Vector Operators in Curvilinear Coordinates

The curvilinear operators are the first line of almost every quantitative model that has symmetry to exploit. Separation of variables — the workhorse of mathematical physics — only works because the Laplacian in the right coordinate system splits into independent ordinary differential equations: the spherical Laplacian gives the Legendre equation and the spherical harmonics behind electrostatic multipoles, the CMB power spectrum, and every atomic orbital; the cylindrical Laplacian gives Bessel's equation and the modes of optical fibres and microwave waveguides. Numerical practice leans just as hard on it: finite-element and spectral solvers map irregular geometries onto reference grids through exactly the Jacobian $J=h_1h_2h_3$, and computational fluid dynamics writes the Navier–Stokes equations in body-fitted coordinates so a turbine blade or an artery wall becomes a coordinate surface. In general relativity the same bookkeeping, promoted to a non-diagonal metric $g_{\mu\nu}$, becomes the covariant derivative — the Lamé coefficients are the flat-space ancestor of the Christoffel symbols. Whenever a simulation reports a field on a sphere, a cylinder, or a warped mesh, equations (4)–(7) are running underneath.

Picture the coordinate surfaces as a curved cage drawn on space. Where they cross, three coordinate curves pass through the point and the local unit vectors $\hat{\mathbf{e}}_1,\hat{\mathbf{e}}_2,\hat{\mathbf{e}}_3$ point along them. A unit step $\mathrm{d}u_i=1$ no longer means one metre: it means $h_i$ metres. The scale factors are the local exchange rate between "coordinate units" and "physical metres", and since every differential operator is a rate of change per physical metre, each derivative $\partial/\partial u_i$ is divided by its $h_i$. The first figure is the elementary curvilinear box on which all four operators are evaluated: its edges are $h_i\,\mathrm{d}u_i$, its faces have areas like $h_2h_3\,\mathrm{d}u_2\,\mathrm{d}u_3$, and its volume is $J\,\mathrm{d}u_1\mathrm{d}u_2\mathrm{d}u_3$.

The volume Jacobian $J=h_1h_2h_3$ appears in the divergence and Laplacian because those operators measure flux per unit volume, and both the box volume and the face areas are coordinate-dependent — tracking how a face area changes as you cross the box generates the derivative-of-$h$ terms. Curl, by contrast, is circulation per unit area and involves products of only two scale factors. The second figure makes this concrete for the workhorse case, spherical coordinates, whose scale factors $(1,\,r,\,r\sin\theta)$ give the volume element its famous $r^2\sin\theta$.

Everything begins with the map from coordinates to physical space. Differentiating the position vector along each coordinate curve gives a tangent vector; its length is the scale factor and its direction is the local unit vector. In orthogonal systems the three tangent directions are mutually perpendicular, which keeps every later formula diagonal.

where: $\mathbf{r}$ — position vector as a function of the coordinates; $u_i$ — the three curvilinear coordinates, $i=1,2,3$; $h_i$ — scale factor, the physical length per unit change in $u_i$; $\hat{\mathbf{e}}_i$ — unit tangent along the $u_i$-curve. Reading it: a coordinate step $\mathrm{d}u_i$ moves you $h_i\,\mathrm{d}u_i$ metres along $\hat{\mathbf{e}}_i$. The line element collects the three scale factors into the single object from which every operator descends. Because the triad is orthonormal the metric is diagonal — no cross terms $\mathrm{d}u_i\,\mathrm{d}u_j$.

where: $\mathrm{d}\mathbf{r}$ — infinitesimal physical displacement; $\mathrm{d}s$ — physical arc length. Reading it: the $h_i^2$ are the diagonal entries of the metric tensor; knowing them is knowing the geometry. From the line element the geometric measures follow at once. The box built from the three edge vectors has volume equal to the product of the edge lengths (orthogonality again), and each face area is the product of two of them.

where: $\mathrm{d}V$ — physical volume of the curvilinear box; $\mathrm{d}\mathbf{S}_1$ — vector area element on the face of constant $u_1$. The gradient measures rate of change per physical length, so each partial derivative is divided by one scale factor — it is the only operator that carries no Jacobian.

where: $\phi$ — a scalar field; $\nabla\phi$ — its gradient vector, pointing along steepest ascent with magnitude equal to the physical slope. Reading it: divide each $\partial\phi/\partial u_i$ by $h_i$. The divergence is net outward flux per unit volume. The face areas $h_jh_k$ are themselves coordinate-dependent, so they sit inside the derivatives, with the $1/(h_1h_2h_3)$ prefactor supplied by the box volume.

where: $\mathbf{A}$ — vector field with physical components $A_i=\mathbf{A}\cdot\hat{\mathbf{e}}_i$; $h_jh_k$ — the area factor of the face pierced by $A_i$. Reading it: weight each component by its face area before differentiating. The curl is circulation per unit area, assembling into a determinant whose bottom row holds the covariant components $h_iA_i$ — the projections of $\mathbf{A}$ onto the physical edge vectors.

where: $h_iA_i$ — covariant component, the piece of $\mathbf{A}$ appearing in the circulation integral $\oint\mathbf{A}\cdot\mathrm{d}\mathbf{r}$. Reading it: expand along the top row as for the Cartesian curl, but differentiate $h_iA_i$, not $A_i$. The Laplacian is $\nabla\cdot(\nabla\phi)$, inheriting the Jacobian of the divergence and the $1/h_i$ of the gradient in one self-adjoint second-order operator.

where: $\nabla^2\phi$ — the Laplacian of $\phi$; each bracket is the divergence formula with $A_i=(1/h_i)\partial\phi/\partial u_i$ inserted. Reading it: the coefficient of each second derivative is $h_jh_k/h_i$, perpendicular face area over parallel edge length. A final identity closes the loop between the tangent basis $\hat{\mathbf{e}}_i$ and the gradient (normal) basis $\nabla u_i$, and fixes the handedness of the triad.

where: $\nabla u_i$ — gradient of the coordinate function, normal to the constant-$u_i$ surface; $\varepsilon_{ijk}$ — Levi-Civita symbol enforcing a right-handed ordering. In an orthogonal system tangent and normal directions coincide, so $\hat{\mathbf{e}}_i$ and $\nabla u_i$ differ only by the factor $h_i$.

Every named coordinate system is one line of the master formulas with its scale factors substituted — the general result is only specialized, never re-derived.

1 · From the master formula to the spherical Laplacian (confidence builder)

Specialize the general Laplacian (7) to spherical coordinates, then test it on $\phi=r^2$ and $\phi=1/r$.

Step 1. Insert $(h_r,h_\theta,h_\phi)=(1,r,r\sin\theta)$, for which $J=r^2\sin\theta$. The three coefficients $h_jh_k/h_i$ become $h_\theta h_\phi/h_r=r^2\sin\theta$, $h_\phi h_r/h_\theta=\sin\theta$, and $h_r h_\theta/h_\phi=1/\sin\theta$.

Step 2. Assemble equation (7) and pull the constant $\sin\theta$ out of the radial term to reach the standard form $\nabla^2\phi=\dfrac1{r^2}\partial_r(r^2\partial_r\phi)+\dfrac1{r^2\sin\theta}\partial_\theta(\sin\theta\,\partial_\theta\phi)+\dfrac1{r^2\sin^2\theta}\partial_\phi^2\phi$.

Step 3. Test on $\phi=r^2$ (spherically symmetric, so only the radial term survives): $\nabla^2 r^2=\dfrac1{r^2}\partial_r(r^2\cdot 2r)=\dfrac1{r^2}\partial_r(2r^3)=\dfrac1{r^2}(6r^2)=\mathbf{6}$. Cross-check in Cartesian, where $r^2=x^2+y^2+z^2$ gives $\nabla^2 r^2=2+2+2=\mathbf{6}$ — the two agree exactly.

Step 4. Test on the Coulomb potential $\phi=1/r$: $\nabla^2(1/r)=\dfrac1{r^2}\partial_r\!\big(r^2\cdot(-1/r^2)\big)=\dfrac1{r^2}\partial_r(-1)=\mathbf{0}$ for $r\ne0$. The spherical Laplacian gives ∇²r² = 6 and ∇²(1/r) = 0 away from the origin — the latter is why 1/r is the free-space potential of a point charge.

2 · Cylindrical divergence and a flux check with numbers

Take the radial field $\mathbf{A}=\rho\,\hat{\mathbf{e}}_\rho$. Compute its divergence, then verify the Divergence Theorem on a cylinder of radius $R=2$ and height $L=5$.

Step 1. Here $A_\rho=\rho$, $A_\phi=A_z=0$, with $(h_\rho,h_\phi,h_z)=(1,\rho,1)$. Equation (5) gives $\nabla\cdot\mathbf{A}=\dfrac1\rho\partial_\rho(\rho\cdot\rho)=\dfrac1\rho(2\rho)=\mathbf{2}$, a uniform divergence.

Step 2. Volume integral: $\displaystyle\int_V\nabla\cdot\mathbf{A}\,\mathrm{d}V=2\times(\pi R^2 L)=2\times20\pi=\mathbf{40\pi}$, using the cylinder volume $\pi R^2 L=\pi\cdot4\cdot5$.

Step 3. Surface flux: the field is purely radial, so the flat top and bottom (normals along $\pm\hat{\mathbf{e}}_z$) contribute nothing. The curved side has outward normal $\hat{\mathbf{e}}_\rho$, area $2\pi R L$, and there $A_\rho=R$, so the flux is $R\times2\pi R L=2\pi R^2 L=2\pi\cdot4\cdot5=\mathbf{40\pi}$.

Step 4. The two independent computations agree: ∫∇·A dV = ∮A·dS = 40π, confirming both the cylindrical divergence formula and Gauss's theorem.

3 · Why a point source has zero divergence

The inverse-square field $\mathbf{A}=(k/r^2)\,\hat{\mathbf{e}}_r$ models a point source of strength $k$. Show its divergence vanishes for $r>0$ yet its flux through any enclosing sphere is $4\pi k$.

Step 1. With $A_r=k/r^2$ the spherical divergence gives $\nabla\cdot\mathbf{A}=\dfrac1{r^2}\partial_r\!\big(r^2\cdot\tfrac{k}{r^2}\big)=\dfrac1{r^2}\partial_r(k)=\mathbf{0}$ for every $r\ne0$ — the Jacobian's $r^2$ exactly cancels the field's $1/r^2$, the signature of an inverse-square law. Yet the flux through a sphere of radius $R$ is $\Phi=(k/R^2)(4\pi R^2)=\mathbf{4\pi k}$, independent of $R$. The divergence is zero away from the origin, where the source hides as a delta-function singularity; the finite flux $4\pi k$ is Gauss's law for a point charge with $k=q/(4\pi\varepsilon_0)$.

The scale factors $h_i$ carry a name — the Lamé coefficients — after the mathematician who turned Descartes' straight axes into a supple, curved coordinate machinery fit for continuous media.

Texts. Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 3 (Vector Analysis) and the coordinate-systems sections · Stone & Goldbart, Mathematics for Physics, Ch. on vector calculus and curvilinear coordinates · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 10 · Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Ch. 6 · Morse & Feshbach, Methods of Theoretical Physics, Vol. 1, Ch. 1 (the classic table of separable coordinate systems). The key self-consistency checks for the formulas in this article are: the gradient follows from $\mathrm{d}\phi=\nabla\phi\cdot\mathrm{d}\mathbf{r}$; the divergence reductions to $(1,\rho,1)$ and $(1,r,r\sin\theta)$ yield the standard cylindrical and spherical forms; the Laplacian is confirmed by composing gradient and divergence and checking $\nabla^2 r^2=6$, $\nabla^2(1/r)=0$, the cylindrical flux $40\pi$, and the point-source flux $4\pi k$; the spherical Jacobian is $r^2\sin\theta$ and the cylindrical Jacobian is $\rho$.

Prerequisites. Scalar and Vector Fields · The Gradient Operator · Divergence of a Vector Field · Curl of a Vector Field · The Divergence (Gauss) Theorem · Stokes' Theorem. Leads to. The Laplacian Operator · The Del Operator and Vector Differential Identities · Surface Integrals and Flux · Volume Integrals.

Ask the tutor: Ask me to specialize any of these operators to your coordinate system — cylindrical, spherical, parabolic, or a mesh of your own — and watch the scale factors do all the work.