Page 77 - Intro to Tensor Calculus

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Figure 1.3-5. Elliptic cylindrical coordinates in the plane z =0.

7. Elliptic coordinates (ξ, η, φ)

s
2
ξ − η 2
p h 1 =
2
2
2
x = (1 − η )(ξ − 1) cos φ 1 ≤ ξ< ∞ ξ − 1
p s
2
2
y = (1 − η )(ξ − 1) sin φ − 1 ≤ η ≤ 1 ξ − η 2
2
h 2 = 2
z = ξη 0 ≤ φ< 2π 1 − η
p
2
2
h 3 = (1 − η )(ξ − 1)
The coordinate curves, illustrated in the figure 1.3-6, are formed by the intersection of the coordinate
surfaces
x 2 y 2 z 2
+ + = 1 Prolate ellipsoid
2
2
ξ − 1 ξ − 1 ξ 2
z 2 x 2 y 2
− − = 1 Two-sheeted hyperboloid
η 2 1 − η 2 1 − η 2
y = x tan φ Planes
8. Bipolar coordinates (u, v, z)
a sinh v 2 2
x = , 0 ≤ u< 2π h = h 2
1
cosh v − cos u 2
2
a sin u h = a
y = , −∞ <v < ∞ 2 (cosh v − cos u) 2
cosh v − cos u 2
z = z −∞ <z < ∞ h =1
3