Curvilinear coordinates

The formula

The scale factor of each coordinate:

h_i = | ∂r/∂q_i |

Reading: nudge coordinate q_i by a small amount and h_i tells you how many meters the point actually travels — degrees and radians are labels, h_i·dq_i is distance.

The line element, which is where every other formula comes from:

ds² = h₁²·dq₁² + h₂²·dq₂² + h₃²·dq₃²

Reading: Pythagoras still works locally, provided each coordinate step is first converted to a true length. This clean sum-of-squares form only holds when the coordinate surfaces meet at right angles.

The three systems that cover most engineering work:

Cartesian  (x, y, z):    h = (1,  1,  1)
Cylindrical (r, φ, z):   h = (1,  r,  1)
Spherical  (ρ, θ, φ):    h = (1,  ρ,  ρ·sinθ)     θ measured from the z-axis

Reading: a degree of azimuth is worth more meters the farther out you stand — that is the entire content of h_φ = r.

Given the scale factors, the vector calculus operators are mechanical. With J = h₁h₂h₃:

grad:  (∇f)_i = (1/h_i) · ∂f/∂q_i
div:   ∇·A = (1/J) · [ ∂(h₂h₃A₁)/∂q₁ + ∂(h₃h₁A₂)/∂q₂ + ∂(h₁h₂A₃)/∂q₃ ]
curl:  (∇×A)₁ = (1/(h₂h₃)) · [ ∂(h₃A₃)/∂q₂ − ∂(h₂A₂)/∂q₃ ]   (cycle 1→2→3 for the rest)
lap:   ∇²f = (1/J) · Σ_i  ∂/∂q_i [ (J/h_i²) · ∂f/∂q_i ]

Reading: one recipe generates the gradient, divergence, curl, and Laplacian in any orthogonal system — the pages of "vector operators in spherical coordinates" printed inside textbook covers are all this recipe evaluated three times.

Where you meet it

On the antenna range. A pattern measurement is a scan over θ and φ, and total radiated power is the integral of intensity times the patch area ρ²·sinθ·dθ·dφ. That sinθ is the product of two scale factors. Forget it and every gain number you compute from a measured pattern is wrong — worst when energy sits near the poles of the measurement sphere, where the missing sinθ overcounts patch areas without bound; a pattern integrated without it can misreport gain by many dB.
On the spin table. Write velocity in polar coordinates and differentiate honestly — the unit vectors rotate with the part — and the acceleration comes out as a_r = r̈ − r·φ̇² and a_φ = r·φ̈ + 2·ṙ·φ̇. The centripetal term and the Coriolis term are not extra physics bolted on; they are what the moving basis vectors cost you.
In the thermal report for anything round. Steady radial conduction through a cylindrical case obeys (1/r)·d/dr(r·dT/dr) = 0, which integrates to a logarithmic temperature profile, not a linear one. That equation is the scale-factor Laplacian with the symmetry crossed out.
At the CFD or FEA review board. Any axisymmetric model — nozzle, motor case, radome — is solved on a cylindrical or spherical grid, and the solver documentation is full of scale-factor bookkeeping. The reviewer question "what did you do at the axis?" exists because h_φ = 0 there.

How it works

Start with the one physical idea: a coordinate is a label, and the scale factor converts label changes into meters. On a 1 m radius sphere, stepping 1° in θ moves the probe 17.45 mm no matter where you are, because h_θ = ρ. Stepping 1° in φ moves it 17.45·sinθ mm — 8.73 mm at 30° off the pole, zero at the pole itself, because h_φ = ρ·sinθ. Once that picture is in your head, the operator formulas stop being incantations.

The payoff for the machinery is separation of variables. Match the coordinate surfaces to the boundary — a cylinder wall becomes r = const, a radome becomes ρ = const — and the boundary condition applies along one coordinate at a time, so a 3-D PDE splits into three ODEs ^[1]^[3]. The Helmholtz equation separates in exactly 11 orthogonal systems, Laplace's in 13 ^[3]^[10]; that short list is why Bessel functions, Legendre polynomials, and spherical harmonics exist. They are not exotic functions — they are what the Laplacian's scale factors do to a sine wave in cylindrical and spherical frames.

The gotchas, in the order they actually bite:

Unit vectors move. ê_r at one point and ê_r at another are different vectors. Differentiating a vector's components while holding the basis constant silently drops the −r·φ̇² and 2·ṙ·φ̇ terms in dynamics, and it is why ∇·(A_r ê_r) ≠ ∂A_r/∂r. A radially spreading field ê_ρ/ρ² has zero divergence away from the origin — the ρ² growth of the flux tube's cross-section exactly cancels the 1/ρ² decay, and only the scale-factor form of the divergence shows it.
The formulas above require orthogonality. They assume the metric is diagonal — coordinate surfaces intersecting at right angles everywhere. Skewed grids (a swept wing's body-fitted mesh, a sheared structural frame) need the full metric tensor with cross terms, and the tidy 1/h_i recipes no longer apply ^[2]^[3].
Coordinate singularities are grid problems, not physics problems. At r = 0 and along θ = 0, one scale factor vanishes: the azimuth becomes meaningless and components like A_φ can jump discontinuously even when the underlying field is perfectly smooth. Solvers regularize the axis; hand analyses need L'Hôpital or a Cartesian patch. A 1/r blowing up in your algebra at the axis of a smooth field is the coordinates failing, not the hardware.
The θ/φ convention swap. Physicists and most engineering references measure θ from the z-axis with φ as azimuth ^[1]; many math texts swap the two letters. A pattern file handed between an RF group and a math-convention plotting script will render transposed and look plausible. Check which angle carries the sinθ before trusting anyone's integral.
Degrees in, radians out. Scale factors assume angular coordinates in radians. An arc length or Laplacian evaluated with degrees is off by a factor of 57.3 — big enough to catch, small enough that people first blame the transducer.

Sanity check worth keeping: integrate the spherical area element h_θ·h_φ·dθ·dφ = ρ²·sinθ·dθ·dφ over the full sphere and you must get 4πρ². If a script or a derivation fails that, stop and fix the scale factors before anything downstream.

History

The bent-grid idea entered mathematics through one specific curve. Bonaventura Cavalieri worked the area inside an Archimedean spiral in his 1635 treatise using distance-and-angle reasoning, the earliest use Julian Coolidge's standard history could find ^[8]^[9]. Newton was the first to treat distance-and-angle as a general way of locating any point in the plane rather than a trick for one curve, and Jacob Bernoulli put a systematic version in Acta Eruditorum in 1691, naming the pole and the polar axis and computing curvature from a polar equation ^[8]^[9]. Alexis Clairaut raised the possibility of three-dimensional versions; Euler was the first to actually build them ^[8]^[9].

The general machinery — arbitrary orthogonal surfaces, scale factors, operators — belongs to Gabriel Lamé (1795–1870), a French engineer-mathematician who worked bridges and vaults as readily as analysis ^[4]^[5]. Studying heat conduction in the 1830s, he organized problems around families of isothermal surfaces — surfaces of constant temperature — and realized the surfaces themselves could serve as coordinates, with the geometry entering through what we now call scale factors ^[5]^[6]. He spent two decades developing the idea across memoirs on heat and elasticity, then consolidated it in his 1859 Leçons sur les coordonnées curvilignes et leurs diverses applications, the book that gave the subject its name ^[4]^[6]^[7]. The scale factors are still called Lamé coefficients in his honor ^[2]^[3]. The subject's later fate is the usual one for good machinery: absorbed into differential geometry as the diagonal special case of the metric tensor, which is exactly how a modern CFD solver carries it.

Related tools

/tools/rf-power-density — the 4πρ² sphere area behind spherical spreading is the surface integral of h_θ·h_φ
/tools/antenna-gain-beamwidth — gain lives on the sphere; the solid-angle math carries the sinθ scale factor
/tools/heat-conduction — radial conduction through walls is the cylindrical and spherical Laplacian at work
/tools/pressure-vessel-hoop — hoop stress is stress resolved in cylindrical components
/tools/torsion-shaft — shear stress varying linearly with r is a cylindrical-coordinate statement
/tools/convert-angle — the degrees-to-radians conversion the scale factors assume you already did

Sources

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