |
| R,θ | R | RdθdR | polar sector | κRθ, see Eq. (C.4) | |
| R,z | 1 | dRdz | meridional sheet | κRz, see Eq. (B.2) | |
| Spherical coordinates | θ,φ | r2sinφ | r2sinφdφdθ | spherical cap | – |
| r,φ | r | rdrdφ | meridional sector | κrφ, see the Appendix D | |
| r,θ | rsinφ | rsinφdrdθ | piece of cone | – | |
|
|
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| Axial symmetry* | R | R | dR | disk | κR, see Eq. (C.6) |
| r | 1 | dr | cone | – | |
| φ | 1 | dφ | spherical cap | – | |
Notes. The formula for the hyperkernel κq1q2, when known in closed form, is indicated in the last column (otherwise “–”).
(*)
Under axial symmetry, a single variable is necessary (see Sect. 3.6).
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