Cylindrical coordinates

The formula

Going from cylindrical (r, φ, z) to Cartesian and back:

x = r·cos φ        r = √(x² + y²)
y = r·sin φ        φ = atan2(y, x)
z = z              z = z

Reading: r is the straight-line distance from the z-axis (not from the origin), φ is the angle swept around the axis from +x, and z rides along unchanged. Same point, different bookkeeping — the conversion is exact both ways.

The volume element picks up a factor of r:

dV = r · dr · dφ · dz

Reading: a wedge of angle dφ is physically wider the farther out you go, so every volume, mass, or flux integral in cylindrical coordinates carries that extra r. Integrate it over a full cylinder and ∫∫∫ r dr dφ dz = π·R²·h falls out, as it should.

The Laplacian — the operator sitting inside heat conduction, electrostatics, pressure diffusion, and vibrating-membrane problems:

∇²f = (1/r)·∂/∂r( r·∂f/∂r ) + (1/r²)·∂²f/∂φ² + ∂²f/∂z²

Reading: the radial term is not a bare second derivative; the geometry of spreading out from an axis is baked into that (1/r)·∂/∂r(r·...) structure.

Acceleration of a particle in the r-φ plane:

a_r = r̈ − r·φ̇²          a_φ = r·φ̈ + 2·ṙ·φ̇

Reading: even at constant radius and constant spin rate, a_r = −r·ω² — the centripetal term. The 2·ṙ·φ̇ piece is the Coriolis term, and it shows up any time something moves radially on a rotating machine.

Where you meet it

The RF bench, holding a piece of coax. The field between inner and outer conductor depends only on r, which is why the characteristic impedance collapses to a log of the diameter ratio: Z₀ = (60/√εr)·ln(D/d). For solid PTFE (εr ≈ 2.1), 50 Ω demands D/d ≈ 3.35 — every 50-ohm PTFE cable ever extruded is built around that number.
The pressure-vessel stress review. Thin-wall hoop stress σ = p·r/t and the thick-wall Lamé equations are cylindrical-coordinate results. The review board argues about σ_hoop, σ_axial, and σ_radial because the vessel's geometry made those the natural directions.
The rotating-machinery test cell. Shaft torsion τ = T·r/J, balancing planes, and accelerometer placement all get called out as radius, clocking angle, and axial station — r, φ, z by other names. "90 degrees off top-dead-center, mid-span, on the case" is a cylindrical coordinate.
The pipe-flow calc. Laminar flow in a round pipe gives the parabolic Hagen–Poiseuille profile u(r) = u_max·(1 − r²/R²) precisely because the problem only depends on r. Try that in x-y-z and you'll bury a one-variable problem under three.

How it works

The whole value proposition is symmetry. If the hardware is axisymmetric — a shaft, a tube, a solenoid, a tank — then nothing physical depends on φ, the ∂/∂φ terms vanish, and a 3D field problem drops to two dimensions or one. That is what "separable" buys you: heat conduction in a cylinder splits into a radial equation, an angular equation, and an axial equation you can solve independently and multiply back together. The radial piece of the Laplace and Helmholtz equations turns into Bessel's equation, which is why Bessel functions J and Y are the sines and cosines of anything round — drumheads, waveguide modes, heated billets ^[1]^[9].

The trap that separates people who've used cylindrical coordinates from people who've been burned by them: the unit vectors move. r̂ and φ̂ point different directions at different points, unlike Cartesian x̂, ŷ, ẑ. Differentiate a vector by naively differentiating its components and you silently drop the centripetal and Coriolis terms — the −r·φ̇² and 2·ṙ·φ̇ in the acceleration above are exactly the derivatives of the rotating unit vectors. The same disease infects divergence and curl, which is why the cylindrical del operators look nothing like their Cartesian cousins.

Second trap: the Jacobian r. Every integral needs r·dr·dφ·dz, not dr·dφ·dz. Forget it and your computed mass, flow rate, or antenna gain comes out wrong by a factor that varies with geometry, so it doesn't even fail consistently.

Third: the axis is singular. At r = 0, φ is undefined and every 1/r in the operators blows up. The physics is fine on the axis; the coordinates aren't. CFD and FEA codes handle it with special axis treatments or by keeping a small core out of the mesh, and closed-form solutions handle it by throwing away the Y Bessel solutions that diverge at r = 0. If your hand derivation produces an infinity on the centerline of a solid shaft, you kept a term the boundary conditions should have killed.

Fourth, the small stuff that causes real errors: use atan2(y, x), not atan(y/x) — the plain arctangent puts half your points in the wrong quadrant. And watch the symbols. ISO 80000-2 writes the triplet as (ρ, φ, z) ^[1], most American engineering texts write (r, θ, z), and in a fluids or E&M problem ρ is already taken by density or charge density. Pick a convention on page one of the calc and state it; reviewers have chased phantom errors that were nothing but a ρ meaning two things.

Range conventions matter too: r ≥ 0 and φ covering one full turn — either 0 to 2π or −π to π — make the mapping unique. Let a curve-fit or a control loop wander across the ±π seam without unwrapping the angle and you get a 2π jump that looks like a step disturbance.

History

The pieces arrived over about two centuries. The first recorded use of polar-style coordinates shows up around 1625 in work from the circle of Grégoire de Saint-Vincent, a Jesuit mathematician whose big collection finally saw print in 1647 ^[2]^[4]. Independently, Bonaventura Cavalieri published in 1635 (corrected 1653) and put radius-and-angle coordinates to work on the Archimedean spiral — the historian Julian Coolidge names him the first writer to actually employ them ^[2]^[3]. Newton, in his Method of Fluxions, was the first to treat radius-and-angle as a general way to fix any point in the plane rather than a trick for one curve, and Jacob Bernoulli pushed the calculus through by working out radius of curvature in polar form ^[2]^[3].

The jump to three dimensions took another generation. Alexis Clairaut raised the possibility of polar coordinates in space, but Coolidge is blunt about the credit: Clairaut "merely mentions the possibility"; it was Euler who first developed the 3D systems ^[2]^[3].

What made cylindrical coordinates an engineering workhorse rather than a curiosity was heat. Joseph Fourier's memoir on heat propagation in solids, read to the Paris Institute on 21 December 1807 and expanded into the Théorie analytique de la chaleur of 1822, attacked conduction problems by separating variables — and his treatment of the heated cylinder forced the radial functions we now call Bessel functions onto the page years before Bessel ^[5]^[6]. Friedrich Bessel then studied those functions systematically in 1824, in a treatise on planetary perturbations, and his name stuck to them ^[7]^[8]. Every cylindrical-coordinate field solution since — waveguide modes, quenched shafts, borehole pressure transients — runs on that machinery.

Related tools

/tools/coax-impedance-dimensions
/tools/coax-loss
/tools/skin-depth
/tools/pressure-vessel-hoop
/tools/torsion-shaft
/tools/pipe-flow-velocity
/tools/pipe-pressure-drop
/tools/tank-volume-vertical
/tools/moment-of-inertia-shapes

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →