Cylindrical Coordinates

Polar coordinates with an extra height

To locate a point in 3-D, cylindrical coordinates use:

• r — the perpendicular distance from the z-axis. Always non-negative.
• θ — the azimuthal angle around the z-axis, measured counter-clockwise from the positive x-axis (when viewed from above). Range: 0 to 2π.
• z — the same z as Cartesian, the height above (or below) the xy-plane.

The relationship to Cartesian:

x = r cos(θ)
y = r sin(θ)
z = z

r = √(x² + y²)
θ = atan2(y, x)
z = z

Geometrically: project the point onto the xy-plane to read off (r, θ) just as you would in 2-D polar; read off z as you would in Cartesian. A surface of constant r is a cylinder around the z-axis; a surface of constant θ is a half-plane through the z-axis; a surface of constant z is a horizontal plane.

Deriving the volume element r dr dθ dz

An infinitesimal cell in cylindrical coordinates is a slightly curved box. Its three sides are:

• The radial side has length dr — a straight segment perpendicular to the axis.
• The azimuthal side is an arc of the circle of radius r, subtending angle dθ. Length: r dθ.
• The vertical side has length dz.

Multiplying:

dV = dr · r dθ · dz = r dr dθ dz

Equivalently, from the Jacobian:

J = ∂(x, y, z)/∂(r, θ, z) =
    | cos θ    −r sin θ    0 |
    | sin θ     r cos θ    0 |
    |   0          0       1 |

|det J| = r (cos²θ + sin²θ) = r

The factor r captures the geometric truth that azimuthal arcs grow with radius. Near the axis (r small), an angular slice covers little area; far from the axis, the same angular slice covers a lot.

Worked example — volume of a cylinder

The volume of a cylinder of radius R and height H, with its axis along z, is:

V = ∫₀^H ∫₀^{2π} ∫₀^R r dr dθ dz
  = ∫₀^H dz · ∫₀^{2π} dθ · ∫₀^R r dr
  = H · 2π · (R²/2)
  = π R² H

The classic textbook formula falls out without any tricks. Doing the same in Cartesians requires limits like √(R² − x²) and an arctan substitution.

Worked example — moment of inertia of a solid cylinder

The moment of inertia of a uniform solid cylinder (mass M, radius R, height H) about its axis is the integral of r² dm over the volume:

I = ∫∫∫ r² ρ_m dV
  = ρ_m ∫₀^H ∫₀^{2π} ∫₀^R r² · r dr dθ dz
  = ρ_m · H · 2π · (R⁴/4)
  = ρ_m · 2π H · R⁴ / 4

Density ρ_m = M / (π R² H), so:
I = (M / (π R² H)) · (2π H · R⁴ / 4) = (1/2) M R²

The famous (1/2) M R² result for a solid cylinder rolling down an incline. The integration takes three lines because the symmetry aligns perfectly with the coordinate system.

Hagen–Poiseuille flow — the classical pipe-flow application

Consider steady, laminar, incompressible flow of a viscous fluid through a circular pipe of radius R, length L, driven by a pressure difference ΔP = P_in − P_out. Symmetry implies the velocity has only a z-component v_z(r), independent of θ and z.

The Navier–Stokes momentum balance in cylindrical coordinates reduces to:

(1/r) d/dr (r dv_z/dr) = − ΔP / (μ L)

where μ is the viscosity. Integrating once:

r dv_z/dr = − ΔP r² / (2μL) + C₁

Regularity at r = 0 (the velocity gradient must be finite on the axis) forces C₁ = 0. Integrating again, with no-slip v_z(R) = 0:

v_z(r) = (ΔP / (4μL)) (R² − r²)

A parabolic velocity profile, fastest at the centre and zero at the wall. The volumetric flow rate is the integral of v_z over the cross-section:

Q = ∫₀^{2π} ∫₀^R v_z(r) r dr dθ
  = 2π · (ΔP / (4μL)) ∫₀^R (R² − r²) r dr
  = 2π · (ΔP / (4μL)) · (R⁴/4)
  = π R⁴ ΔP / (8 μ L)

The Hagen–Poiseuille equation. The R⁴ scaling — halving the bore cuts flow 16-fold — is why arteries narrowed by atherosclerosis or oil pipelines coated with deposits become disproportionately ineffective. Without cylindrical coordinates this derivation requires brutal Cartesian integrals; with them, three lines.

Cartesian vs cylindrical vs spherical Jacobians

	Cartesian	Cylindrical	Spherical
Coordinates	(x, y, z)	(r, θ, z)	(ρ, θ, φ)
Volume element	dx dy dz	r dr dθ dz	ρ² sin(φ) dρ dθ dφ
Jacobian determinant	1	r	ρ² sin(φ)
Surface r = const	—	Cylinder	—
Surface (radial) = const	Plane	Cylinder	Sphere
Best symmetry	Boxes, slabs	Pipes, wires, drums	Atoms, planets, monopoles
Singular set	None	z-axis (r = 0)	Origin and poles
Radial Laplacian	∂²/∂x²	(1/r) ∂/∂r(r ∂/∂r)	(1/ρ²) ∂/∂ρ(ρ² ∂/∂ρ)

Cylindrical sits between Cartesian and spherical: it shares Cartesian's z behaviour and polar's radial behaviour. Whenever a problem decomposes neatly along an axis, cylindrical coordinates are usually the lightest tool.

Gradient, divergence, curl, Laplacian in cylindrical coordinates

∇f = (∂f/∂r) r̂ + (1/r)(∂f/∂θ) θ̂ + (∂f/∂z) ẑ

∇·F = (1/r) ∂(r F_r)/∂r
     + (1/r) ∂F_θ/∂θ
     + ∂F_z/∂z

∇×F = (1/r ∂F_z/∂θ − ∂F_θ/∂z) r̂
     + (∂F_r/∂z − ∂F_z/∂r) θ̂
     + (1/r) (∂(r F_θ)/∂r − ∂F_r/∂θ) ẑ

∇²f = (1/r) ∂/∂r (r ∂f/∂r)
    + (1/r²) ∂²f/∂θ²
    + ∂²f/∂z²

Each formula carries its 1/r metric factors for the same reason as in spherical coordinates: angular derivatives must be divided by the arc-length scale r dθ to give per-distance rates.

Bessel functions and cylindrical wave equations

Substituting f(r, θ, z, t) = R(r) Θ(θ) Z(z) T(t) into the wave equation ∇²f = (1/c²) ∂²f/∂t² and separating variables produces:

• For Θ: e^{imθ} with integer m (periodicity around the axis).
• For Z: sines and cosines (or exponentials) along the axis.
• For T: oscillation in time.
• For R: r² R'' + r R' + (k²r² − m²) R = 0 — Bessel's equation. Solutions: J_m(kr), Y_m(kr), the Bessel functions of the first and second kinds.

This is why drumheads (governed by the 2-D wave equation in polar coordinates) ring at frequencies determined by zeros of Bessel functions, and why optical fibres support discrete propagation modes labelled by integer m.

Where cylindrical coordinates earn their keep

• Fluid mechanics. Pipe flow, blood flow in arteries, oil through pipelines, coolant in nuclear reactors. The Navier–Stokes equations in cylindrical coordinates are the starting point.
• Electromagnetism. The field of an infinite straight wire (E ∝ 1/r); coaxial cables (TEM modes); cylindrical capacitors; solenoids and toroids.
• Mechanical engineering. Stress in pressure vessels, rotating shafts, bearings; moments of inertia of cylindrical components.
• Acoustics. Vibrating drumheads, resonance modes in cylindrical cavities (microwave ovens, organ pipes).
• Plasma physics. Tokamak reactors and most magnetic confinement devices have axial symmetry, reducing 3-D problems to 2-D in the (r, z) cross-section.
• Astronomy. Accretion disks, galactic disks, and rotating stars are often modelled in cylindrical coordinates aligned with the rotation axis.

Common mistakes

• Forgetting the r in r dr dθ dz. Using just dr dθ dz gives wrong areas and volumes — your "volume of a cylinder" comes out R H instead of π R² H.
• Confusing r with the spherical ρ. Cylindrical r = distance from the axis (in the xy-plane). Spherical ρ = distance from the origin. They differ by ρ² = r² + z².
• Treating r̂ and θ̂ as constant. They rotate as θ changes, which is why divergence and curl pick up extra (1/r) terms.
• Using arctan instead of atan2. Plain arctan loses quadrant information; atan2(y, x) gives the correct θ in (−π, π].
• Picking cylindrical when the problem is spherical. A point charge has spherical symmetry; using cylindrical coordinates inflates the integrals into Bessel functions when a one-line spherical-Gauss argument suffices.
• Imposing no-slip in the wrong place. In Hagen–Poiseuille flow the wall is r = R, not r = 0. Setting v(0) = 0 is wrong (the centreline is the fastest, not slowest, point).