Page 79 - Intro to Tensor Calculus

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9. Conical coordinates (u, v, w)

uvw 2 2 2 2 2
x = , b >v >a >w , u ≥ 0 h =1
1
ab
2
2
2
r u (v − w )
2
2
2
2
u (v − a )(w − a ) h =
2
y = 2 (v − a )(b − v )
2
2
2
2
2
a a − b 2
2
2
2
r u (v − w )
2
2
2
2
2
u (v − b )(w − b ) h =
z = 3 (w − a )(w − b )
2
2
2
2
2
b b − a 2
The coordinate curves, illustrated in the figure 1.3-8, are formed by the intersection of the coordinate
surfaces
2
2
2
x + y + z = u 2 Spheres
x 2 y 2 z 2
+ + =0, Cones
2
2
v 2 v − a 2 v − b 2
x 2 y 2 z 2
+ + =0, Cones.
2
2
w 2 w − a 2 w − b 2
Figure 1.3-8. Conical coordinates.
10. Prolate spheroidal coordinates (u, v, φ)

2
x = a sinh u sin v cos φ, u ≥ 0 h = h 2 2
1
2
2
2
2
y = a sinh u sin v sin φ, 0 ≤ v ≤ π h = a (sinh u +sin v)
2
2
2
2
2
z = a coshu cos v, 0 ≤ φ< 2π h = a sinh u sin v
3
The coordinate curves, illustrated in the figure 1.3-9, are formed by the intersection of the coordinate
surfaces
x 2 y 2 z 2
+ + =1, Prolate ellipsoids
(a sinh u) 2 (a sinh u) 2 (a cosh u) 2
z 2 x 2 y 2
− − =1, Two-sheeted hyperboloid
(a cos v) 2 (a sin v) 2 (a sin v) 2
y = x tan φ, Planes.