Page 80 - Intro to Tensor Calculus

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Figure 1.3-9. Prolate spheroidal coordinates

11. Oblate spheroidal coordinates (ξ, η, φ)
2
x = a cosh ξ cos η cos φ, ξ ≥ 0 h = h 2 2
1
π π 2 2 2 2
y = a cosh ξ cos η sin φ, − ≤ η ≤ h = a (sinh ξ +sin η)
2
2 2
2 2 2 2
z = a sinh ξ sin η, 0 ≤ φ ≤ 2π h = a cosh ξ cos η
3
The coordinate curves, illustrated in the figure 1.3-10, are formed by the intersection of the coordinate
surfaces
x 2 y 2 z 2
+ + =1, Oblate ellipsoids
(a cosh ξ) 2 (a cosh ξ) 2 (a sinh ξ) 2
x 2 y 2 z 2
+ − =1, One-sheet hyperboloids
(a cos η) 2 (a cos η) 2 (a sin η) 2
y = x tan φ, Planes.

12. Toroidal coordinates (u, v, φ)
a sinh v cos φ h = h 2
2
x = , 0 ≤ u< 2π 1 2
cosh v − cos u a 2
2
a sinh v sin φ h =
2
y = , −∞ <v < ∞ (cosh v − cos u) 2
cosh v − cos u 2 2
a sin u 2 a sinh v
z = , 0 ≤ φ< 2π h = 2
3
cosh v − cos u (cosh v − cos u)
The coordinate curves, illustrated in the figure 1.3-11, are formed by the intersection of the coordinate
surfaces
a cos u 2 a 2
2 2
x + y + z − = 2 , Spheres
sin u sin u
2 2
p cosh v 2 a
2
2
x + y − a + z = 2 , Torus
sinh v sinh v
y = x tan φ, planes