Cylindrical polar co-ordinates

Cylindrical polar co-ordinates

1. Why Cylindrical (vs Spherical)

For solids with circular symmetry about an axis (usually $z$-axis) but not spherical — e.g., cylinder $x²+y² ≤ a²$, cone $z = r$, paraboloid $z = r²$ — cylindrical $(r,θ,z)$ is simpler than spherical or Cartesian. It is the 3D extension of polar in $xy$.

2. Transformation and Jacobian

with $x²+y² = r²$ and volume element:

(Jacobian $= r$). Do not confuse with spherical's $r² sinφ$ — cylindrical keeps $z$.

3. Limits

• Cylinder $x²+y² ≤ a², 0≤z≤h$: $r 0→a, θ 0→2π, z 0→h$.
• Cone $z = √(x²+y²) = r$, $0≤z≤h$: for each $z$, $r 0→z$, $θ 0→2π$.
• Paraboloid $z = r², 0≤z≤h$: $r 0→√z$ or $z r²→h$.
• Between cylinders $a≤r≤b$.

4. Worked Examples

(a) Volume of cylinder radius a height h: V= ∫_{0}^{2π}∫_{0}^{a}∫_{0}^{h} r dz dr dθ = π a² h ✓

(b) ∭_{cylinder} (x²+y²) dV : = ∫_{0}^{2π}∫_{0}^{a}∫_{0}^{h} r²· r dz dr dθ =2π h·a⁴/4 = π a⁴ h/2 ✓
    Without cylindrical, Cartesian limits messy.

(c) ∭_{paraboloid z = x²+y² below z=4} 1 dV (volume under paraboloid): D: r 0→2? Since z=r² → top z=4 ⇒ r=2
    Using order r→z: V= ∫_{0}^{2π}∫_{0}^{2}∫_{r²}^{4} r dz dr dθ =2π·∫_{0}^{2}(4 -r²)r dr =8π ✓
    Using order z→r: V= ∫_{0}^{4}∫_{0}^{√z} 2π r dr dz =π·∫_{0}^{4} z dz =8π same.

(d) Mass of cylinder with density ρ = k z: Mass= ∫_{0}^{2π}∫_{0}^{a}∫_{0}^{h} k z·r dz dr dθ = π a² k h²/2 ✓

(e) Moment of inertia of cylinder about z-axis: I_z = ∭ (x²+y²) dM ; with ρ constant ⇒ (1/2) M a² (standard).

(f) Cone volume? Cone r=z? Actually cone vertex at origin opening upward: V= ∫_{0}^{h} π r(z)² dz = π∫_{0}^{h} z² dz = π h³/3 ✓ matches cylindrical.

5. Choose Cylindrical vs Spherical vs Cartesian (exam tip)

• $x²+y² ≤ a²$ with $z$ free / linear in $z$ → cylindrical.
• $x²+y²+z² ≤ a²$ or cone with apex at origin → spherical (sometimes also cylindrical for cone).
• Box or plane $z = a - x - y$ → Cartesian.

6. Exam Points

• Convert $∭_{V} f(x,y,z) dV$ over cylinder/cone/paraboloid to $∭ f(r,θ,z) r dr dθ dz$ — 7 marks, very frequent.
• Evaluate $∭ (x²+y²)$, $∭ z$ over cylinder/cone — 7 marks, frequent.
• Volume via cylindrical — 5 marks, frequent.
• State $dV = r dr dθ dz$ vs $r² sinφ dr dφ dθ$ — 3 marks.