Page 365 - Intro to Tensor Calculus

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2
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9. Prolate spheroidal coordinates (u, v, φ)= (x ,x ,x )
u ≥ 0 2 2
1
x = a sinh u sin v cos φ, h = h 2
2
2
2
2
h = a (sinh u +sin v)
y = a sinh u sin v sin φ, 0 ≤ v ≤ π 2
2 2 2 2
3
z = a coshu cos v, 0 ≤ φ< 2π h = a sinh u sin v
The coordinate curves are formed by the intersection of the coordinate surfaces
2 2 2
x y z
+ + =1, Prolate ellipsoids
(a sinh u) 2 a sinh u) 2 a cosh u) 2
2 2 2
x y z
− − =1, Two-sheeted hyperpoloid
(a cos v) 2 (a sin v) 2 (a cos v) 2
y = x tan φ, Planes.
1 cosh u sinh u 2
2
= 2 2 = − cosv sin vsinh u
11 33 2 2
sin v + sinh u
sin v + sinh u

2 cos v sin v 1
= = cos v sin v
22 2 2 12 2 2
sin v + sinh u
sin v + sinh u

1 cosh u sinh u 2
= − = cosh u sinh u
22 2 2 21 2 2
sin v + sinh u
sin v + sinh u
2
1 sin v cosh u sinh u 3 cosh u
= − =
33 2 2 31
sin v + sinh u sinh u

2 cos v sin v 3 cos v
= − =
11 2 2 32
sin v + sinh u sin v
3
1
2
10. Oblate spheroidal coordinates (ξ, η, φ)= (x ,x ,x )
ξ ≥ 0 2 2
1
x = a cosh ξ cos η cos φ, h = h 2
π π 2 2 2 2
− ≤ η ≤ h = a (sinh ξ +sin η)
2 2
y = a cosh ξ cos η sin φ, 2
2 2 2 2
3
z = a sinh ξ sin η, 0 ≤ φ ≤ 2π h = a cosh ξ cos η
The coordinate curves are formed by the intersection of the coordinate surfaces
2 2 2
+ + =1, Oblate ellipsoids
x y z
(a cosh ξ) 2 (a cosh ξ) 2 (a sinh ξ) 2
2 2 2
+ − One-sheet hyperboloids
x y z
(a cos η) 2 (a cos η) 2 (a sin η) 2 =1,
y = x tan φ, Planes.
2
1 cosh ξ sinh ξ 2
= = cos η sin ηcosh ξ
11 2 2 33 2 2
sin η + sinh ξ
sin η + sinh ξ

2 cos η sin η 1
= = cos η sin η
22 2 2 12 2 2
sin η + sinh ξ
sin η + sinh ξ
1 cosh ξ sinh ξ 2

= − 2 2 = cosh ξ sinh ξ
22 21 2 2
sin η + sinh ξ
sin η + sinh ξ
2
1 cos η cosh ξ sinh ξ 3 sinh ξ
= − =
33 2 2 31
sin η + sinh ξ cosh ξ
2 cos η sin η 3 sin η

= − = −
11 2 2 32
sin η + sinh ξ cosη