The mathematical prerequisites now lead back to Maxwell's equations and classical electromagnetism. This chapter records precisely what its claims require. Definitions and physical postulates have a different status from proved consequences. Where a formerly broader sentence is false, the appropriate mathematical answer is a counterexample or a corrected theorem.
Chapters 2–6 provide proofs in the smooth Cartesian setting and explicitly identify external general theorems when a more general setting is mentioned. The ledger below covers the equations, prose claims, formula derivations and mathematical content of the displayed examples in the Maxwell article as inspected on October 15, 2026. It also records corrections to that inspected version. Historical assertions and physical interpretations are listed separately; they cannot be proved from calculus.
7.1. Definitions, operators, and geometric claims
| Article claim | Precise statement and proof route |
|---|---|
| The four SI equations; \(\rho,J,\varepsilon_0,\mu_0,\nabla\). | 6.1 defines the fields, sources, constants, regularity and equations. The equations are adopted physics. Chapter 2 constructs the operators from partial derivatives. |
| Maxwell describes everything in classical electromagnetism. | Qualify this: it determines constraints and evolution for prescribed sources and suitable data. Particle motion needs the Lorentz law; matter needs constitutive laws. 6.1 distinguishes them. Initial and boundary conditions matter for uniqueness, as 4.2 explains. |
| Divergence is the sum of the three diagonal partials; it measures net outflow per volume. | The formula is a definition. The local flux limit, not a picture of arrows, proves its geometric meaning in chapter 3. \(\operatorname{div}E=\rho/\varepsilon_0\) identifies electric charge density with this local source strength within the model. |
| Positive divergence means a source; negative means a sink. | This refers to positive or negative limiting flux. It does not assert that every individual arrow points away from or toward the observation point. Adding a large constant vector leaves divergence unchanged. |
| The three-component curl formula; curl is local rotation. | The Cartesian formula is defined in chapter 2. The oriented small-loop circulation limit is proved in chapter 3. |
| Curl measures how rapidly all neighbours differ; it is perpendicular to the field and the direction of variation. | False for arbitrary fields. For \(F=(x,-y,0)\), neighbours vary but \(\operatorname{curl}F=0\). For \(F=(-y,x,1)\), \(\operatorname{curl}F=(0,0,2)\) and \(F\cdot\operatorname{curl}F=2\), so they are not perpendicular. Direct substitution proves both counterexamples. Curl measures the antisymmetric part of the first derivative, not all spatial change. |
| For \(E=E_0\cos(kz-\omega t)\hat x\), curl is \(-kE_0\sin(kz-\omega t)\hat y\); the arrows have a quarter-cycle offset. | 6.9 computes it, using the proved sine derivative. \(\sin\theta=\cos(\theta-\pi/2)\) gives the offset, with its sign fixed by the displayed curl. At zero field magnitude a direction is undefined. |
| A plane wave has identical fields on transverse planes; the diagram samples one line. | The explicit ansatz in 6.9 depends only on \(z,t\). Thus all \((x,y)\) values at the same \(z,t\) give the same result. It extends through all space and is an idealization. |
| Divergence and curl are coordinate independent; cylindrical/spherical formulas use scale factors. | Chapter 3 states the precise coordinate transformation rules and charts. The physical geometric operations are invariant; their components and formulas change. Curvilinear charts are used only where smooth and invertible: cylindrical coordinates exclude the axis, and spherical coordinates exclude the origin and polar chart singularities. Curl also uses a fixed orientation; reflections require the corresponding axial-vector convention. |
| Polar coordinates are planar; cylindrical and spherical coordinates are three-dimensional. | These are the coordinate maps defined in chapter 3. “Spherical polar” is another name for spherical coordinates, not the two-dimensional polar map. |
| Faraday: changing magnetic fields imply circulating electric fields; the displacement term couples time-dependent electric fields to magnetic curl. | The differential equations are postulates. Their exact fixed-loop/fixed-surface integral meaning follows from Stokes' theorem and differentiation under the integral. Devices with moving matter need the further coupling and material assumptions; their full engineering behaviour is not derived here. |
| Without displacement current, vacuum electromagnetic waves would not exist. | For the source-free equations with that term deleted, the nonzero-frequency plane-wave coefficient equations are \(k\times D=0\), \(k\cdot D=0\), \(k\times A=\omega D\), \(k\cdot A=0\). The first two make \(D=0\): the first forces \(D\) parallel to \(k\), and the second kills its parallel coefficient. The last two then give \(A=0\). Thus the modified system has no nonzero monochromatic vacuum plane waves with \(k,\omega\ne0\). This precise result does not claim every static field is zero. |
| No magnetic monopoles means magnetic lines always close. | The first phrase describes the adopted zero-divergence law. The second is false. \(B=(0,0,1)\) is divergence free; its integral curves \((x_0,y_0,z_0+s)\) are unbounded straight lines, not closed loops. Chapter 3 distinguishes local differential facts from global topology. |
7.2. Static-field claims and singular sources
| Article claim | Precise statement and proof route |
|---|---|
| A point charge has \(E=q\hat r/(4\pi\varepsilon_0r^2)\). | Chapter 5 defines the point-source interpretation, derives the field with symmetry and boundary hypotheses, and distinguishes it from a smooth field at the origin. Chapter 3 computes the sphere area and radial flux. |
| Gauss's law alone forces the Coulomb field; spherical symmetry is automatic. | Symmetry of the solution requires a symmetry or uniqueness argument plus the boundary condition. Even both static vacuum equations outside a charge permit adding a constant vector field. For example \(E_C+\hat x\) has the same point charge, divergence and curl as \(E_C\), but is not spherical. The isolated decaying solution has the Coulomb form; source symmetry alone cannot justify discarding other fields. |
| Integrate \(q\delta^3\) over a ball and use the divergence theorem. | A delta source is not an ordinary function or Riemann density. Chapter 5 supplies the punctured-domain flux/weak-source formulation. Applying the ordinary smooth divergence theorem across the singular origin without that formulation is invalid. |
| The inverse-square law gives one quarter of the field at twice the radius; radial equipotentials are perpendicular to field arrows. | The first is the substitution \((2r)^{-2}=r^{-2}/4\). For \(\phi=q/(4\pi\varepsilon_0r)\), differentiation gives \(E=-\nabla\phi\); tangent vectors to a level sphere have zero dot product with its radial normal. At \(q=0\) the field is zero and no field direction is defined. |
| The Biot–Savart element has the stated cross product, direction and inverse-square kernel. | Chapter 5 derives the volume-current formula under smoothness, stationarity, source conservation and decay, then explains the closed thin-wire model. The norm of its numerator is \(I|d\ell'|\,|x-x'|\sin\alpha\), so the kernel scales as inverse square for fixed element and direction. A standalone open “current element” is a summand of a conserved current model, not generally an independent steady Maxwell solution. |
| An infinite straight wire has \(|B|=\mu_0|I|/(2\pi r)\) and right-hand-rule direction. | Chapter 5 evaluates the straight-wire integral and derives the azimuthal field. Its infinite-length hypothesis differs from a compact loop. Its field is singular on a zero-radius ideal wire. |
| A finite-radius uniform-current wire has finite surface field \(\mu_0I/(2\pi R)\). | Chapter 5 treats an infinite straight cylinder of radius \(R>0\), fixes cylindrical symmetry and matches interior and exterior fields. The surface is not a zero-radius ideal wire. Finite local integrability explains why nearby volume-current elements do not make its surface value infinite. |
| A \(1/s^2\) singularity is integrable in three dimensions because the volume factor is \(s^2\,ds\,d\Omega\). | Chapter 5 bounds the local integral using spherical coordinates established in chapter 3. The assertion is local, near \(s=0\): \(\int_{0}^{\eta}s^{-2}s^2ds=\eta\). On all space the same radial integral diverges at infinity, so global convergence needs a separate decay/source-support argument. |
| A uniformly charged sphere has finite field at its surface. | Here “uniformly charged” must specify a bounded volume density, or separately a spherical surface layer. The volume-density result is proved below using spherical Gauss flux; the surface-layer model needs its interface interpretation. A point charge at the surface is a different source and is singular there. |
| \(\operatorname{div}B=0\) implies \(B=\operatorname{curl}A\). | Chapter 3 states the local/global distinction. A global potential requires domain hypotheses. The radial divergence-free field \(B=x/|x|^3\) on \(\mathbb R^3\setminus\{0\}\) has nonzero flux through a surrounding sphere and cannot be a globally defined smooth curl there. The static construction in chapter 5 directly builds a suitable potential for its declared sources. |
| The Coulomb gauge reduces the static vector equation to Poisson's equation; its integral is “the solution.” | Use \(\operatorname{curl}\operatorname{curl}A=\nabla\operatorname{div}A-\Delta A\), \(\operatorname{div}A=0\), and \(\operatorname{curl}B=\mu_0J\). This gives \(\Delta A=-\mu_0J\). Chapter 5 proves the integral construction and its hypotheses. Harmonic additions and gauge freedom preclude uniqueness without boundary/decay and gauge conditions. Chapter 4 proves the scalar uniqueness principle used for decaying solutions. |
Supplementary proof. A uniformly charged solid sphere
Let a sphere of radius \(R>0\) have total charge \(q\), constant volume density \(\rho_0=3q/(4\pi R^3)\) inside, and zero density outside. Require a radial field regular at the centre and no added external field. Then
\[E(x)=\begin{cases}\dfrac{q}{4\pi\varepsilon_0R^3}x,&|x|\le R,\\[4pt]\dfrac{q}{4\pi\varepsilon_0}\dfrac{x}{|x|^3},&|x|\ge R.\end{cases}\]Worked example for Supplementary proof
Counterexample: A uniform surface shell differs from a uniform volume charge
Proof.
For \(0<r<R\), the enclosed charge is \(\rho_0(4\pi r^3/3)=qr^3/R^3\), since integrating the chapter 3 spherical volume element gives \(4\pi r^3/3\). Gauss flux and radial symmetry give \(4\pi r^2E_r(r)=qr^3/(\varepsilon_0R^3)\). Thus \(E_r=qr/(4\pi\varepsilon_0R^3)\). For \(r>R\), enclosed charge is \(q\), giving \(E_r=q/(4\pi\varepsilon_0r^2)\). The two values agree at \(R\); the interior Cartesian formula extends continuously to zero. Its divergence is \(3q/(4\pi\varepsilon_0R^3)=\rho_0/\varepsilon_0\) and its curl is zero. The exterior formula has zero divergence and curl by the chapter 5 Coulomb calculation. At the interface the field is continuous, so integration by parts on the interior and exterior gives cancelling interface terms: there is no surface delta in divergence or curl. The equations therefore hold classically away from the density jump and weakly across it. In particular the finite surface magnitude is \(|q|/(4\pi\varepsilon_0R^2)\). This proof uses a bounded volume density; it does not replace the distinct surface-charge or point-charge models.
□7.3. Wave, polarisation, and propagation claims
| Article claim | Precise statement and proof route |
|---|---|
| Taking curls gives wave equations for both fields. | 6.5 proves both, including source terms and mixed-derivative hypotheses. Source-free wave equations are necessary, not sufficient: the two explicit counterexamples there fail Maxwell while satisfying the wave equations. |
| Comparing coefficients identifies speed \(1/\sqrt{\mu_0\varepsilon_0}\). | Chapter 4's one-dimensional solution, its three-dimensional cone proof, and 6.7 establish the actual propagation statement. Merely naming the coefficient a speed supplies no proof of finite domain of dependence. |
| Oscillatory vacuum plane waves are transverse. | 6.9 proves \(k\cdot A=k\cdot D=0\) for nonzero wave vector and frequency. A field independent of \(x,y\) can still have a constant longitudinal part; derivatives only eliminate a varying longitudinal part. The pure monochromatic ansatz excludes that background. |
| \(B=(k/\omega)\hat z\times E=(1/c)\hat z\times E\). | 6.9 derives dispersion from both curl equations. The second equality assumes \(k>0,\omega=ck>0\), so \(\hat z\) is the propagation direction. Faraday alone does not derive the vacuum dispersion relation. |
| Real \(E,B\) are perpendicular even for elliptical polarisation. | Take real parts of \(D=n\times A/c\) to obtain the real cross-product relation, then use the real triple-product identity. 6.9 proves it and distinguishes bilinear from Hermitian complex pairings. A bilinear complex dot-product equality alone is insufficient reasoning about real parts. |
| Magnetic fields have no independent degrees of freedom once the electric field is known. | True for the nonzero-frequency oscillatory amplitude of a single pure plane wave with its frequency and direction fixed. False as a general Maxwell statement: add a constant magnetic field, still satisfying the same vacuum equations. Arbitrary initial data and combinations of directions require both fields and constraints. |
| The general transverse ansatz is encoded by two amplitudes and two phases; complex notation packages cosines and sines. | 6.8 gives the real-part calculation and links the exponential construction from elementary arithmetic. 6.10 treats the full component ansatz, with a zero-amplitude phase undefined and irrelevant. The common phase fixes the time/position origin of the oscillation; the relative phase determines its polarisation shape. |
| Linear for relative phase \(0,\pi\); circular for equal amplitudes in quadrature; otherwise elliptical. | 6.10 proves the exhaustive rank classification. A missing qualification is that zero amplitudes give a line or point regardless of phase. A circle requires equal nonzero amplitudes. A nondegenerate ellipse has axes given by the eigenvalues of the stated two-by-two matrix, not generally by the component amplitudes. |
| The example \(A=a\hat x+ib\hat y\) gives an axis-aligned ellipse and a displayed helix. | Direct real-part evaluation gives \((a\cos\theta,-b\sin\theta)\). 6.10 proves its semiaxes and supplies the helix parametrisation across distinct observation points. Rotation at one point and joining arrows at different points are different constructions. |
| All polarisation states are Maxwell solutions; none is mathematically preferred. | 6.9 checks all four equations for every complex transverse amplitude. This is admissibility in the vacuum equations. Sources, media and boundary conditions can select particular states and are additional data. |
| A constant-phase surface has velocity \(\omega/k=c\); the pattern translates rigidly. | 6.11 solves the phase equation and proves translation directly. A general solution need not be a single rigidly translating pattern: opposing waves, multiple directions and standing waves also solve the equations. |
| “Speed of light” means only phase velocity in the classical picture. | Too narrow. In this nondispersive vacuum model it is also the right-moving packet/envelope velocity and the local domain-of-dependence speed. 6.11 proves the profile result; 6.7 proves local propagation. Other dispersion relations can distinguish phase, group and signal speeds. |
| Electromagnetic energy flow follows from Maxwell. | 6.6 proves the Poynting differential and fixed-volume identities. The physical assignment of energy and matter power is distinguished from the algebraic identity. A nonzero ideal plane wave has infinite total energy, although its local density is finite. |
| Classical wave energy can take arbitrary positive values continuously. | Scaling a vacuum solution by \(a\in\mathbb R\) multiplies the quadratic density by \(a^2\), proved in 6.6. For a finite-energy solution it likewise scales total energy. Classical Maxwell equations impose no integer-valued energy restriction. |
7.4. Physical and historical boundaries
Mathematics proves implications of a model. Agreement of that model with an experiment, the identification of a field wave with light, and the selection of numerical constants require observation. The source-free equations predict \(c^2=1/(\mu_0\varepsilon_0)\); concluding that this is the speed of optical light compares the prediction with measured optical behaviour. Alphabeta's experiment-record definition, wave-propagation report, induction report, and inverse-square report keep empirical evidence separate from theorems.
The statement \(\mu_0=4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}\) was exact in the former SI. Since the SI revision effective May 20, 2019, \(c,h,e\) have fixed defining values and \(\mu_0\) is experimentally determined; the displayed old value is now an approximation. See the BIPM SI Brochure, ninth edition, section 2.2.1 and Alphabeta's SI conventions. This affects the numerical status, not the algebraic definition \(c=(\mu_0\varepsilon_0)^{-1/2}\).
Maxwell's paper A dynamical theory of the electromagnetic field was published in 1865. Its abstract and text identify the electromagnetic interpretation of light. The modern four-vector-equation presentation is a later notation for the theory; it should not be described as a verbatim display from Maxwell's paper. Maxwell's historical account included a medium hypothesis. The modern vacuum equations can be formulated without that medium; that mathematical fact should not be substituted for Maxwell's own historical commitments.
Planck's quantum work dates to 1900 and Einstein's light-quantum/photoelectric work to 1905, rather than a single “forty years after Maxwell” event. The Nobel Foundation's Planck award account and Einstein award account document their quantum and photoelectric contributions. The word “photon” was introduced by Gilbert N. Lewis in 1926, not by Planck in 1900 or Einstein in 1905; Lewis's original paper is The conservation of photons. The historical term and Einstein's earlier quantum hypothesis are distinct facts.
Photon energy \(E=hf\) and photon helicity are quantum claims, not consequences of the smooth classical Maxwell system proved here. For a quantized free harmonic mode of frequency \(f\), excitation energy above its ground state is \(nhf\), with occupation number \(n=0,1,2,\ldots\); its harmonic-oscillator total energy is \((n+\tfrac12)hf\). Thus “the energy in a mode is always an integer multiple of \(hf\)” needs the excitation-energy convention. It also describes the eigenvalues in the free-mode model, not an assertion that an arbitrary quantum state's expected energy is an integer multiple. A superposition with noninteger expected occupation is permitted. Neither the quantization rule nor its physical applicability is established by this classical series. The MIT course 8.04 Quantum Physics I develops the quantum oscillator and the quantum premises; readers must use that separate theory for a proof.
Likewise the quantum statement about the two photon helicities \(\pm\hbar\) requires a quantized massless field and its angular-momentum representation. The classical result we proved is exactly that there are two independent transverse amplitude components and two circular polarisation senses among their real solutions. Identifying these with photon helicity eigenstates requires further quantum structure. The revised Maxwell article keeps those topics outside its claimed classical derivations; this ledger does not pretend to prove them from its four equations.
7.5. How to use the linked library
The Alphabeta pages below were inspected on October 15, 2026. They offer general formulations beyond this series' elementary smooth branch. The links target the individual definitions and results relevant to the Maxwell article, rather than suggesting that reading a chapter title supplies an unstated theorem.
- Framework: laboratory geometry, SI conventions, sources, fields, smooth/weak vacuum Maxwell postulate, superposition, and separate Lorentz coupling.
- Integral constraints: fixed test geometry, integral Maxwell equivalence, distributional sources, distributional vacuum equations, charge compatibility, constraint propagation, and parameter integrals.
- Waves: real plane waves, complex encoding, necessary wave equation, dispersion and transversality, polarisation, compatible initial-value problem, and Kirchhoff representation, uniqueness and domain of dependence. Our shrinking-ball proof establishes the propagation claim without assuming that representation formula.
- Energy: energy and flux definitions, Poynting balance, period averaging, plane-wave energy averages, and coherent interference. The last three support subsequent energy calculations, not extra hypotheses needed for the perpendicularity proof.
- Scope: material closure, classical validity limits, and radiation-reaction limits. Their qualifications prevent treating a prescribed-source smooth-field model as a complete theory of matter or point-particle self-force.
The static-field and coordinate items are linked where their mathematical constructions are proved in chapter 3 and chapter 5. Nothing in this reading route turns an adopted physical postulate into a mathematical theorem, or a locally proved statement into a global theorem without its hypotheses.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. Checking the field at, inside, and outside a charged sphere
Applies to: Supplementary proof.
Choose \(R=2\) and \(q=4\pi\varepsilon_0\) in the stated volume-charge model. Then \(E(x)=x/8\) for \(|x|\le2\), and \(E(x)=x/|x|^3\) outside. At the surface both have magnitude \(1/4\); at the centre the first is zero and regular. The interior divergence is \(3/8=\rho_0/\varepsilon_0\), since \(\rho_0=3\varepsilon_0/8\). At radius \(r\le2\), flux is \(4\pi r^2(r/8)=\pi r^3/2\), the enclosed charge divided by \(\varepsilon_0\); at larger radii flux is \(4\pi=q/\varepsilon_0\). The equal interface values cancel the interior and exterior boundary terms in integration by parts, so no surface delta source is present. This checks the proposition’s bounded volume density, its finite boundary value, and its weak interpretation at the density jump.
Worked example 2. Testing the corrected local and static claims
The ledger’s smooth counterexamples can be checked independently: \(F=(x,-y,0)\) has a nonzero derivative matrix but zero curl; \(G=(-y,x,1)\) has curl \((0,0,2)\) and dot product \(G\cdot\operatorname{curl}G=2\); \(B=e_z\) has zero divergence and integral curves \((x_0,y_0,z_0+s)\), which never close. These distinguish the local differential facts from the stronger false claims.
For \(q\ne0\), adding the constant field \(e_x\) to Coulomb’s field changes neither its distributional divergence nor its curl, but prevents decay at infinity and breaks spherical symmetry. Thus the source alone does not impose isolation. On the punctured space, \(B=x/|x|^3\) is divergence free yet has flux \(4\pi\) through a unit sphere. If it were a globally smooth curl there, applying the patched Stokes theorem to that closed sphere would give flux zero, contradiction. This checks the separate domain hypothesis needed for a global vector potential. Finally the local kernel bound \(\int_{|x|<\eta}|x|^{-2}dV=4\pi\eta\) is finite, while extending the radial integral to infinity diverges; local and global convergence therefore require different hypotheses.
Worked example 3. A standing wave and an independent magnetic background
Superpose the two vacuum solutions \(E_+=\cos(kz-ckt)e_x,B_+=c^{-1}\cos(kz-ckt)e_y\) and \(E_-=\cos(kz+ckt)e_x,B_-=-c^{-1}\cos(kz+ckt)e_y\), \(k>0\). Each satisfies the plane-wave amplitude conditions, with propagation directions \(\pm e_z\). Their sum is \(E=2\cos kz\cos ckt\,e_x\), \(B=2c^{-1}\sin kz\sin ckt\,e_y\), a standing wave rather than a rigid translating pattern. Adding a constant \(b_0e_z\) to its magnetic field leaves every Maxwell derivative unchanged while the electric field is identical. Thus the single-plane-wave amplitude relation does not determine a general solution’s magnetic background.
For one travelling wave of electric amplitude \(a\), \(u=\varepsilon_0a^2\cos^2(kz-ckt)\), \(S=cu\,e_z\), using \(c^{-2}=\mu_0\varepsilon_0\). The period averages are \(\varepsilon_0a^2/2\) and \(c\varepsilon_0a^2e_z/2\); averaging \(\cos^2\) over one period gives \(1/2\). The local energy is finite, but integrating over all transverse space diverges whenever \(a\ne0\). This checks the ledger’s energy, averaging, and ideal-plane-wave qualifications.
Further proof details
The infinite-total-energy assertion has a concrete lower bound. At any fixed time and nonzero amplitude \(a\), choose a positive-length interval \(I\) of \(z\)-values around a phase maximum so that \(\cos^2(kz-ckt)\ge1/4\) on \(I\). On \([-L,L]^2\times I\), the energy is at least \((\varepsilon_0a^2/4)(2L)^2|I|\), tending to infinity with \(L\). Thus the whole-space integral is infinite, even though the local density is bounded. For period averages, \(\cos^2\theta=(1+\cos2\theta)/2\), and the primitive \(\sin2\theta/2\) has equal values at endpoints a period apart; the average is exactly \(1/2\), proving the displayed energy and flux averages.
Worked example 4. Excitation energy and an expected value
Within the separately adopted quantum oscillator spectrum, let \(|0\rangle,|1\rangle\) be normalized orthogonal energy eigenvectors with energies \(hf/2\) and \(3hf/2\), where \(h,f>0\). The normalized state \(\psi=\sqrt{3/4}|0\rangle+\sqrt{1/4}|1\rangle\) has mean total energy \((3/4)(hf/2)+(1/4)(3hf/2)=3hf/4\); its mean excitation above the ground state is \(hf/4\). Individual excitation outcomes are 0 and \(hf\), while their expected value is not an integer multiple of \(hf\). This illustrates the ledger’s distinction once the quantum spectrum and Born law are adopted; neither premise is derived from classical Maxwell equations.
Counterexamples and limits of the hypotheses
Each construction below refutes the explicitly stated stronger claim. It does not refute the related theorem with its original hypotheses.
Counterexample 1. A uniform surface shell differs from a uniform volume charge
Related statements: Supplementary proof.
Claim refuted. “Uniformly charged sphere” always gives a continuous electric field at its surface.
Proof.
Instead of the bounded volume density in the supplementary proposition, put a uniform surface charge \(\sigma=q/(4\pi R^2)\) on \(S_R\), with \(R>0,q\ne0\). Its weak source is \(\rho(\varphi)=\sigma\int_{S_R}\varphi\,dA\). The radial field \(E=0\) for \(r<R\), \(E=q e_r/(4\pi\varepsilon_0r^2)\) for \(r>R\), has zero classical divergence and curl away from the sphere. For a compactly supported test function, integrate by parts in the exterior region. Its inner outward normal is \(-e_r\), so
\[-\int E\cdot\nabla\varphi\,dV=\frac{q}{4\pi\varepsilon_0R^2}\int_{S_R}\varphi\,dA=\varepsilon_0^{-1}\rho(\varphi).\]The tangential jump is zero, so the corresponding boundary term in the weak curl is zero; equivalently this field is \(-\nabla\Phi\) for the continuous potential \(\Phi=q/(4\pi\varepsilon_0R)\) inside and \(q/(4\pi\varepsilon_0r)\) outside. The two one-sided field limits at \(R\) are 0 and \(\sigma e_r/\varepsilon_0\), different. Assigning a value on the surface cannot make the field continuous and does not change the distribution. This surface-layer model therefore refutes the unrestricted phrase while leaving the proved volume-density result intact.
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