Page 74 - Intro to Tensor Calculus

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Figure 1.3-1. Cylindrical coordinates.

2. Cylindrical coordinates (r, θ, z)

x = r cos θ r ≥ 0 h 1 =1
y = r sin θ 0 ≤ θ ≤ 2π h 2 = r

z = z −∞ <z < ∞ h 3 =1

The coordinate curves, illustrated in the figure 1.3-1, are formed by the intersection of the coordinate
surfaces
2
2
2
x + y = r , Cylinders
y/x =tan θ Planes
z = Constant Planes.
3. Spherical coordinates (ρ, θ, φ)

x = ρ sin θ cos φ ρ ≥ 0 h 1 =1
y = ρ sin θ sin φ 0 ≤ θ ≤ π h 2 = ρ
z = ρ cos θ 0 ≤ φ ≤ 2π h 3 = ρ sin θ

The coordinate curves, illustrated in the figure 1.3-2, are formed by the intersection of the coordinate
surfaces
2
2
2
x + y + z = ρ 2 Spheres
2
2
2
x + y =tan θz 2 Cones
y = x tan φ Planes.
4. Parabolic cylindrical coordinates (ξ, η, z)
p
2
x = ξη −∞ <ξ < ∞ h 1 = ξ + η 2
1 2 2 p
2
y = (ξ − η ) −∞ <z < ∞ h 2 = ξ + η 2
2
z = z η ≥ 0 h 3 =1