Curvilinear Coordinates and Scale Factors

You have now built the same machinery twice. For spherical coordinates: stretch factors $(1, r, r\sin\theta)$ , volume element $r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi$ , a divergence with $r^2$ tucked inside the radial derivative. For cylindrical: factors $(1, s, 1)$ , element $s\,\mathrm{d}s\,\mathrm{d}\phi\,\mathrm{d}z$ , an $s$ inside the derivative. The parallel is too perfect to be coincidence, and this article names the pattern: every orthogonal coordinate system is completely characterized by three functions $h_1, h_2, h_3$ — the scale factors, or Lamé coefficients — that convert coordinate changes into real lengths. Hand me the three $h$ 's and I will hand you back the displacement, the volume element, the gradient, the divergence, the curl, and the Laplacian, by turning a single crank. This is more than tidy bookkeeping. It is the difference between memorizing six formulas per coordinate system and owning one machine that manufactures them all — including for systems you have never met, the parabolic and elliptical and toroidal grids that turn specific hard boundary-value problems into easy ones. And it is a first, honest step toward one of the deep ideas of physics: that geometry is encoded in how coordinate displacements translate into distances. Promote the three scale factors to a general metric and you are standing at the door of general relativity. The plan: define the scale factors from the position vector, harvest the elements and operators once and for all, verify that spherical and cylindrical drop out as special cases, and then take the machine somewhere new.

Key takeaways

An orthogonal curvilinear system $(u_1,u_2,u_3)$ has coordinate lines that cross at right angles everywhere; each coordinate carries a scale factor $h_i=\big|\partial\mathbf{r}/\partial u_i\big|$ converting $\mathrm{d}u_i$ into a real length $h_i\,\mathrm{d}u_i$ .
Displacement: $\mathrm{d}\boldsymbol{\ell}=h_1\mathrm{d}u_1\,\hat{\mathbf{e}}_1+h_2\mathrm{d}u_2\,\hat{\mathbf{e}}_2+h_3\mathrm{d}u_3\,\hat{\mathbf{e}}_3$ ; volume element $\mathrm{d}\tau=h_1h_2h_3\,\mathrm{d}u_1\mathrm{d}u_2\mathrm{d}u_3$ (the Jacobian, factorized).
Cartesian is $(1,1,1)$ , cylindrical $(1,s,1)$ , spherical $(1,r,r\sin\theta)$ — all previous formulas are one machine with different fuel.
The master operators: gradient divides by $h_i$ ; divergence and Laplacian carry the transverse factors inside the derivatives, e.g. $\nabla\cdot\mathbf{F}=\frac{1}{h_1h_2h_3}\big[\partial_1(h_2h_3F_1)+\cdots\big]$ — geometry of the cell faces, not convention.
The machinery works only for orthogonal systems; skewed grids need the full metric tensor, the road to general relativity.
Key idea: three scale factors — the local exchange rates between coordinate changes and real lengths — completely determine an orthogonal coordinate system: every element and every differential operator is manufactured from $h_1,h_2,h_3$ by one universal recipe.

The core insight

Three scale factors $h_i=\big|\partial\mathbf{r}/\partial u_i\big|$ — the local exchange rates between coordinate changes and real distances — completely determine an orthogonal coordinate system: displacement, volume element, gradient, divergence, curl, and Laplacian all follow from $h_1,h_2,h_3$ by one universal recipe.

Where this lives today

The most literal descendants of Lamé's machinery are the body-fitted grids of computational physics. Airflow over a wing, plasma in a tokamak, the electromagnetic field in a curved waveguide — production solvers generate a curvilinear coordinate system shaped to the hardware, compute the scale factors (numerically, cell by cell), and run exactly the master formulas of this article on the resulting grid. When a simulation paper says "generalized coordinates," it means equations (2) through (6) with $h$ 's tabulated by mesh software. The most spectacular descendant is transformation optics. Maxwell's equations keep their form under coordinate transformations, with the geometry reappearing as effective material properties — so a desired bending of light can be specified as a coordinate transformation and then built as a metamaterial whose permittivity and permeability mimic the transformed metric. The invisibility-cloak literature that erupted after 2006 is scale-factor engineering: space is not actually curved, but light is made to believe it is. And in general relativity the promotion is complete — there the metric is not a computational convenience but the gravitational field itself, and this article's $h_1,h_2,h_3$ are the friendly diagonal corner of that grand structure.

The physical picture

Think of a coordinate system as a currency, and space as the land you are traveling through. A step of "one unit" in coordinate $u_i$ buys you a different amount of actual distance depending on where you stand — one degree of longitude is 111 km at the equator and nothing at the pole. The scale factor $h_i$ is the local exchange rate: real distance per unit of coordinate. Cartesian coordinates are the special currency whose rate is everywhere 1; all the exotic behaviour of spherical and cylindrical formulas — the $r^2\sin\theta$ , the lonely $s$ — is nothing but exchange rates showing up in the arithmetic.

Figure

An orthogonal curvilinear grid — coordinate lines curve, but cross at right angles, and the brick edges are h du

Orthogonality is the property that keeps everything simple: at every point, the three coordinate directions are mutually perpendicular, so the infinitesimal cell is an honest (curvilinear) brick — edge lengths $h_1\mathrm{d}u_1$ , $h_2\mathrm{d}u_2$ , $h_3\mathrm{d}u_3$ , volume the product of the three. All the coordinate systems of daily physics — Cartesian, cylindrical, spherical, plus the exotic parabolic, elliptic, and toroidal grids — are orthogonal. (Skewed systems, where grid lines cross at changing angles, need a full metric tensor with off-diagonal entries; that heavier machinery belongs to general relativity, and knowing the boundary of the light machinery is part of owning it.)

Figure

One machine, three fuels — the familiar systems as scale-factor triples

The scale factors, defined and harvested

Let $(u_1,u_2,u_3)$ label points, with the position vector $\mathbf{r}(u_1,u_2,u_3)$ known in Cartesian terms. Nudge one coordinate: the point moves by $\partial\mathbf{r}/\partial u_i\,\mathrm{d}u_i$ . The direction of that move defines the unit vector $\hat{\mathbf{e}}_i$ ; its length per unit coordinate defines the scale factor:

h_i \equiv \left|\frac{\partial\mathbf{r}}{\partial u_i}\right|, \qquad \hat{\mathbf{e}}_i=\frac{1}{h_i}\frac{\partial\mathbf{r}}{\partial u_i} \qquad (1)

where: orthogonality means $\hat{\mathbf{e}}_i\cdot\hat{\mathbf{e}}_j=\delta_{ij}$ at every point, with the triad ordered right-handed. The definition is completely mechanical — differentiate, take the magnitude — and it packages all the geometry of the system. Note the units: if $u_i$ is a length (like $r$ or $z$ ), $h_i$ is dimensionless; if $u_i$ is an angle, $h_i$ carries the metres. The scale factor is precisely what keeps every formula dimensionally honest. The general displacement adds the three perpendicular nudges, and the elements follow by multiplying edges exactly as they did on the sphere and the cylinder:

\mathrm{d}\boldsymbol{\ell}=h_1\,\mathrm{d}u_1\,\hat{\mathbf{e}}_1+h_2\,\mathrm{d}u_2\,\hat{\mathbf{e}}_2+h_3\,\mathrm{d}u_3\,\hat{\mathbf{e}}_3, \qquad \mathrm{d}\tau=h_1h_2h_3\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3 \qquad (2)

where: the area element on a surface of constant $u_1$ is $h_2h_3\,\mathrm{d}u_2\,\mathrm{d}u_3\,\hat{\mathbf{e}}_1$ , and cyclically. The product $h_1h_2h_3$ is the Jacobian of the transformation, factorized into meaningful pieces — one exchange rate per direction — rather than delivered as an opaque determinant.

Derivation AComputing scale factors — spherical coordinates as the worked case

The goal is to show that definition (1) is a two-line calculation in practice, by extracting the spherical factors $(1,\ r,\ r\sin\theta)$ from the position vector — no geometric ingenuity required, just differentiation. The strategy: write $\mathbf{r}$ in Cartesian components as a function of $(r,\theta,\phi)$ , differentiate with respect to each coordinate in turn, and take magnitudes. Watch the trigonometric identities collapse each magnitude to a single clean factor — that collapse is orthogonality quietly certifying itself.

Step 1. The position vector: $\mathbf{r}=r\sin\theta\cos\phi\,\hat{\mathbf{x}}+r\sin\theta\sin\phi\,\hat{\mathbf{y}}+r\cos\theta\,\hat{\mathbf{z}}$ .

Step 2. Differentiate with respect to $r$ : $\partial\mathbf{r}/\partial r=(\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta)$ . Its magnitude: $\sin^2\theta(\cos^2\phi+\sin^2\phi)+\cos^2\theta=1$ , so $h_r=1$ . Why it had to be 1: $r$ is itself a distance, so a unit change of $r$ is a unit of real length.

Step 3. Differentiate with respect to $\theta$ : $\partial\mathbf{r}/\partial\theta=(r\cos\theta\cos\phi,\ r\cos\theta\sin\phi,\ -r\sin\theta)$ . Magnitude: $r\sqrt{\cos^2\theta+\sin^2\theta}=r$ , so $h_\theta=r$ .

Step 4. Differentiate with respect to $\phi$ : $\partial\mathbf{r}/\partial\phi=(-r\sin\theta\sin\phi,\ r\sin\theta\cos\phi,\ 0)$ . Magnitude: $r\sin\theta$ , so $h_\phi=r\sin\theta$ . Why this is the axis distance: the $\phi$ -motion is a circle around the $z$ -axis, and $r\sin\theta$ is exactly the perpendicular distance to that axis — the geometry of the spherical article, recovered by blind differentiation.

Step 5. Orthogonality check (the license for everything downstream): the three derivative vectors have pairwise dot products $\big(\partial_r\mathbf{r}\big)\cdot\big(\partial_\theta\mathbf{r}\big)=r\sin\theta\cos\theta-r\sin\theta\cos\theta=0$ , and similarly for the other pairs — all zero, everywhere. Verified: reproduces the spherical results of Griffiths §1.4; the same two-line procedure applied to $(s\cos\phi,\ s\sin\phi,\ z)$ yields the cylindrical triple $(1,\ s,\ 1)$ .

The master operators

With lengths under control, the differential operators follow from their coordinate-free definitions — the same definitions used on the sphere and cylinder, now run once for all systems. The gradient is "rate of change per unit length in each direction," so each derivative is divided by its exchange rate:

\nabla T=\frac{1}{h_1}\frac{\partial T}{\partial u_1}\,\hat{\mathbf{e}}_1+\frac{1}{h_2}\frac{\partial T}{\partial u_2}\,\hat{\mathbf{e}}_2+\frac{1}{h_3}\frac{\partial T}{\partial u_3}\,\hat{\mathbf{e}}_3 \qquad (3)

where: the quickest honest route to (3) is to demand $\mathrm{d}T=\nabla T\cdot\mathrm{d}\boldsymbol{\ell}$ for the displacement (2): the chain rule gives $\mathrm{d}T=\sum_i\partial_iT\,\mathrm{d}u_i$ , and matching coefficients against $\sum_i(\nabla T)_i\,h_i\,\mathrm{d}u_i$ forces $(\nabla T)_i=\partial_iT/h_i$ . Insert $(1,r,r\sin\theta)$ and the spherical gradient reappears; insert $(1,s,1)$ for the cylindrical one. The divergence is flux out per unit volume, computed on the curvilinear brick — and here the transverse scale factors are forced inside the derivatives, because opposite faces of the brick have different areas:

\nabla\cdot\mathbf{F}=\frac{1}{h_1h_2h_3}\left[\frac{\partial}{\partial u_1}\big(h_2h_3F_1\big)+\frac{\partial}{\partial u_2}\big(h_3h_1F_2\big)+\frac{\partial}{\partial u_3}\big(h_1h_2F_3\big)\right] \qquad (4)

where: $h_2h_3$ is precisely the area factor of the faces the $F_1$ flux crosses. Feed the machine spherical fuel: $h_2h_3=r^2\sin\theta$ , and the first term becomes $\frac{1}{r^2}\partial_r(r^2F_r)$ — the formula you met before, now inevitable rather than memorized.

Go deeper · Derivation B · 1 minThe master divergence from flux through the curvilinear brick

Derivation BThe master divergence from flux through the curvilinear brick

The goal is equation (4), proved once for every orthogonal system simultaneously — after this, the spherical and cylindrical cell arguments you have already seen become two specializations of a single computation. The strategy is the standard one: compute the net flux out of the brick with edges $h_i\,\mathrm{d}u_i$ , divide by its volume. The step to watch is the same one as always, now in full generality: both the field component and the face area vary from one face to the opposite one, and the product rule bundles them into a single derivative of $(\text{area factor})\times(\text{component})$ .

Step 1. The two faces perpendicular to $\hat{\mathbf{e}}_1$ sit at $u_1$ and $u_1+\mathrm{d}u_1$ . Each has area $(h_2\,\mathrm{d}u_2)(h_3\,\mathrm{d}u_3)$ — evaluated at its own location, where $h_2$ and $h_3$ generally differ. The net outflow through this pair is

\big[F_1h_2h_3\big]_{u_1+\mathrm{d}u_1}\mathrm{d}u_2\,\mathrm{d}u_3-\big[F_1h_2h_3\big]_{u_1}\mathrm{d}u_2\,\mathrm{d}u_3=\frac{\partial}{\partial u_1}\big(h_2h_3F_1\big)\,\mathrm{d}u_1\,\mathrm{d}u_2\,\mathrm{d}u_3

Why the whole product is differentiated: a constant field still leaks net flux if the far face is bigger — geometry and field variation enter on exactly the same footing, and only the product $h_2h_3F_1$ knows both.

Step 2. The other two face-pairs contribute cyclically: $\partial_2(h_3h_1F_2)$ and $\partial_3(h_1h_2F_3)$ , each times $\mathrm{d}u_1\mathrm{d}u_2\mathrm{d}u_3$ .

Step 3. Divide the total by the brick's volume $h_1h_2h_3\,\mathrm{d}u_1\mathrm{d}u_2\mathrm{d}u_3$ to get flux per volume — equation (4).

Step 4. Harvest the special cases. Cartesian $(1,1,1)$ : all factors disappear, $\nabla\cdot\mathbf{F}=\partial_xF_x+\partial_yF_y+\partial_zF_z$ . Cylindrical $(1,s,1)$ : $h_2h_3=s$ gives $\frac{1}{s}\partial_s(sF_s)+\frac{1}{s}\partial_\phi F_\phi+\partial_zF_z$ . Spherical $(1,r,r\sin\theta)$ : $h_2h_3=r^2\sin\theta$ gives $\frac{1}{r^2}\partial_r(r^2F_r)+\frac{1}{r\sin\theta}\partial_\theta(\sin\theta F_\theta)+\frac{1}{r\sin\theta}\partial_\phi F_\phi$ . Why this harvest matters: three previously separate derivations are now one — and any new orthogonal system you ever meet is pre-solved. *Verified: reproduces Griffiths Appendix A and Boas Ch. 10; the three specializations match the formulas derived independently in the spherical and cylindrical articles.*

The remaining two operators come from composing what we have. The Laplacian is divergence-of-gradient — insert (3) into (4):

\nabla^2T=\frac{1}{h_1h_2h_3}\left[\frac{\partial}{\partial u_1}\!\left(\frac{h_2h_3}{h_1}\frac{\partial T}{\partial u_1}\right)+\frac{\partial}{\partial u_2}\!\left(\frac{h_3h_1}{h_2}\frac{\partial T}{\partial u_2}\right)+\frac{\partial}{\partial u_3}\!\left(\frac{h_1h_2}{h_3}\frac{\partial T}{\partial u_3}\right)\right] \qquad (5)

where: each slot carries the ratio (transverse area)/(own factor) — one $h$ from the gradient's division, two from the divergence's areas. This is the formula behind every separation-of-variables solution in E2: the shapes of $h_1,h_2,h_3$ decide whether the Laplace equation splits into ordinary differential equations, and the classical coordinate systems are famous precisely because theirs do. The curl follows the same grammar, tidiest as a determinant:

\nabla\times\mathbf{F}=\frac{1}{h_1h_2h_3}\begin{vmatrix} h_1\hat{\mathbf{e}}_1 & h_2\hat{\mathbf{e}}_2 & h_3\hat{\mathbf{e}}_3\\[2pt] \partial/\partial u_1 & \partial/\partial u_2 & \partial/\partial u_3\\[2pt] h_1F_1 & h_2F_2 & h_3F_3 \end{vmatrix} \qquad (6)

where: each component reproduces the circulation-per-area computation on the corresponding brick face — for instance $(\nabla\times\mathbf{F})_1=\frac{1}{h_2h_3}\big[\partial_2(h_3F_3)-\partial_3(h_2F_2)\big]$ . Note the pattern across all four operators: gradient divides by one $h$ ; divergence and curl differentiate components pre-multiplied by $h$ 's. Once that asymmetry makes sense (lengths per coordinate versus areas swept by faces and edges), the formulas stop being memorization and start being readable.

Which formula answers which question

you want	use	notes
the scale factors of any system	$h_i=\big\|\partial\mathbf{r}/\partial u_i\big\|$	two lines of differentiation
lengths, areas, volumes	equation (2)	multiply the swept edges
slope of a scalar	equation (3)	divide by $h_i$
sources of a field	equation (4)	transverse areas inside the derivative
Laplace or Poisson equation	equation (5)	the separability engine of E2
swirl of a field	equation (6)	determinant with $h$ 's in rows one and three

Limiting cases

Case	Result	Meaning
$h=(1,1,1)$	all operators collapse to Cartesian	the machine's calibration check
$h=(1,s,1)$	cylindrical formulas	one curved direction
$h=(1,r,r\sin\theta)$	spherical formulas	two curved directions
drop $u_3$ , $h=(1,s)$	plane polar	2D electrostatics and central-force orbits
all $h_i$ constant but unequal	rescaled Cartesian (anisotropic stretch)	units conversion is a "coordinate system"
non-orthogonal grid	machinery fails; needs metric tensor $g_{ij}$	the door to general relativity

Common trapScale factors are functions, not constants — and the operators know it

The single most damaging habit from Cartesian life is treating differentiation and multiplication by geometry as interchangeable. In equation (4) the derivative acts on the product $h_2h_3F_1$ ; pulling the scale factors out front — writing $\frac{h_2h_3}{h_1h_2h_3}\partial_1F_1$ — silently deletes the geometric leak terms and produces a "divergence" that fails on the simplest tests (it would give $\nabla\cdot\hat{\mathbf{r}}/r^2\neq0$ off the origin, breaking Gauss's law). The same discipline applies to the Laplacian's nested structure and to the curl determinant, where the $h$ 's inside row three must be differentiated by row two before anything is divided. Two quick self-audits catch most slips: units (every term of a gradient must be field-per-length; every term of a divergence, field-per-length with the field's own units) and the Cartesian limit (set all $h=1$ and confirm your expression collapses correctly). If either audit fails, a scale factor is on the wrong side of a derivative.

Worked examples

1 · The length of a helical antenna wire (confidence builder)

A wire is wound as a helix of radius $R=3.0\ \mathrm{cm}$ and pitch $p=2.0\ \mathrm{cm}$ (vertical rise per turn), for $N=25$ turns. Find the wire's length.

Step 1. Choose cylindrical coordinates $(s,\phi,z)$ , scale factors $(1,s,1)$ . The helix is $s=R$ (constant), $z=\dfrac{p}{2\pi}\phi$ , with $\phi$ running from $0$ to $2\pi N$ .

Step 2. The line element from equation (2), with $\mathrm{d}s=0$ : $\mathrm{d}\ell=\sqrt{(h_\phi\,\mathrm{d}\phi)^2+(\mathrm{d}z)^2}=\sqrt{R^2+\big(\tfrac{p}{2\pi}\big)^2}\,\mathrm{d}\phi$ . Note the scale factor at work: the $\mathrm{d}\phi$ contributes arc $R\,\mathrm{d}\phi$ , not $\mathrm{d}\phi$ .

Step 3. The integrand is constant, so $L=2\pi N\sqrt{R^2+\big(\tfrac{p}{2\pi}\big)^2}=N\sqrt{(2\pi R)^2+p^2}$ — per turn, the wire is the hypotenuse of a rectangle "circumference by pitch," unrolled.

Step 4. Numbers: $2\pi R=0.1885\ \mathrm{m}$ , so per turn $\sqrt{0.1885^2+0.020^2}=\sqrt{0.03553+0.00040}=0.1896\ \mathrm{m}$ , and $L=25\times0.1896=\boxed{4.74\ \mathrm{m}}$ . Check the limits: $p\to0$ gives $25$ flat circles, $L=4.71\ \mathrm{m}$ — the pitch adds barely 3 cm, as the nearly-flat winding suggests.

2 · Running the machine on an unfamiliar system (exam level)

Parabolic cylindrical coordinates $(u,v,z)$ are defined by $x=\tfrac12(u^2-v^2)$ , $y=uv$ , $z=z$ . (a) Find the scale factors and verify orthogonality. (b) Compute the area of the patch $0\le u\le1.0$ , $0\le v\le2.0$ in the $z=0$ plane (coordinates in metres).

Step 1. Differentiate the position vector. $\partial\mathbf{r}/\partial u=(u,\ v,\ 0)$ and $\partial\mathbf{r}/\partial v=(-v,\ u,\ 0)$ , with $\partial\mathbf{r}/\partial z=(0,0,1)$ .

Step 2. Orthogonality: $(u,v,0)\cdot(-v,u,0)=-uv+uv=0$ , and both are orthogonal to $\hat{\mathbf{z}}$ — the system qualifies, so all master formulas apply.

Step 3. Scale factors: $h_u=\sqrt{u^2+v^2}$ , $h_v=\sqrt{u^2+v^2}$ , $h_z=1$ . (Equal $h_u=h_v$ is the signature of a conformal map in the plane.) At the point $(u,v)=(1.0,\ 2.0)$ : $h_u=h_v=\sqrt{5}=2.24$ — one coordinate unit there buys $2.24$ m of real distance.

Step 4. The area element on $z=0$ is $h_uh_v\,\mathrm{d}u\,\mathrm{d}v=(u^2+v^2)\,\mathrm{d}u\,\mathrm{d}v$ . Integrate:

A=\int_0^{1}\!\!\int_0^{2}(u^2+v^2)\,\mathrm{d}v\,\mathrm{d}u=\int_0^1\Big[2u^2+\tfrac{8}{3}\Big]\mathrm{d}u=\tfrac{2}{3}+\tfrac{8}{3}=\boxed{\tfrac{10}{3}\approx3.3\ \mathrm{m^2}}

Step 5. Sanity check: the patch's coordinate rectangle is $1\times2$ , but the exchange rate $u^2+v^2$ averages well above 1 over the region (it reaches $5$ at the far corner), so a real area of $3.3\ \mathrm{m^2}$ — bigger than the naive $2$ — is exactly the right kind of number. A machine you can audit is a machine you can trust on systems you have never seen before.

Origins · 1 minGabriel Lamé and how this was discovered

Gabriel Lamé

1795–1870 · École Polytechnique and the Sorbonne

Lamé was the nineteenth century's great believer in coordinates fitted to problems. An engineer-mathematician who spent a decade building railways and schools in Russia before returning to Paris chairs, he attacked heat conduction in crystals and ellipsoids where the standard grids fail — and, in doing so, built the general theory of curvilinear coordinates, publishing the definitive Leçons on the subject in 1859. The scale factors of this article are called the Lamé coefficients in his honour, and the curvilinear Laplacian machinery is essentially his. His fingerprints are all over solid mechanics too: the Lamé parameters of elasticity theory measure every material's stiffness to this day. Gauss praised him as the finest French mathematician of his generation; characteristically, Lamé thought of himself as an engineer. Fitting coordinates to geometry, he insisted, was not a trick but a method — the metric viewpoint that Riemann would soon make universal.

History

1828

— Gauss's Disquisitiones circa superficies curvas builds the geometry of surfaces from the line element — distance data alone.

1833

— Lamé, studying heat flow in crystals, begins fitting coordinates to boundaries and tracking the length-per-coordinate factors.

1854

— Riemann's habilitation lecture generalizes the line element to arbitrary metrics in any dimension — curvilinear coordinates outgrow flat space.

1859

— Lamé's Leçons sur les coordonnées curvilignes codifies scale factors and the operator machinery; the coefficients take his name.

1915

— Einstein's general relativity promotes the metric from bookkeeping to dynamics: geometry itself becomes the gravitational field.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §1.4 and Appendix A (curvilinear coordinates) · Jackson, Classical Electrodynamics, 3rd ed., front-cover vector formulas and §3 · Zangwill, Modern Electrodynamics, §1.2 and Appendix C · Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Ch. 10 (curvilinear coordinates) · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., §3.10 · Morse & Feshbach, Methods of Theoretical Physics, Ch. 5 (the separable systems)

Provenance. Scale factor definitions and the gradient/divergence/curl/Laplacian master formulas follow Griffiths §1.4 and Appendix A, and Boas Ch. 10; verifier should confirm the spherical $h_\theta=r,\,h_\phi=r\sin\theta$ by direct differentiation of $\mathbf{r}=r\sin\theta\cos\phi\,\hat{\mathbf{x}}+\dots$ , and check the spherical/cylindrical Laplacians against Jackson's inside-cover formulas. All limiting cases ( $h_i=1$ Cartesian) are recomputable by substitution. Worked-example numbers on this page (helix length $L=N\sqrt{(2\pi R)^2+p^2}=4.74$ m for $R=3$ cm, $p=2$ cm, $N=25$ ; parabolic-cylindrical $h_u=h_v=\sqrt{u^2+v^2}$ and patch area $10/3\ \mathrm{m^2}$ ) are illustrative arithmetic recomputed from the boxed formulas; the parabolic-cylindrical scale factors match Morse & Feshbach's tabulation.

Builds on: Scalar and Vector Fields · The Del Operator (Nabla) · The Gradient Operator · Line Integrals · Surface Integrals and Flux Leads to: The Divergence (Gauss) Theorem · The Curl (Stokes) Theorem · Second Derivatives and the Laplacian

How do I compute the scale factors of a system I have never seen?

Mechanically, in three moves: write the position vector in Cartesian components as a function of the new coordinates, differentiate with respect to each coordinate, and take magnitudes: $h_i=|\partial\mathbf{r}/\partial u_i|$ . While you are there, dot the derivative vectors pairwise — all zero means the system is orthogonal and every master formula applies; any nonzero dot product means stop, this machinery does not cover it. The parabolic-coordinates worked example runs the full procedure in four lines. With the three $h$ 's in hand you never derive anything else for that system: elements and operators are read off equations (2) through (6).

Why do some scale factors carry units of length and others not?

Because coordinates themselves come in two species: lengths (like $r$ , $s$ , $z$ ) and pure numbers (like angles). A displacement $h_i\,\mathrm{d}u_i$ must always be a length, so the scale factor supplies whatever units the coordinate lacks — $h_\theta=r$ carries metres because $\mathrm{d}\theta$ carries none, while $h_r=1$ is unitless because $\mathrm{d}r$ is already a length. This is more useful than it sounds: dimensional analysis becomes a proofreading tool. Every term of a gradient must have units of field-per-length; if one term of your expression has field-per-radian instead, a scale factor has been dropped, and the units tell you exactly which one.

What exactly goes wrong for non-orthogonal coordinates?

The brick stops being a brick. When coordinate directions meet at oblique (and position-dependent) angles, the cell is a parallelepiped: its volume involves the angles, displacement components mix, and a single number per direction can no longer encode the geometry. The honest bookkeeping is the metric tensor $g_{ij}=\frac{\partial\mathbf{r}}{\partial u_i}\cdot\frac{\partial\mathbf{r}}{\partial u_j}$ — a symmetric matrix whose diagonal holds the squared scale factors and whose off-diagonal entries measure the skew. Orthogonal systems are precisely those with diagonal $g$ , where $g_{ii}=h_i^2$ and everything in this article follows. The general- $g$ machinery (index gymnastics, Christoffel symbols) is the standard toolkit of general relativity — this article is its diagonal, flat-space apprenticeship.

Is the product h₁h₂h₃ the same thing as the Jacobian determinant?

For orthogonal systems, yes — up to sign. The Jacobian determinant $\det(\partial(x,y,z)/\partial(u_1,u_2,u_3))$ measures the volume of the parallelepiped spanned by the three derivative vectors; when those vectors are mutually perpendicular, that volume is simply the product of their lengths, $h_1h_2h_3$ . So the mysterious $|J|$ of multivariable calculus factorizes, for orthogonal systems, into three meaningful exchange rates: the $r^2\sin\theta$ of spherical coordinates is not one opaque determinant but $(1)\times(r)\times(r\sin\theta)$ , one factor per direction. For skewed systems the determinant picks up the angle factors and the neat factorization is lost.

Why do the gradient and the divergence treat the scale factors so differently?

Because they answer different geometric questions. The gradient asks "how fast does the scalar change per metre in each direction?" — a length question, so each coordinate derivative gets divided by its own exchange rate, one $h$ per component. The divergence asks "how much net flux leaves per unit volume?" — an area and volume question: the flux through a face involves the face's area ( $h_2h_3$ for an $\hat{\mathbf{e}}_1$ -face, inside the derivative because areas vary from face to face), and the whole is divided by the brick volume $h_1h_2h_3$ . Curl, an area-and-edge question, mixes the two patterns. Once you attach each $h$ to the geometric object it measures — edge, face, or volume — the four master formulas become readable sentences rather than arbitrary strings.

Will I ever actually use a coordinate system beyond the big three?

Sooner than you might think, and always for the same reason: separability. Laplace's and Helmholtz's equations split into ordinary differential equations only when the boundaries are coordinate surfaces, and classical physics assembled a whole zoo for exactly this purpose — parabolic coordinates for the field near a charged parabolic tip (and, famously, the quantum Stark effect), prolate spheroidal for the two-centre problem of the hydrogen molecular ion, elliptic cylindrical for slots and strips, toroidal for fusion plasmas. The Helmholtz equation separates in precisely eleven orthogonal systems, all catalogued in Morse and Feshbach. You will not memorize them — but with this article's machine, you can operate any of them the day a problem demands it.

How does this connect to the metric of special and general relativity?

Directly — this is the shallow end of that pool. Write the squared line element from equation (2): $\mathrm{d}\ell^2=h_1^2\,\mathrm{d}u_1^2+h_2^2\,\mathrm{d}u_2^2+h_3^2\,\mathrm{d}u_3^2$ . That quadratic form is a metric: the rule that converts coordinate differences into distances. Relativity makes two upgrades: time joins the coordinates (with its notorious minus sign), and the coefficients — now written $g_{\mu\nu}$ , allowed off-diagonal entries — become dynamical, determined by matter through Einstein's equations rather than chosen for convenience. But the conceptual move is the one this article already made: geometry lives not in the coordinates themselves, which are arbitrary labels, but in the conversion table between coordinates and lengths. Master scale factors and the metric tensor is a generalization, not a revolution.