Page 363 - Intro to Tensor Calculus

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2
3
1
5. Elliptic cylindrical coordinates (ξ, η, z)= (x ,x ,x )
q
ξ ≥ 0 h 1 = 2 2
x =cosh ξ cos η sinh ξ +sin η
q
h 2 = sinh ξ +sin η
y = sinh ξ sin η 0 ≤ η ≤ 2π 2 2
−∞ <z < ∞
h 3 =1
z = z
The coordinate curves are formed by the intersection of the coordinate surfaces
2 2
x y
+ = 1 Elliptic cylinders
2 2
cosh ξ sinh ξ
2 2
x y
− = 1 Hyperbolic cylinders
2 2
cos η
sin η
z = Constant Planes.
1 sinh ξ cosh ξ 2 sin η cos η

= 2 = 2
11 2 22 2
sinh ξ +sin η sinh ξ +sin η
1 − sinh ξ cosh ξ 2 − sin η cos η

= =
22 2 2 11 2 2
sinh ξ +sin η sinh ξ +sin η

1 1 sin η cos η 2 2 sinh ξ cosh ξ
= = = =
12 21 2 2 12 21 2 2
sinh ξ +sin η sinh ξ +sin η
3
2
1
6. Elliptic coordinates (ξ, η, φ)= (x ,x ,x )
s
2 2
p h 1 = ξ − η
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 2 2
h 2 = ξ − η
2
1 − η
z = ξη 0 ≤ φ< 2π
p
2
2
h 3 = (1 − η )(ξ − 1)
The coordinate curves are formed by the intersection of the coordinate surfaces
2 2 2
+ + = 1 Prolate ellipsoid
x y z
2
2
ξ − 1 ξ − 1 ξ 2
2 2 2
z x y
− − = 1 Two-sheeted hyperboloid
2 2 2
η 1 − η 1 − η
Planes
y = x tan φ

1 ξ ξ 2 2 2
= − + = −1+ ξ η 1 − η
2
11 −1+ ξ 2 ξ − η 2 33 2 2
ξ − η
2 η η 1
= − η
2
22 1 − η 2 ξ − η 2 12 = − 2 2
1 ξ −1+ ξ 2
2 ξ − η
= − = ξ
2
2
2
2
22 (1 − η )(ξ − η ) 21 ξ − η 2
1 ξ −1+ ξ 1 − η 3 ξ
2 2
= − =
2
33 ξ − η 2 31 −1+ ξ 2
2
2 η 1 − η 3 η
= = − 2
2
2
2
11 (−1+ ξ )(ξ − η ) 32 1 − η