Page 75 - Intro to Tensor Calculus

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Figure 1.3-2. Spherical coordinates.

The coordinate curves, illustrated in the figure 1.3-3, are formed by the intersection of the coordinate
surfaces
ξ 2
2 2
x = −2ξ (y − ) Parabolic cylinders
2
η 2
2
2
x =2η (y + ) Parabolic cylinders
2
z = Constant Planes.

Figure 1.3-3. Parabolic cylindrical coordinates in plane z =0.

5. Parabolic coordinates (ξ, η, φ)

x = ξη cos φ ξ ≥ 0 h 1 = p ξ + η 2
2
y = ξη sin φ η ≥ 0 h 2 = p ξ + η 2
2
1 2 2
z = (ξ − η ) 0 <φ < 2π h 3 = ξη
2