Page 362 - Intro to Tensor Calculus

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2
3
1
3. Parabolic cylindrical coordinates (ξ, η, z)=(x ,x ,x )
p
−∞ <ξ < ∞ 2 2
x = ξη h 1 = ξ + η
1 2 2 p
2
y = (ξ − η ) −∞ <z < ∞ h 2 = ξ + η 2
2
η ≥ 0 h 3 =1
z = z
The coordinate curves are formed by the intersection of the coordinate surfaces
2
2 2 ξ
x = −2ξ (y − ) Parabolic cylinders
2
2
2
2
x =2η (y + η ) Parabolic cylinders
2
z = Constant Planes.
1 ξ 1 −ξ

= =
2
2
11 ξ + η 2 22 ξ + η 2

2 η 1 1 η
= = =
2
2
22 ξ + η 2 12 21 ξ + η 2
2 −η 2 2 ξ

= = =
2
2
11 ξ + η 2 21 12 ξ + η 2
1
2
3
4. Parabolic coordinates (ξ, η, φ)=(x ,x ,x )
p
2
x = ξη cos φ ξ ≥ 0 h 1 = ξ + η 2
p
2
y = ξη sin φ η ≥ 0 h 2 = ξ + η 2
1 2 2
z = (ξ − η )
2 0 <φ < 2π h 3 = ξη
The coordinate curves are formed by the intersection of the coordinate surfaces
2
2 2 2 ξ
x + y = −2ξ (z − ) Paraboloids
2
2
2
2
2
x + y =2η (z + η ) Paraboloids
2
y = x tan φ Planes.
2
1 ξ 1 −ξη
= =
2
2
11 ξ + η 2 33 ξ + η 2

2 η 1 1 η
= = =
2
22 ξ + η 2 21 21 ξ + η 2
2

1 −ξ 2 2 ξ
= = =
2
2
22 ξ + η 2 21 12 ξ + η 2

2 −η 3 3 1
= = =
2
11 ξ + η 2 32 23 η
2
2 −ηξ 3 3 1
= = =
2
33 ξ + η 2 13 31 ξ