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Vector PEC scattering, dipole limits and controlled scalar diffraction

1 · Prerequisites

2 · Summary

Objective. This page is the scattering and optics capstone of the classical-electromagnetism track. It fixes a general monochromatic phasor class and its exact real-field calculus, solves the smooth exterior vector scattering problem for a stationary perfect conductor and states its ideal boundary condition and uniqueness statement, defines the ideal harmonic dipole model with its differential and total model cross sections and its actual-source fidelity bounds, derives the leading Thomson-model coefficient as a controlled first variation of the relativistic force law, and adopts a polarization-insensitive scalar aperture model with quantified Fraunhofer and Fresnel errors. Everything is conditional on the adopted postulates, the named laboratory chart and the SI conventions; no measured cross section or optical record is claimed.

Phasor encoding and its real calculus. A frequency- $ω$ source-free vacuum phasor pair on an open set is a pair of $C^{\infty}$ complex fields $(\tilde{E}, \tilde{B})$ satisfying $div \tilde{E} = div \tilde{B} = 0$ , $curl \tilde{E} = i ω \tilde{B}$ and $curl \tilde{B} = - i ω c^{- 2} \tilde{E}$ (Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding); the definition fixes the $e^{- i ω t}$ encoding, the componentwise complex divergence and curl, the units and the excluded content only. The companion mathematical lemma proves the exact real-part calculus, the injectivity of the encoding at the two phases $0$ and $π / (2 ω)$ , and the equivalence of the phasor system with the real source-free system $div E = div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = c^{- 2} \partial_{t} E$ on $D \times R$ (Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system). The mathematical harmonic-response lemma classifies all solutions of $\ddot{z} = Re (w e^{- i ω t})$ as $z_{p} + a + b t$ , with the unique global extension of any local solution, the periodicity criterion $b = 0$ , the single-pair test and the period means $a + b (t + T / 2)$ (Periodic responses of $\ddot{z} = Re (w e^{- i ω t})$ and the affine classification of all solutions); it is the tool that lets the Thomson preparation be imposed on the linear response without asserting a global nonlinear family.

Vector scattering by a smooth PEC obstacle. For a bounded $C^{\infty}$ obstacle with connected exterior, one frequency and a $C^{\infty}$ incident source-free phasor pair, the exterior vector Helmholtz problem with the prescribed tangential scattered datum has a unique outgoing smooth solution (via Existence and uniqueness for the exterior vector Helmholtz problem with prescribed tangential trace and outgoing radiation, which carries the Axiom of Choice). The physics theorem Monochromatic vector scattering by a smooth PEC obstacle: phasor Maxwell fields, the PEC trace condition and outgoing-correction uniqueness turns that solution into a scattering statement: the scattered and total pairs are source-free phasor pairs, the associated real fields solve the zero-source smooth vacuum Maxwell equations, the ideal PEC condition $n \times E_{t o t} = 0$ holds on $\partial O \times R$ , and the outgoing scattered correction with the prescribed tangential trace is unique. The total field is explicitly not required to be outgoing, and no induced charge or current is determined.

Dipole limits and the Thomson coefficient. The page defines the ideal harmonic electric-dipole model by its leading transverse far-zone fields and its model flux, with the declared absence of magnetic-dipole and higher-multipole content and the actual-source comparison through the moment-mismatch and error bounds of the dipole theorem (The ideal harmonic electric-dipole radiation model and its adopted status). In a plane wave of amplitude $e$ , the model differential and total cross sections are $d σ / d Ω = ω^{4} ∣ n \times p_{0} ∣^{2} / (16 π^{2} ε_{0}^{2} c^{4} ∣ e ∣^{2})$ and $σ = ω^{4} ∣ p_{0} ∣^{2} / (6 π ε_{0}^{2} c^{4} ∣ e ∣^{2})$ , with the scalar-polarizability branch, and the actual-source fidelity bounds with the mismatch $M$ , the conserved compact-window realization and the scope exclusions (Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds). The Thomson item then takes a uniformly convergent solution family of the adopted relativistic external-force law in the field $η (E_{i n c}, B_{i n c})$ , proves the first-order equation $m {\ddot{r}}_{1} = q E_{i n c} (x_{0}, \cdot)$ , extracts the periodic first-order response under the explicit preparation $r_{1}^{'} (0) = Re (i w / ω)$ with $w = (q / m) e e^{i κ \cdot x_{0}}$ , and substitutes the induced amplitude $p_{0} = - \frac{q^{2}}{m ω^{2}} e e^{i κ \cdot x_{0}}$ into the dipole cross section to obtain $σ_{T} = \frac{q^{4}}{6 π ε_{0}^{2} m^{2} c^{4}}$ , independent of the frequency (Controlled weak-field first variation of the relativistic force law and the leading Thomson-model cross section). The result is a conditional leading-model coefficient, not an assertion about the nonlinear particle's emitted field.

Controlled scalar diffraction. The page adopts a scalar aperture model: a finite complex aperture distribution on a compact planar aperture, the outgoing Helmholtz field $u = G_{k} * α$ , its real reading $U = Re (u e^{- i ω t})$ and the squared field $I = \frac{1}{2} ∣ u ∣^{2}$ , declared as a polarization-insensitive optical approximation rather than a consequence of the Maxwell postulates (The adopted scalar aperture-diffraction model). The diffraction theorem then proves the off-support smoothness and Helmholtz equation, the uniform Fraunhofer estimate at the closed radius $r \geq 2 a$ with the explicit outgoing residual, the quantified Fresnel phase and amplitude errors on the declared small-angle range, the period-average identity and the difference-of-moduli intensity transfer, together with the trace caveat that the aperture datum is not the exact screen boundary data (Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer). The companion page carries the worked uniform rectangular aperture.

Scope and evidence. Every item on this page is a conditional model statement within the named conventions. The PEC condition is an adopted idealization; the dipole and Thomson items are conditional on their prescribed sources and solution families; the aperture model is an adopted scalar approximation with inherited error terms; and no item makes a measured, apparatus, sampling or statistical claim.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding

Physical scope

Definition of the frequency- $ω$ complex phasor encoding $E = Re (\tilde{E} e^{- i ω t})$ , $B = Re (\tilde{B} e^{- i ω t})$ of a pair of $C^{\infty}$ complex fields on an open spatial region, and of the class of pairs satisfying the four source-free vacuum phasor equations with the SI vacuum constants. Excluded: any assertion that the associated real fields solve Maxwell's equations, any outgoing, boundary, existence, uniqueness, energy, flux or constitutive statement, and every empirical claim; only the $C^{\infty}$ open-set class is defined.

Definition

Let $ω > 0$ , let $D \subseteq R^{3}$ be a nonempty open set, and let $c > 0$ , $μ_{0} > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ be the vacuum constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values; put $T : = 2 π / ω$ .

(a) [Phasor encoding.] For every pair of $C^{\infty}$ maps $\tilde{E}, \tilde{B} : D \to C^{3}$ ( $C^{k}$ Euclidean maps and diffeomorphisms) define the associated real fields $E, B : D \times R \to R^{3}$ componentwise by

$E (x, t) : = Re (\tilde{E} (x) e^{- i ω t}), B (x, t) : = Re (\tilde{B} (x) e^{- i ω t}) ((x, t) \in D \times R),$

with the complex exponential and real part of The complex exponential by its power series and Real and imaginary parts, complex conjugation, and modulus. This is the encoding convention of Complex plane-wave fields: real parts of complex exponentials specialized from plane-wave amplitudes to a general $C^{\infty}$ amplitude, and the complex fields are not themselves physical fields.

(b) [Source-free class.] The pair $(\tilde{E}, \tilde{B})$ is called a frequency- $ω$ source-free vacuum phasor pair on $D$ when it is $C^{\infty}$ and satisfies, on $D$ ,

$div \tilde{E} = 0, div \tilde{B} = 0, curl \tilde{E} = i ω \tilde{B}, curl \tilde{B} = - i ω ε_{0} μ_{0} \tilde{E} = - i ω c^{- 2} \tilde{E},$

with the divergence and curl of Divergence and curl of a $C^{1}$ vector field applied componentwise to the complex fields. The word "source-free" records that the displayed equations carry no charge or current term; the associated real fields are read as an electric and a magnetic field of The electric and magnetic fields as fixed-time spatial vector fields, $ω$ is the angular frequency and $T$ the period.

(c) [Status and scope.] This item fixes the encoding, the four phasor equations and the terminology only. It asserts no equation for the associated real fields, no outgoing or boundary condition, no existence or uniqueness statement, no energy or flux reading and no empirical content. In particular the deduction that the associated real fields satisfy the smooth-branch vacuum Maxwell equations with zero sources is not part of this definition; for the pairs used on the exterior-scattering page it is established by the exterior-scattering theorem recorded there. Only $C^{\infty}$ pairs on an open set are defined, so no boundary trace, weak or distributional class is introduced here.

Well-definedness and conventions

The componentwise reading of the two complex fields is the one of $C^{k}$ Euclidean maps and diffeomorphisms: $\tilde{E}$ and $\tilde{B}$ are $C^{\infty}$ exactly when the twelve real coordinate maps $Re {\tilde{E}}_{k}$ , $Im {\tilde{E}}_{k}$ , $Re {\tilde{B}}_{k}$ , $Im {\tilde{B}}_{k}$ ( $k = 1, 2, 3$ ) are $C^{\infty}$ on the open set $D$ . The real and imaginary parts and the complex exponential of Real and imaginary parts, complex conjugation, and modulus and The complex exponential by its power series are total on $C$ , so the associated real fields are defined for every $C^{\infty}$ pair, with no equation assumed and no regularity asserted of them by clause (a). Clause (b) is the subclass cut out by the four displayed equations: the divergence and curl of a $C^{\infty}$ complex field are the componentwise extensions of the divergence and curl of Divergence and curl of a $C^{1}$ vector field to $Re$ and $Im$ parts, evaluated on the open set $D$ , and the right-hand sides use only complex scalar multiplication by the constants $i ω$ and $i ω ε_{0} μ_{0}$ . In particular $ε_{0} μ_{0} = c^{- 2}$ by Framework-local SI units, the vacuum constants and the exact/measured status of their values, a positive scalar parameter; all four equations are pointwise, so no selection of data and no choice principle enters the definition.

The units are inherited from The electric and magnetic fields as fixed-time spatial vector fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values: $\tilde{E}$ and $E$ carry $V m^{- 1}$ , $\tilde{B}$ and $B$ carry the tesla $T$ , the wavevector-like spatial variable $x$ carries $m$ , $t$ carries $s$ , and $ω$ carries $s^{- 1}$ . The complex fields are auxiliary amplitudes encoding the real fields; clause (b) is a condition on those amplitudes, not a new physical assertion about the real fields.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system

Statement

Let $D \subseteq R^{3}$ be open, $c > 0$ and $ω > 0$ , and let $\tilde{E}, \tilde{B} : D \to C^{3}$ . Define $E, B : D \times R \to R^{3}$ by $E (x, t) : = Re (\tilde{E} (x) e^{- i ω t})$ and $B (x, t) : = Re (\tilde{B} (x) e^{- i ω t})$ componentwise (The complex exponential by its power series, Real and imaginary parts, complex conjugation, and modulus; $C^{k}$ Euclidean maps and diffeomorphisms for the classes). Then: (a) [Calculus] if $\tilde{E}, \tilde{B}$ are $C^{1}$ , then $E, B$ are $C^{1}$ and at every $(x, t) \in D \times R$ : $\partial_{t} E (x, t) = Re (- i ω \tilde{E} (x) e^{- i ω t})$ , and likewise for $B$ ; for each $j \in {1, 2, 3}$ , $\partial_{j} E (x, t) = Re (\partial_{j} \tilde{E} (x) e^{- i ω t})$ , and likewise for $B$ ; and with the divergence and curl of Divergence and curl of a $C^{1}$ vector field acting on the spatial variable, $div E (x, t) = Re ((div \tilde{E}) (x) e^{- i ω t})$ , $curl E (x, t) = Re ((curl \tilde{E}) (x) e^{- i ω t})$ , and likewise for $B$ . (b) [Injectivity] if $w \in C^{3}$ and $Re (w e^{- i ω t}) = 0$ for all $t \in R$ , then $w = 0$ ; more generally, if $w : D \to C^{3}$ is continuous and $Re (w (x) e^{- i ω t}) = 0$ for all $(x, t) \in D \times R$ , then $w \equiv 0$ on $D$ . (c) [Equivalence] if $\tilde{E}, \tilde{B}$ are $C^{1}$ , the phasor system $div \tilde{E} = 0, div \tilde{B} = 0, curl \tilde{E} = i ω \tilde{B}, curl \tilde{B} = - i ω c^{- 2} \tilde{E}$ holds on $D$ if and only if the real system $div E = 0, div B = 0, curl E = - \partial_{t} B, curl B = c^{- 2} \partial_{t} E$ holds on $D \times R$ .

Facts & Assumptions

Given: An open set $D \subseteq R^{3}$ , reals $c > 0$ and $ω > 0$ , complex fields $\tilde{E}, \tilde{B} : D \to C^{3}$ , and the associated real fields $E, B : D \times R \to R^{3}$ of the Statement. Components are indexed by $k, j \in {1, 2, 3}$ .

[F1]

Complex exponential and Euler form: $\exp$ is entire with $\exp^{'} = \exp$ ; for real $θ$ one has $e^{i θ} = \cos θ + i \sin θ$ , so $e^{- i ω t} = \cos ω t - i \sin ω t$ and $e^{- i π / 2} = \cos (π / 2) - i \sin (π / 2) = - i$ ; $\sin^{'} = \cos$ and $\cos^{'} = - \sin$ on $R$ (The complex exponential by its power series, The complex exponential is entire and its complex derivative is itself, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The derivatives of sine and cosine are cosine and minus sine, Quarter-turn values and shifts by pi/2 and pi).

[F2]

Real and imaginary parts: $Re (a + b i) = a$ , $Im (a + b i) = b$ ; $Re$ and $Im$ are $R$ -linear; $(a + b i) (s + t i) = (a s - b t) + (a t + b s) i$ ; in particular $Re ((a + b i) (c - i s)) = a c + b s$ and $Re (- i ω (a + b i) (c - i s)) = ω b c - ω a s$ for real $a, b, ω, c, s$ (Real and imaginary parts, complex conjugation, and modulus, The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ).

[F3]

Complex differentiability and real differentiability: complex differentiability at a point is equivalent to real total differentiability there with the derivative given by multiplication by the complex derivative; hence $\exp$ is real totally differentiable everywhere with real derivative the multiplication map $h \mapsto e^{z} h$ , and the chain rule for total derivatives applies to real-differentiable compositions (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations, The complex exponential is entire and its complex derivative is itself, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F4]

One-variable and componentwise rules: sums, scalar multiples and products of differentiable real functions are differentiable with the usual derivatives (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ); partial derivatives are coordinate-line directional derivatives, so $\partial_{j} g (x)$ is the derivative at $0$ of $h \mapsto g (x + h e_{j})$ (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ); a map $U \to R^{q}$ on an open $U$ is $C^{k}$ when every component's iterated partials up to order $k$ exist and are continuous, and a complex field is $C^{1}$ exactly when its six real coordinate maps $Re {\tilde{E}}_{k}, Im {\tilde{E}}_{k}$ are $C^{1}$ , with complex partials $\partial_{j} {\tilde{E}}_{k} = \partial_{j} (Re {\tilde{E}}_{k}) + i \partial_{j} (Im {\tilde{E}}_{k})$ ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

The divergence of a $C^{1}$ real field $F$ on open $U \subseteq R^{n}$ is $div F = \sum_{k} \partial_{k} F_{k}$ , and the curl on $U \subseteq R^{3}$ is $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ ; both are linear in $F$ with real scalars, and on complex fields both are applied componentwise to real and imaginary parts (Divergence and curl of a $C^{1}$ vector field, Divergence and curl are linear and satisfy the scalar product rules).

[F6]

Continuity toolkit: continuity is the metric notion of Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, and on $R^{m}$ the Euclidean topology agrees with the product-of- $R$ topology (For $n \geq 1$ the product topology on $n$ copies of the usual topology of $R$ is the metric topology of $d_{\infty}$ on $R^{n}$ , and hence also of $d_{1}$ and $d_{2}$ , so $R^{n}$ as a product and $R^{n}$ as a metric space are one space); projections from a product are continuous and a map into a product is continuous exactly when all its components are (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice); finite sums, products, maxima, minima and quotients of continuous real-valued maps on a topological space are continuous (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined); and a function differentiable at a point of its domain is continuous there (A function differentiable at $c$ is continuous at $c$ ). For the pullbacks used here, fix $(x_{*}, t_{*})$ . The Euclidean product distance bounds both $∣ x - x_{*} ∣$ and $∣ t - t_{*} ∣$ ; choosing the $δ$ from continuity of $g : D \to R$ at $x_{*}$ or of $h : R \to R$ at $t_{*}$ therefore proves directly that $(x, t) \mapsto g (x)$ and $(x, t) \mapsto h (t)$ are continuous. Finite sums and products of these pullbacks are continuous by the cited algebra lemma.

Proof

technique · direct; expand the encoding in real components, differentiate along the time and coordinate directions with the one-variable rules, read the divergence and curl off the componentwise formulas, and obtain the equivalence from the injectivity of the real-part evaluation proved at the two phases $t=0$ and $t=\pi/(2\omega)$

1.1givenF1F2F4

[Encoding] For clauses (a) and (c), assume the stated $C^{1}$ hypotheses. For every $k$ and every $(x, t) \in D \times R$ , [F1] and [F2] give $E_{k} (x, t) = Re ({\tilde{E}}_{k} (x) e^{- i ω t}) = (Re {\tilde{E}}_{k} (x)) \cos ω t + (Im {\tilde{E}}_{k} (x)) \sin ω t$ , and the same display with ${\tilde{B}}_{k}$ in place of ${\tilde{E}}_{k}$ gives $B_{k} (x, t)$ ; moreover, since $\tilde{E}$ and $\tilde{B}$ are $C^{1}$ , their six real coordinate maps and all first partials are continuous on $D$ and the complex partials are $\partial_{j} {\tilde{E}}_{k} = \partial_{j} (Re {\tilde{E}}_{k}) + i \partial_{j} (Im {\tilde{E}}_{k})$ , by [F4].

1.2givenF1F2

[Injectivity] Let $w \in C^{3}$ satisfy $Re (w e^{- i ω t}) = 0$ for every real $t$ . At $t = 0$ , [F1] gives $e^{0} = 1$ , so $Re w = 0$ ; at $t = π / (2 ω)$ one has $e^{- i π / 2} = - i$ by [F1], so $0 = Re (- i w) = Im w$ by [F2]; hence $w = Re w + i Im w = 0$ . If $w : D \to C^{3}$ satisfies $Re (w (x) e^{- i ω t}) = 0$ for all $(x, t)$ , applying the preceding conclusion at each fixed $x \in D$ gives $w (x) = 0$ for every $x$ , so $w \equiv 0$ ; no continuity of $w$ is needed for this conclusion.

2.1step 1.1F1F2F3F4

[Time derivative] Fix $x$ and $k$ . The map $t \mapsto {\tilde{E}}_{k} (x) e^{- i ω t}$ is the composition of the real-linear map $t \mapsto - i ω t$ (which is differentiable on $R$ with derivative $- i ω$ and real-linear, so totally differentiable with that derivative) with $\exp$ , which is real totally differentiable everywhere with derivative the multiplication $h \mapsto e^{z} h$ by [F3], followed by multiplication by the constant ${\tilde{E}}_{k} (x)$ ; the chain and product rules of [F3], [F4] give derivative $- i ω {\tilde{E}}_{k} (x) e^{- i ω t}$ . Taking real parts by [F2] and inserting the display of step 1.1, $\partial_{t} E_{k} (x, t) = Re (- i ω {\tilde{E}}_{k} (x) e^{- i ω t}) = ω (Im {\tilde{E}}_{k} (x)) \cos ω t - ω (Re {\tilde{E}}_{k} (x)) \sin ω t$ for every $(x, t)$ ; the same computation with $\tilde{B}$ gives $\partial_{t} B_{k} (x, t) = Re (- i ω {\tilde{B}}_{k} (x) e^{- i ω t})$ .

2.2step 1.1F2F4

[Spatial derivatives] Fix $(x, t)$ and indices $j, k$ . The map $h \mapsto E_{k} (x + h e_{j}, t)$ equals $h \mapsto Re ({\tilde{E}}_{k} (x + h e_{j}))$ times $\cos ω t$ plus $h \mapsto Im ({\tilde{E}}_{k} (x + h e_{j}))$ times $\sin ω t$ on a neighbourhood of $0$ small enough that $x + h e_{j} \in D$ , by step 1.1. By [F4] the coordinate-line maps $h \mapsto Re {\tilde{E}}_{k} (x + h e_{j})$ and $h \mapsto Im {\tilde{E}}_{k} (x + h e_{j})$ are differentiable at $h = 0$ with derivatives $\partial_{j} Re {\tilde{E}}_{k} (x)$ and $\partial_{j} Im {\tilde{E}}_{k} (x)$ , so the sum, scalar-multiple and product rules of [F4] give $\partial_{j} E_{k} (x, t) = (\partial_{j} Re {\tilde{E}}_{k} (x)) \cos ω t + (\partial_{j} Im {\tilde{E}}_{k} (x)) \sin ω t$ , which by [F2] and step 1.1 equals $Re ((\partial_{j} {\tilde{E}}_{k}) (x) e^{- i ω t})$ ; the same computation with $\tilde{B}$ gives $\partial_{j} B_{k} (x, t) = Re ((\partial_{j} {\tilde{B}}_{k}) (x) e^{- i ω t})$ .

3.1step 2.2F2F4F5

[Divergence and curl] By [F4] the divergence and curl act componentwise on real and imaginary parts of complex fields; hence, summing step 2.2 over $k$ and using $R$ -linearity of $Re$ [F2], $div E (x, t) = \sum_{k} \partial_{k} E_{k} = Re ((div \tilde{E}) (x) e^{- i ω t})$ , and for the third curl coordinate, in the naming of [F5], the alternating sum of the displayed partials and [F2] give $(curl E)_{z} (x, t) = \partial_{x} E_{y} - \partial_{y} E_{x} = Re ((\partial_{x} {\tilde{E}}_{y} - \partial_{y} {\tilde{E}}_{x}) (x) e^{- i ω t}) = Re ((curl \tilde{E})_{z} (x) e^{- i ω t})$ , the other two coordinates being identical by cyclic relabelling of the indices; the linearity of [F5] organizes these componentwise identities into the displayed formulas, and the same argument with $\tilde{B}$ gives those for $B$ .

4.1step 2.1step 2.2step 3.1F1F4F6

[Regularity] Each of the displayed first partials of $E$ and of $B$ in steps 2.1, 2.2 and 3.1 is a finite sum of products of the maps $(x, t) \mapsto g (x)$ for $g$ among $Re {\tilde{E}}_{k}, Im {\tilde{E}}_{k}, \partial_{j} Re {\tilde{E}}_{k}, \partial_{j} Im {\tilde{E}}_{k}$ (and the $\tilde{B}$ analogues), each continuous on $D \times R$ by [F6] since its one-variable factor is continuous on $D$ by [F4], together with the maps $(x, t) \mapsto \cos ω t$ and $(x, t) \mapsto \sin ω t$ , continuous on $D \times R$ by [F1], [F6]; the fields $E_{k}, B_{k}$ themselves are of the same form by step 1.1. By the sum-and-product clause of [F6], all these functions are continuous on $D \times R$ ; hence every first partial of $E$ and $B$ exists and is continuous there, so $E$ and $B$ are $C^{1}$ by [F4], with the partials displayed in steps 2.1, 2.2 and 3.1. This proves (a).

4.2step 2.1step 3.1givenalgebra

[(i) implies (ii)] Assume the phasor system of (c) holds on $D$ . Then for every $(x, t)$ : $div E = Re ((div \tilde{E}) e^{- i ω t}) = 0$ and $div B = 0$ by step 3.1 and $div \tilde{E} = div \tilde{B} = 0$ ; $curl E = Re ((curl \tilde{E}) e^{- i ω t}) = Re (i ω \tilde{B} e^{- i ω t}) = - \partial_{t} B$ by steps 3.1, 2.1 and $curl \tilde{E} = i ω \tilde{B}$ ; and $curl B = Re ((curl \tilde{B}) e^{- i ω t}) = Re (- i ω c^{- 2} \tilde{E} e^{- i ω t}) = c^{- 2} \partial_{t} E$ by steps 3.1, 2.1 and $curl \tilde{B} = - i ω c^{- 2} \tilde{E}$ . Hence the real system holds on $D \times R$ .

4.3step 1.2step 2.1step 3.1givenalgebra

[(ii) implies (i)] Assume the real system of (c) holds on $D \times R$ . The fields $u : = div \tilde{E}$ , $v : = div \tilde{B}$ , $w : = curl \tilde{E} - i ω \tilde{B}$ and $z : = curl \tilde{B} + i ω c^{- 2} \tilde{E}$ are continuous maps on $D$ , with $u, v$ scalar-valued and $w, z$ valued in $C^{3}$ by step 1.1 and [F4], [F5], and steps 2.1 and 3.1 give $Re (u (x) e^{- i ω t}) = div E = 0$ , $Re (v (x) e^{- i ω t}) = div B = 0$ , $Re (w (x) e^{- i ω t}) = curl E + \partial_{t} B = 0$ and $Re (z (x) e^{- i ω t}) = curl B - c^{- 2} \partial_{t} E = 0$ for all $(x, t)$ ; the two-phase calculation of step 1.2, also valid for a single complex scalar, then forces $u = v = w = z = 0$ on $D$ , which is exactly the phasor system.

5.1step 1.2step 2.1step 2.2step 3.1step 4.1step 4.2step 4.3given∎

[Conclusion] Steps 2.1, 2.2, 3.1 and 4.1 prove the calculus and $C^{1}$ clauses of (a); step 1.2 proves the injectivity clause (b); steps 4.2 and 4.3 prove both directions of the equivalence (c). All assertions are pointwise identities on the open domain $D \times R$ , no existence theory for partial differential equations, no boundary condition and no choice principle are used, and the argument rests only on the cited complex, real-analysis and topological continuity suppliers.

Remarks

The two evaluation points $t = 0$ and $t = π / (2 ω)$ are the whole content of injectivity; no integration or orthogonality argument is needed, and the field version of (b) does not use the continuity hypothesis, which is recorded only because the Statement carries it.
The equivalence (c) concerns the four source-free equations only. It says nothing about boundary conditions, energy, uniqueness of solutions or existence; those belong to the consuming items.
The proof uses no choice principle: all steps are pointwise computations with explicit finite data, and the continuity conclusion only composes explicitly given continuous maps.

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Periodic responses of $\ddot{z} = Re (w e^{- i ω t})$ and the affine classification of all solutions

Statement

Let $ω > 0$ , $T : = 2 π / ω$ , $n \geq 1$ and $w \in C^{n}$ ; use the real part and complex exponential of Real and imaginary parts, complex conjugation, and modulus and The complex exponential by its power series. Define $z_{p} : R \to R^{n}$ by $z_{p} (t) : = Re (- w ω^{- 2} e^{- i ω t})$ componentwise (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set for the derivative below). Then:

(a) $z_{p}$ is $C^{\infty}$ ( $C^{k}$ Euclidean maps and diffeomorphisms) and satisfies $z_{p}^{''} (t) = Re (w e^{- i ω t})$ for every $t \in R$ ; it is $T$ -periodic.

(b) every $C^{2}$ solution $z : R \to R^{n}$ of $z^{''} (t) = Re (w e^{- i ω t})$ on $R$ has the form $z (t) = z_{p} (t) + a + b t$ with unique $a, b \in R^{n}$ .

(c) such a solution is $T$ -periodic if and only if $b = 0$ ; the $T$ -periodic solutions are exactly $z_{p} (t) + a$ , $a \in R^{n}$ , and their oscillating part has complex amplitude $- w / ω^{2}$ .

(d) [Local solutions and exact global extension] Let $I$ be any nonempty open interval, $t_{*} \in I$ , and let $z : I \to R^{n}$ be $C^{2}$ with the same harmonic equation on $I$ . Then $z = z_{p} + a + b t$ on $I$ for a unique pair $a, b \in R^{n}$ , with $b = z^{'} (t_{*}) - z_{p}^{'} (t_{*})$ and $a = z (t_{*}) - z_{p} (t_{*}) - b t_{*}$ . The same explicit formula defines its unique global harmonic-ODE extension $Z$ on $R$ . This extension is $T$ -periodic exactly when $b = 0$ , equivalently $z^{'} (t_{*}) = z_{p}^{'} (t_{*})$ ; when $b = 0$ its mean over every complete period is $a$ . For general drift, the mean of its global extension over $[t, t + T]$ is $a + b (t + T / 2)$ . Thus periodic preparation can be imposed on the global response without requiring a driven nonlinear family to exist on $R$ . If $I$ contains at least one pair $t, t + T$ , equality at that pair forces $b = 0$ . If $I$ contains no such pair, a condition expressed only as $z (t + T) = z (t)$ wherever both arguments belong to $I$ is vacuous and does not imply periodicity of the global extension. All affine-drift local solutions remain classified, and the original global branches (a)-(c) remain intact.

Facts & Assumptions

Given: A real $ω > 0$ with $T = 2 π / ω$ , an integer $n \geq 1$ , a vector $w \in C^{n}$ , and the map $z_{p} : R \to R^{n}$ of the Statement with components $z_{p, j} (t) = Re (- w_{j} ω^{- 2} e^{- i ω t})$ .

[F1]

Complex exponential and Euler form: $\exp$ is entire with $\exp^{'} = \exp$ and satisfies the addition law $\exp (z + u) = \exp z \exp u$ ; for real $θ$ one has $e^{i θ} = \cos θ + i \sin θ$ , and the complex exponential extends the real one; $\cos 0 = 1$ , $\sin 0 = 0$ , and $\sin (m π) = 0$ for every integer $m$ , so $\sin 2 π = 0$ ; both sine and cosine have period $2 π$ (The complex exponential by its power series, The complex exponential is entire and its complex derivative is itself, $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The zero sets of sine and cosine and the least positive common period 2 pi, The derivatives of sine and cosine are cosine and minus sine).

[F2]

Real and imaginary parts: for real $a, b$ one has $Re (a + b i) = a$ and $Im (a + b i) = b$ ; $Re$ and $Im$ are $R$ -linear; complex multiplication in real coordinates is $(a + b i) (c + s i) = (a c - b s) + (a s + b c) i$ ; conjugation is a real-field automorphism with $\overline{z + u} = \overline{z} + \overline{u}$ and $\overline{z u} = \overline{z} \overline{u}$ (Real and imaginary parts, complex conjugation, and modulus, The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ).

[F3]

One-variable calculus rules on $R$ : $\sin^{'} = \cos$ and $\cos^{'} = - \sin$ ; sums, scalar multiples and products of differentiable functions are differentiable, with $(f + g)^{'} = f^{'} + g^{'}$ , $(λ f)^{'} = λ f^{'}$ and $(f g)^{'} = f^{'} g + f g^{'}$ ; the chain rule holds for compositions; the identity map has derivative $1$ and $t \mapsto ω t$ has derivative $ω$ (The derivatives of sine and cosine are cosine and minus sine, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[F4]

Shift and period formulas: $\sin (x + π) = - \sin x$ and $\cos (x + π) = - \cos x$ for every real $x$ , and both sine and cosine have period $2 π$ : $\sin (x + 2 π) = \sin x$ and $\cos (x + 2 π) = \cos x$ (Quarter-turn values and shifts by pi/2 and pi, The zero sets of sine and cosine and the least positive common period 2 pi).

[F5]

Zero-derivative corollary (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant): if $J \subseteq R$ is order-convex (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), $f : J \to R$ is continuous on $J$ and differentiable at every interior point of $J$ with $f^{'} = 0$ there, then $f$ is constant on $J$ ; if $f, g : J \to R$ are continuous on $J$ and differentiable at every interior point of $J$ with $f^{'} = g^{'}$ there, then $f - g$ is constant on $J$ . The interior of $R$ is $R$ , the interior of every nonempty open interval is itself, and every point of a nondegenerate interval that lies in its interior is an interior point (Interior, closure, boundary and exterior of a subset of $R$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[F6]

Integral calculus: if $G : [a, b] \to R$ is differentiable at every point of $[a, b]$ (one-sided at the endpoints) and $f : = G^{'}$ is Riemann integrable on $[a, b]$ , then $\int_{a}^{b} f = G (b) - G (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ); a real function continuous on $[a, b]$ is Riemann integrable on $[a, b]$ (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion); a function differentiable at a point of its domain that is a limit point of that domain is continuous there, so a function differentiable on an interval is continuous on it (A function differentiable at $c$ is continuous at $c$ ).

[F7]

Class conventions: a map $U \to R^{q}$ on an open $U$ is $C^{k}$ when every component is, and $C^{\infty}$ when it is $C^{k}$ for every $k$ ( $C^{k}$ Euclidean maps and diffeomorphisms); the componentwise rules of [F3] show that sums, scalar multiples, products and compositions of smooth real functions are smooth.

Proof

technique · direct; expand the complex encoding in real components, use the elementary antiderivative $\frac{d}{dt}e^{-i\omega t}=-i\omega e^{-i\omega t}$ componentwise, classify solutions by the zero-derivative corollary, and compute the period means with the fundamental theorem of calculus

1.1givenF1F2F3F7

Write $w_{j} = α_{j} + i β_{j}$ with $α_{j} = Re w_{j}$ , $β_{j} = Im w_{j}$ real and fix the component index $j$ ; by [F1] $e^{- i ω t} = \cos ω t - i \sin ω t$ , so the product rule of [F2] with $(a, b) = (α_{j}, β_{j})$ and $(c, s) = (\cos ω t, - \sin ω t)$ gives $z_{p, j} (t) = Re (- w_{j} ω^{- 2} e^{- i ω t}) = - ω^{- 2} (α_{j} \cos ω t + β_{j} \sin ω t)$ for every real $t$ . The maps $t \mapsto ω t$ , $\sin$ and $\cos$ are smooth, and the sum, scalar-multiple and product rules of [F3] give the derivatives of finite combinations, so each $z_{p, j}$ is $C^{\infty}$ and hence $z_{p}$ is $C^{\infty}$ by [F7].

2.1step 1.1F1F2F3

Differentiating the display of step 1.1 componentwise with the rules of [F3] gives $z_{p, j}^{'} (t) = ω^{- 1} (α_{j} \sin ω t - β_{j} \cos ω t)$ and $z_{p, j}^{''} (t) = α_{j} \cos ω t + β_{j} \sin ω t = Re ((α_{j} + i β_{j}) (\cos ω t - i \sin ω t)) = Re (w_{j} e^{- i ω t})$ for every $j$ and every real $t$ , the middle equality by the product rule of [F2] and the last by [F1]; hence $z_{p}^{''} (t) = Re (w e^{- i ω t})$ componentwise.

2.2step 1.1F4given

Since $ω (t + T) = ω t + 2 π$ , the display of step 1.1 and the period formulas $\cos (ω t + 2 π) = \cos ω t$ , $\sin (ω t + 2 π) = \sin ω t$ of [F4] give $z_{p, j} (t + T) = z_{p, j} (t)$ for every $j$ and every $t \in R$ , so $z_{p}$ is $T$ -periodic; with steps 1.1 and 2.1 this is claim (a).

3.1step 2.1F3F5F6given

Let $z : R \to R^{n}$ be $C^{2}$ with $z^{''} (t) = Re (w e^{- i ω t})$ on $R$ and set $h : = z - z_{p}$ ; by step 2.1, $h^{''} = 0$ on $R$ . Fix $j$ ; the component $h_{j}$ is $C^{2}$ , so $h_{j}^{'}$ is continuous on $R$ by [F6] and differentiable at every point of $R$ , which is the interior of $R$ by [F5], with $(h_{j}^{'})^{'} = h_{j}^{''} = 0$ ; [F5] applied on the order-convex interval $R$ therefore makes $h_{j}^{'}$ a constant $b_{j} \in R$ . The map $h_{j} - b_{j} i d$ is continuous on $R$ by [F6] and differentiable at every point with derivative $h_{j}^{'} - b_{j} = 0$ , the derivative of $b_{j} i d$ being $b_{j}$ by [F3]; [F5] makes $h_{j} - b_{j} i d$ a constant $a_{j}$ , so $h_{j} = a_{j} + b_{j} t$ and $z = z_{p} + a + b t$ with $a = (a_{1}, \dots, a_{n})$ , $b = (b_{1}, \dots, b_{n})$ . If also $z = z_{p} + a^{'} + b^{'} t$ on $R$ , evaluating at $t = 0$ gives $a = a^{'}$ and then differentiating at $t = 0$ gives $b = b^{'}$ , so the pair $(a, b)$ is unique; this is claim (b).

4.1step 2.2step 3.1given

Let $z = z_{p} + a + b t$ be a solution as in step 3.1. If $b = 0$ , then $z = z_{p} + a$ is $T$ -periodic by step 2.2. Conversely, if $z$ is $T$ -periodic, then for every $t \in R$ the equality $z (t + T) = z (t)$ together with $z_{p} (t + T) = z_{p} (t)$ of step 2.2 gives $b (t + T) = b t$ , hence $b T = 0$ and $b = 0$ because $T > 0$ . The $T$ -periodic solutions are therefore exactly the maps $z_{p} + a$ with $a \in R^{n}$ ; their oscillating part is $z_{p}$ itself, whose defining formula $z_{p} (t) = Re (- w ω^{- 2} e^{- i ω t})$ exhibits the complex amplitude vector $- w / ω^{2}$ ; this is claim (c).

4.2step 2.1step 2.2step 3.1F5given

Let $I$ be a nonempty open interval, $t_{*} \in I$ , and let $z : I \to R^{n}$ be $C^{2}$ with $z^{''} = Re (w e^{- i ω t})$ on $I$ ; put $h : = z - z_{p}$ on $I$ . Fixing $j$ and repeating the argument of step 3.1 on the order-convex interval $I$ , every interior point of which lies in $I$ where the assumed derivatives exist, gives a constant $b_{j}$ on $I$ equal to $h_{j}^{'}$ , then a constant $a_{j}$ with $h_{j} = a_{j} + b_{j} t$ on $I$ ; hence $z = z_{p} + a + b t$ on $I$ . Evaluating at $t_{*}$ gives $b = z^{'} (t_{*}) - z_{p}^{'} (t_{*})$ and $a = z (t_{*}) - z_{p} (t_{*}) - b t_{*}$ , and conversely those values reproduce $z$ on $I$ , so the pair $(a, b)$ is unique. Define $Z (t) : = z_{p} (t) + a + b t$ for all real $t$ with these $(a, b)$ ; by steps 2.1 and 2.2, $Z$ is a $C^{\infty}$ solution on $R$ extending $z$ . If $\tilde{Z}$ is any global $C^{2}$ solution on $R$ agreeing with $z$ on $I$ , then step 3.1 writes $\tilde{Z} = z_{p} + \tilde{a} + \tilde{b} t$ with a unique pair $(\tilde{a}, \tilde{b})$ ; restricting to $I$ and using the uniqueness just proved gives $\tilde{a} = a$ and $\tilde{b} = b$ , so $\tilde{Z} = Z$ and the extension is unique.

5.1step 4.1step 4.2given

For $Z$ as in step 4.2, step 4.1 gives that $Z$ is $T$ -periodic if and only if $b = 0$ , which by step 4.2 is equivalent to $z^{'} (t_{*}) = z_{p}^{'} (t_{*})$ . If $t$ and $t + T$ both lie in $I$ , then $Z (t + T) - Z (t) = b T$ while $Z$ agrees with $z$ at those two arguments, so equality $z (t + T) = z (t)$ forces $b T = 0$ and $b = 0$ . If $I$ contains no such pair, then the condition " $z (t + T) = z (t)$ whenever both $t$ and $t + T$ lie in $I$ " has empty scope and is satisfied vacuously; the extension with any fixed nonzero $b$ obeys it and is not $T$ -periodic, so the condition determines nothing about periodicity of the global extension.

6.1step 5.1F3F4F6algebra∎

Fix $t \in R$ and a component index $j$ and define $G_{j} (s) : = - ω^{- 3} (α_{j} \sin ω s - β_{j} \cos ω s)$ for $s \in [t, t + T]$ ; by the rules of [F3], $G_{j}$ is differentiable at every point of $[t, t + T]$ with $G_{j}^{'} (s) = - ω^{- 2} (α_{j} \cos ω s + β_{j} \sin ω s) = z_{p, j} (s)$ . The map $z_{p, j}$ is differentiable on $R$ by steps 1.1 and 2.1, hence continuous on $[t, t + T]$ by [F6], hence Riemann integrable on $[t, t + T]$ by [F6]; the fundamental theorem [F6] therefore gives $\int_{t}^{t + T} z_{p, j} = ω^{- 3} [β_{j} (\cos (ω (t + T)) - \cos ω t) - α_{j} (\sin (ω (t + T)) - \sin ω t)] = 0$ , the vanishing because $\cos (ω t + 2 π) = \cos ω t$ and $\sin (ω t + 2 π) = \sin ω t$ by [F4] and $ω T = 2 π$ . Summing over $j$ and dividing by $T$ gives mean zero for $z_{p}$ over $[t, t + T]$ . For the affine part put $P (s) : = a s + b s^{2} / 2$ , differentiable with $P^{'} (s) = a + b s$ by [F3]; the same fundamental theorem, the integrand $s \mapsto a + b s$ being continuous, gives $\frac{1}{T} \int_{t}^{t + T} (a + b s) d s = \frac{P (t + T) - P (t)}{T} = a + \frac{b ((t + T)^{2} - t^{2})}{2 T} = a + b (t + \frac{T}{2})$ . Adding the two computations, the mean of $Z$ over $[t, t + T]$ is $a + b (t + T / 2)$ for every real $t$ , and it equals $a$ for every $t$ exactly when $b = 0$ , which by step 5.1 is exactly the periodicity condition; then the mean over every complete period is $a$ . This completes the proof of (d) and of the lemma.

Remarks

The proof uses no choice principle: the exponential, sine and cosine interfaces, the real-linear algebra of $Re$ and $Im$ , the zero-derivative corollary and the fundamental theorem are all constructive, and the only selections made are of finitely many components of the fixed vectors $a$ and $b$ .
The affine drift is deliberately retained: the lemma does not assert that every solution family is periodic, only that a specified periodic preparation selects the solutions with $b = 0$ , and that this selection can be imposed on the unique global extension of any local solution even when no global driven nonlinear family is available.
The local-interval clauses are not cosmetic. For an interval $I$ shorter than $T$ the equation $z (t + T) = z (t)$ is never instantiated inside $I$ , so it cannot test periodicity; the global extension and its mean are the exact objects that the periodic-preparation hypothesis constrains.

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The ideal harmonic electric-dipole radiation model and its adopted status

Physical scope

Explicit ideal radiation-law model of a harmonic electric dipole of complex amplitude $p_{0}$ in vacuum: the leading transverse far-zone fields $E_{d}, B_{d}$ on the punctured space, their flux $S_{d}$ , and the declared absence of magnetic-dipole and higher-multipole content. Excluded: exact Maxwell solutions at finite radius, near-zone and static fields, any actual source construction, boundary conditions, finite total energy, self-force and radiation reaction, material response laws, and every measured or empirical statement; the relation to actual compact smooth sources is only through the explicit error bounds of the dipole theorem.

Definition

Let $ω > 0$ , $k : = ω / c$ with $c$ the vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values, $μ_{0} > 0$ the vacuum permeability of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and $p_{0} \in C^{3}$ (the complex dipole-moment amplitude, in $C m$ ). Define, on the punctured space $R^{3} ∖ {0}$ and all times $t \in R$ ,

$E_{d} (x, t) : = Re (- \frac{μ_{0} ω^{2}}{4 π r} n \times (n \times p_{0}) e^{i (k r - ω t)}), B_{d} (x, t) : = \frac{1}{c} n \times E_{d} (x, t),$

where $r : = ∣ x ∣$ and $n : = x / r$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ ), with the real part and complex exponential of Real and imaginary parts, complex conjugation, and modulus and The complex exponential by its power series and the componentwise real-part encoding convention of Complex plane-wave fields: real parts of complex exponentials (the spatial amplitude here varies with $x$ ; no plane-wave instance is asserted). Call $(E_{d}, B_{d})$ the ideal harmonic electric-dipole radiation model of frequency $ω$ and amplitude $p_{0}$ , and call $S_{d} : = E_{d} \times B_{d} / μ_{0}$ its model flux (Vacuum electromagnetic energy density and energy flux (the Poynting vector)); the model fields are read as an electric and a magnetic field of The electric and magnetic fields as fixed-time spatial vector fields.

Adopted status

This is an explicitly adopted ideal radiation-law model, fixed by the following declarations.

It is the leading transverse far-zone field of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source(c) evaluated at the harmonic transverse moment $V (u) = Re (- ω^{2} p_{0} e^{- i ω u})$ , i.e. at $p^{''}$ for the waveform $p (u) = Re (p_{0} e^{- i ω u})$ : by part (a) of Periodic responses of $\ddot{z} = Re (w e^{- i ω t})$ and the affine classification of all solutions the second derivative of that explicit $C^{\infty}$ waveform is $Re (- ω^{2} p_{0} e^{- i ω u})$ , and substituting $u = t - r / c$ into $μ_{0} n \times (n \times V (u)) / (4 π r)$ gives exactly $E_{d}$ , because $e^{i (k r - ω t)} = e^{- i ω (t - r / c)}$ .
The model contains no independent magnetic-dipole or higher-multipole radiation, no near-zone or static field and no other radiating component.
It is not asserted to solve the vacuum Maxwell equations at finite $r$ , to be the field of any actual source, to satisfy any boundary condition or to have finite total energy on $R^{3} ∖ {0}$ .
For compact smooth conserved sources, comparison with these prescribed model fields uses the actual source's moment $p_{a c t} (u)$ of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source, the harmonic waveform $p_{h} (u) = Re (p_{0} e^{- i ω u})$ , their second-derivative mismatch

$M (u) : = ∣ p_{a c t}^{''} (u) - p_{h}^{''} (u) ∣,$

and the explicit $D, Γ$ bounds of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source (b), (c) and (e). A bound depending only on that source's $D$ and $Γ$ is valid when $M \equiv 0$ on the observation window, in which case the theorem's leading fields for the source's moment coincide with the model fields; for $M \equiv̸ 0$ the comparison carries the additional mismatch term. The source's moment is not automatically the independently prescribed $p_{0}$ . These actual-source comparisons assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) carried by the cited theorem. The explicit ideal fields themselves and their pointwise flux definition remain choice-free.

No measured, empirical or material-response content is adopted: $p_{0}$ is prescribed model data, not a derived polarizability.

Well-definedness and conventions

On the declared domain $r > 0$ , so $r = ∣ x ∣$ is the positive Euclidean norm and $n = x / r$ is a unit vector; $n \times (n \times p_{0})$ and its real part are defined componentwise, and $E_{d}$ , $B_{d}$ are $C^{\infty}$ on $(R^{3} ∖ {0}) \times R$ because $r$ is $C^{\infty}$ there and the real and imaginary parts of $- \frac{μ_{0} ω^{2}}{4 π r} n \times (n \times p_{0}) e^{i (k r - ω t)}$ are smooth combinations of the smooth functions $r^{- 1}$ , $x_{i}$ , $\cos (k r - ω t)$ and $\sin (k r - ω t)$ (Continuity and derivatives of positive-base real powers, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ). The field is transverse in the sense that $E_{d} \cdot n = 0$ and $B_{d} \cdot n = 0$ , because $n \times (n \times p_{0})$ is orthogonal to $n$ and $n \times E_{d}$ is orthogonal to $n$ by The cross product is bilinear, alternating, and orthogonal to both factors applied separately to real and imaginary parts, and $B_{d} = n \times E_{d} / c$ holds by definition with $∣ B_{d} ∣ = ∣ E_{d} ∣ / c$ . The model is meaningful at $p_{0} = 0$ , where both fields and the flux vanish identically. The varying radial amplitude $1 / r$ is not a plane-wave amplitude, and the componentwise real-part convention is that of Complex plane-wave fields: real parts of complex exponentials; the cross product, inner product and norms are those of The cross product in $R^{3}$ and The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . No equation of motion of any source, no determination of $p_{0}$ from any charge distribution and no energy or flux reading beyond the pointwise definition of $S_{d}$ in Vacuum electromagnetic energy density and energy flux (the Poynting vector) is part of this item.

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The adopted scalar aperture-diffraction model

Physical scope

Adopted scalar aperture-diffraction model: a finite complex aperture distribution $α$ on a compact planar aperture, the outgoing scalar Helmholtz field $u = G_{k} * α$ , its real-field reading $U = Re (u e^{- i ω t})$ and the squared-field comparisons $(1 / 2) ∣ u ∣^{2}$ between observation points, in a declared polarization-insensitive regime. Excluded: derivation from the vacuum Maxwell or material-closure postulates, exact Maxwell validity, identification of $α$ with the exact screen boundary traces, well-posedness of simultaneously prescribed $u$ and $\partial_{n} u$ , polarization and vector effects, aperture back-reaction, absolute radiometric calibration, apparatus, sampling and every empirical comparison.

Postulate

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) as carried by Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms and Exact Helmholtz boundary representation with actual traces (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the smooth modal full-Choice branch of the latter is not used. Adopt as a model assumption of the framework the following scalar aperture-diffraction model.

(a) [Data.] Fix an angular frequency $ω > 0$ and wavenumber $k : = ω / c$ , with $c$ the vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Let $S \subset {y \in R^{3} : y_{3} = 0} \cap {y : ∣ y ∣ \leq a}$ be a nonempty compact subset of a declared aperture plane $y_{3} = 0$ with aperture radius $a > 0$ (the screening plane and spatial region are read in the laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields), and let $α$ be a finite complex measure on $R^{3}$ supported in $S$ (A complex measure is a finite-valued countably additive set function, The total variation |nu|(E) from countable measurable partitions) — the aperture distribution — with $α$ read in volts so that the model field below carries the unit $V m^{- 1}$ of an electric-field component and its pairings are those of Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|).

(b) [Model field.] The model assigns to every observation point $x$ with $x_{3} > 0$ the complex scalar amplitude

$u (x) : = \int G_{k} (x - y) d α (y),$

where $G_{k} (z) = e^{i k ∣ z ∣} / (4 π ∣ z ∣)$ for $z \neq 0$ is the outgoing Helmholtz kernel of Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms, together with the real scalar field $U (x, t) : = Re (u (x) e^{- i ω t})$ and the model period-averaged squared field $(1 / 2) ∣ u (x) ∣^{2}$ . All model predictions are declared comparisons of this squared field between observation points; no absolute radiometric power or intensity calibration is adopted.

(c) [Adopted regime and status.] The model is the declared polarization-insensitive scalar approximation of aperture diffraction. It ignores the vector nature and polarization of the electromagnetic field, takes the aperture distribution rather than the exact screen traces as its datum, assigns no back-reaction of the diffracted field on the illumination and no screen interaction beyond the aperture, and is not derived from either branch of The microscopic vacuum Maxwell equations: smooth and weak branches or from the material-closure branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch; it is a declared optical approximation, not an exact electromagnetic model. It does not assert that $u$ or $U$ satisfies the Maxwell equations exactly, and it does not identify $α$ with the unknown exact Maxwell boundary traces: assigning both $u$ and $\partial_{n} u$ on the aperture independently is not a well-posed exact boundary problem, as recorded in Exact Helmholtz boundary representation with actual traces, and the error terms of Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms are the controlled content of the model. No measured intensity, apparatus, sampling model, uncertainty budget, empirical comparison or quantitative radiometric calibration is adopted.

Well-definedness and non-derivation notes

For fixed $x$ with $x_{3} > 0$ and $y$ ranging over the compact support $S$ of $α$ , one has $∣ x - y ∣ \geq x_{3} > 0$ , so $y \mapsto G_{k} (x - y)$ is continuous on $S$ , bounded there by $1 / (4 π x_{3})$ ; the pairing $\int G_{k} (x - y) d α (y)$ is therefore the integral of a bounded continuous complex function against a finite complex measure and is defined by Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|), with $∣ u (x) ∣ \leq ∥ α ∥_{T V} / (4 π x_{3})$ in the total variation of The total variation |nu|(E) from countable measurable partitions. The model permits the compact closed-radius support $∣ y ∣ \leq a$ , including aperture corners: the integral, off-support regularity and closed-radius estimates at $r \geq 2 a$ are established for finite measures supported in $\overline{B_{a} (0)}$ by Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms. The consuming diffraction theorem also supplies direct kernel estimates at that same closed radius.

The integral against a finite complex measure, the real-field reading and the squared field $\frac{1}{2} ∣ u ∣^{2}$ constitute the whole mathematical content adopted here; the physical reading uses the aperture plane and spatial region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, the relation $k = ω / c$ and the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Neither the smooth nor the weak branch of The microscopic vacuum Maxwell equations: smooth and weak branches implies this scalar source model, and the finite-conductivity and material branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch contain no ideal aperture or screen model. The item uses no choice principle beyond those carried by the cited mathematical suppliers; the Countable Choice hypothesis is inherited verbatim from the kernel and boundary-representation suppliers and this postulate adds no choice use of its own. No derivation of the adopted data from the underlying postulates, and no empirical validation of the model, is claimed or implied by this item.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer

Physical scope

Conditional model predictions for scalar aperture diffraction: the outgoing model field $u = G_{k} * α$ , its Fraunhofer and Fresnel approximations with the explicit field-error terms of the outgoing-source lemma, the transfer of those errors to the model squared field $I = \frac{1}{2} ∣ u ∣^{2}$ , and the trace caveat that the aperture datum is not the exact screen boundary data. Excluded: exact Maxwell validity, vector or polarization content, aperture back-reaction, well-posedness of simultaneously prescribed aperture traces, absolute radiometric calibration, apparatus and sampling models, and every measured or empirical claim.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ) as carried by Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms and Exact Helmholtz boundary representation with actual traces (The Axiom of Countable Choice ( ${A C}_{ω}$ )), whose trace qualification is used in clause (d); the full-Choice modal branch of the latter is not used. Adopt the scalar aperture model and its notation from The adopted scalar aperture-diffraction model: frequency $ω > 0$ , $k = ω / c$ , aperture radius $a > 0$ , finite complex aperture distribution $α$ supported in the planar aperture $S \subset {y_{3} = 0} \cap {y : ∣ y ∣ \leq a}$ , model field $u = G_{k} * α$ , real field $U (x, t) = Re (u (x) e^{- i ω t})$ and $T = 2 π / ω$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The complex exponential by its power series, Real and imaginary parts, complex conjugation, and modulus, Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|)). Set $I (x) : = (1 / 2) ∣ u (x) ∣^{2}$ . Then: (a) [Model field and Fraunhofer estimate] $u$ is $C^{\infty}$ on the half-space ${x_{3} > 0}$ and on $R^{3} ∖ S$ , satisfies $(Δ + k^{2}) u = 0$ there, and is outgoing in the sense of Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms: for $x = r n$ with $r \geq 2 a$ and $n \in S^{2}$ , $u (r n) = \frac{e^{i k r}}{4 π r} \int e^{- i k n \cdot y} d α (y) + O (\frac{∥ α ∥_{T V}}{r} [\frac{a}{r} + \frac{k a^{2}}{r}])$ uniformly in direction, and $r (\partial_{r} u - i k u) = O (r^{- 1})$ uniformly. (b) [Fresnel estimate] write $y_{t} = (y_{1}, y_{2})$ and $ξ = (ξ_{1}, ξ_{2})$ for transverse vectors; for the planar aperture and observation $x = (ξ_{1}, ξ_{2}, z)$ with $z > 0$ , $b \geq 0$ , $∣ ξ ∣ \leq b$ and any declared fixed small-angle range $(a + b) / z \leq η_{0} < 1$ , where $0 < η_{0} < 1$ , the Fresnel approximation replaces $R = ∣ x - y ∣$ by $z + ∣ ξ - y_{t} ∣^{2} / (2 z)$ and $1 / R$ by $1 / z$ in the model integral with phase error $O (k (a + b)^{4} / z^{3})$ , relative amplitude error $O ((a + b)^{2} / z^{2})$ and absolute field error $O (∥ α ∥_{T V} z^{- 1} [(a + b)^{2} / z^{2} + k (a + b)^{4} / z^{3}])$ . (c) [Intensity and error transfer] at every $x$ with $x_{3} > 0$ the model period average satisfies $\frac{1}{T} \int_{0}^{T} U (x, s)^{2} d s = I (x)$ (Period averages of products of real parts of complex exponentials). If $u_{a p p r o x}$ is any approximation of $u$ with $∣ u_{a p p r o x} (x) - u (x) ∣ \leq δ (x)$ , then $∣ \frac{1}{2} ∣ u_{a p p r o x} (x) ∣^{2} - I (x) ∣ \leq \frac{1}{2} δ (x) (∣ u (x) ∣ + ∣ u_{a p p r o x} (x) ∣)$ ; if moreover $u (x) \neq 0$ and $∣ u_{a p p r o x} (x) - u (x) ∣ \leq ϵ (x) ∣ u (x) ∣$ , the relative intensity error is at most $2 ϵ (x) + ϵ (x)^{2}$ . In particular the Fraunhofer and Fresnel field errors of (a) and (b) bound the corresponding intensity errors; no new approximation is introduced by passing to $I$ . (d) [Trace and scope] the aperture distribution $α$ is an adopted model datum and is not identified with the exact Maxwell boundary data on the screen; assigning both $u$ and $\partial_{n} u$ on the aperture independently does not produce a well-posed exact boundary problem (Exact Helmholtz boundary representation with actual traces). The construction ignores polarization and aperture back-reaction, makes no claim that $u$ or $U$ satisfies the Maxwell equations exactly, and makes no measured-intensity, apparatus, sampling or empirical claim; only comparisons of $I$ between observation points are read as model predictions.

Facts & Assumptions

Given: The model data of the Statement: $ω > 0$ , $k = ω / c$ , $a > 0$ , the compact planar aperture $S \subset {y_{3} = 0} \cap {∣ y ∣ \leq a}$ , the finite complex measure $α$ supported in $S$ , the model field $u = G_{k} * α$ with $G_{k} (z) = e^{i k ∣ z ∣} / (4 π ∣ z ∣)$ for $z \neq 0$ , the real field $U (x, t) = Re (u (x) e^{- i ω t})$ , and $I = \frac{1}{2} ∣ u ∣^{2}$ . All lengths are in the laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, and $c$ , $ε_{0}$ , $μ_{0}$ are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

[F1]

Model interface: the aperture distribution is a finite complex measure with total variation $∥ α ∥_{T V}$ supported in the compact $S$ , read in volts; for every $x$ with $x_{3} > 0$ the pairing $\int G_{k} (x - y) d α (y)$ is defined by Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|) and equals the model field $u (x)$ ; the same formula defines $u$ at every $x \notin S$ , and $U$ , $I$ are as in the Statement (The adopted scalar aperture-diffraction model, The total variation |nu|(E) from countable measurable partitions, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

Outgoing-source and trace suppliers: assume ${A C}_{ω}$ ; $G_{k}$ is locally integrable, $(- Δ - k^{2}) G_{k} = δ_{0}$ distributionally, and for a finite complex measure supported in a closed ball the field $G_{k} * α$ is $C^{\infty}$ outside its support, solves $(Δ + k^{2}) w = 0$ there, is outgoing with the Fraunhofer and Fresnel error terms of that item, and the aperture data are an explicitly specified scalar source, not the exact Maxwell boundary traces; assigning both aperture values independently is not a well-posed exact boundary problem (Outgoing source Helmholtz field: fundamental solution, Fraunhofer and Fresnel error terms, Exact Helmholtz boundary representation with actual traces, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

Finite-measure integral bound: let $h$ be Borel measurable and bounded on the support of the finite complex measure $α$ , with $M_{h} : = \sup_{supp α} ∣ h ∣ < + \infty$ ; then $h \in L^{1} (α) = L^{1} (∣ α ∣)$ and $∣ \int h d α ∣ \leq \int ∣ h ∣ d ∣ α ∣ \leq M_{h} ∥ α ∥_{T V}$ : the modulus bound for a general integrand is Integrals against signed or complex measures are bounded by total variation, integrability in $L^{1} (α) = L^{1} (∣ α ∣)$ is the definition of Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|), and since $α$ is supported in $S$ the variation $∣ α ∣$ vanishes off $S$ by its defining supremum over partitions (The total variation |nu|(E) from countable measurable partitions) and is finite (The total variation of a signed or complex measure is a positive measure, Every complex measure has finite total variation), so monotonicity of the integral of nonnegative measurable functions gives $\int ∣ h ∣ d ∣ α ∣ \leq \int M_{h} d ∣ α ∣ = M_{h} ∥ α ∥_{T V}$ (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F4]

Period average: for $ω > 0$ , $T = 2 π / ω$ and $u \in C$ , $\frac{1}{T} \int_{0}^{T} Re (u e^{- i ω t})^{2} d t = \frac{1}{2} ∣ u ∣^{2}$ (Period averages of products of real parts of complex exponentials).

[F5]

Modulus and conjugation: $z \overline{z} = ∣ z ∣^{2}$ , $∣ z ∣ \geq 0$ , $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , and $∣ ∣ z ∣ - ∣ w ∣ ∣ \leq ∣ z - w ∣$ for all $z, w \in C$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, Real and imaginary parts, complex conjugation, and modulus).

[F6]

Elementary toolbox: $∣ a + b i ∣ = \sqrt{a^{2} + b^{2}}$ and $e^{i θ} = \cos θ + i \sin θ$ for real $θ$ , with the addition law $e^{z + w} = e^{z} e^{w}$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential), $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ , $∣ \cos θ ∣ \leq 1$ and $\cos^{2} θ + \sin^{2} θ = 1$ (Real and imaginary parts, complex conjugation, and modulus, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine); the Taylor remainder bound $∣ R_{n, a} f (x) ∣ \leq M s^{n + 1} / (n + 1)!$ when $∣ f^{(n + 1)} ∣ \leq M$ on the interval, with $R_{n, a} f = f - T_{n, a} f$ (A uniform derivative bound gives a uniform Taylor remainder bound, Taylor polynomials and their remainders); the real-power derivatives $(x^{α})^{'} = α x^{α - 1}$ on $(0, \infty)$ (Continuity and derivatives of positive-base real powers); the chain and algebra of derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ); the Euclidean norm and inner product with $\sqrt{}$ the nonnegative square root (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , The complex exponential by its power series).

Proof

technique · direct; instantiate the supplier's off-support regularity on the declared closed ball, re-derive the sharp closed-radius Fraunhofer and outgoing estimates from the explicit kernel with elementary Taylor bounds, establish the Fresnel estimates through the exact substitution $R=z\sqrt{1+s}$, and transfer the field errors to the intensity with the modulus inequalities

1.1givenF1F2

[Off-support regularity and the Helmholtz equation] Since $S$ is compact and contained in the closed ball $\overline{B_{a} (0)}$ , $α$ is a finite complex measure supported in that closed ball, exactly as required by [F2]. By [F2] the field $G_{k} * α$ is $C^{\infty}$ outside the support of $α$ and solves $(Δ + k^{2}) w = 0$ there; since the support of $α$ is contained in $S$ , the field $u$ of [F1] is $C^{\infty}$ on $R^{3} ∖ S$ and satisfies $(Δ + k^{2}) u = 0$ there; the half-space ${x_{3} > 0}$ is disjoint from the plane containing $S$ , hence is contained in $R^{3} ∖ S$ , so $u$ is $C^{\infty}$ there as well.

1.2givenF6algebra

[Elementary bounds: phase, radial expansion, kernel residual] First, for every real $θ$ one has $∣ e^{i θ} - 1 ∣ \leq ∣ θ ∣$ : by [F6], $2 (1 - \cos θ) = ∣ e^{i θ} - 1 ∣^{2}$ and $∣ \cos θ ∣ \leq 1$ , and the Taylor bound for $\cos$ at $0$ with $n = 1$ gives $∣ \cos θ - 1 ∣ \leq θ^{2} / 2$ , because $∣ \cos^{''} ∣ \leq 1$ ; hence $∣ e^{i θ} - 1 ∣^{2} = 2 (1 - \cos θ) \leq θ^{2}$ . Second, fix $y \in S$ , $r \geq 2 a$ , $n \in S^{2}$ , and put $R : = ∣ r n - y ∣$ and $h (s) : = ∣ r n - s y ∣$ for $s \in [0, 1]$ ; then $h$ is $C^{2}$ on $[0, 1]$ with $h (0) = r$ , $h (1) = R$ , $h^{'} (0) = - n \cdot y$ , $h^{'} (s) = - (r n - s y) \cdot y / h (s)$ , $h^{''} (s) = ∣ y ∣^{2} / h (s) - ((r n - s y) \cdot y)^{2} / h (s)^{3}$ , so $∣ h^{''} ∣ \leq 2 ∣ y ∣^{2} / h (s) \leq 2 a^{2} / (r - a) \leq 4 a^{2} / r$ by $h (s) \geq r - a \geq r / 2$ ; the Taylor bound at $0$ with $n = 1$ gives $∣ R - r + n \cdot y ∣ \leq 2 a^{2} / r$ , and then $∣ r / R - 1 ∣ = ∣ r - R ∣ / R \leq (a + 2 a^{2} / r) / (r / 2) = 2 a / r + 4 a^{2} / r^{2} \leq 4 a / r$ since $∣ r - R ∣ \leq ∣ n \cdot y ∣ + 2 a^{2} / r \leq a + a = 2 a$ , $R \geq r / 2$ , and $r \geq 2 a$ gives $4 a^{2} / r^{2} \leq 2 a / r$ . Third, with $R$ as above, $\partial_{r} R = (r - n \cdot y) / R$ satisfies $∣ \partial_{r} R ∣ \leq 1$ and $∣ \partial_{r} R - 1 ∣ \leq ∣ R - r + n \cdot y ∣ / R \leq 4 a^{2} / r^{2}$ , and the exact kernel derivatives give $\partial_{r} G_{k} (r n - y) = \partial_{r} R (i k - 1 / R) e^{i k R} / (4 π R)$ and hence $∣ \partial_{r} G_{k} - i k G_{k} ∣ \leq (1 / (4 π R)) [k ∣ \partial_{r} R - 1 ∣ + ∣ \partial_{r} R ∣ / R] \leq 2 k a^{2} / (π r^{3}) + 1 / (π r^{2})$ .

1.3givenF1F6algebra

[Fresnel substitution and estimates] Fix the planar observation $x = (ξ_{1}, ξ_{2}, z)$ with $z > 0$ , $∣ ξ ∣ \leq b$ , and let $y \in S$ with transverse part $y_{t}$ ; put $d_{0} : = a + b$ , $s : = ∣ ξ - y_{t} ∣^{2} / z^{2}$ , so that $0 \leq s \leq η_{0}^{2} < 1$ on the declared range. Decomposing the Euclidean norm by [F6], $R = ∣ x - y ∣ = z \sqrt{1 + s}$ , hence $1 / R = (1 / z) (1 + s)^{- 1 / 2}$ and the Fresnel replacements are $R_{F} : = z (1 + s / 2)$ and $1 / z$ . The Taylor bound of [F6] at $0$ with $n = 1$ gives $∣ \sqrt{1 + s} - 1 - s / 2 ∣ \leq s^{2} / 8$ , because $∣ (d^{2} / d t^{2}) \sqrt{1 + t} ∣ = ∣ ⁣ - ⁣ \frac{1}{4} (1 + t)^{- 3 / 2} ∣ \leq 1 / 4$ on $[0, s]$ ; the Taylor bound with $n = 0$ gives $∣ (1 + s)^{- 1 / 2} - 1 ∣ \leq s / 2$ , because $∣ (d / d t) (1 + t)^{- 1 / 2} ∣ = \frac{1}{2} (1 + t)^{- 3 / 2} \leq 1 / 2$ . Consequently the phase error satisfies $k ∣ R - R_{F} ∣ = k z ∣ \sqrt{1 + s} - 1 - s / 2 ∣ \leq k d_{0}^{4} / (8 z^{3})$ , and the relative amplitude error satisfies $∣ 1 - (1 + s)^{- 1 / 2} ∣ \leq s / 2 \leq d_{0}^{2} / (2 z^{2})$ , where $s \leq d_{0}^{2} / z^{2}$ by $∣ ξ - y_{t} ∣ \leq d_{0}$ .

1.4givenF4algebra

[Period average equals the model intensity] Fix $x$ with $x_{3} > 0$ and write $u = u (x) \in C$ ; then $U (x, s) = Re (u e^{- i ω s})$ , so [F4] applied with the complex number $u$ in both slots gives $\frac{1}{T} \int_{0}^{T} U (x, s)^{2} d s = \frac{1}{2} ∣ u ∣^{2} = I (x)$ .

2.1step 1.2F1F2F3F6algebra

[Fraunhofer estimate at the closed radius] Fix $y \in S$ , $r \geq 2 a$ and $n \in S^{2}$ , and write $R = ∣ r n - y ∣$ . Using only the bounds of step 1.2 and the addition law of the exponential for $e^{i k (r - n \cdot y)} = e^{i k r} e^{- i k n \cdot y}$ from [F6], one has $∣ G_{k} (r n - y) - e^{i k r} e^{- i k n \cdot y} / (4 π r) ∣ = ∣ e^{i k R} (r / R - 1) + e^{i k R} - e^{i k (r - n \cdot y)} ∣ / (4 π r) \leq (1 / (4 π r)) [4 a / r + k \cdot 2 a^{2} / r] \leq (1 / (π r)) [a / r + k a^{2} / r]$ , since $4 / (4 π) = 1 / π$ and $2 / (4 π) \leq 1 / π$ , and because the middle term was bounded by $∣ e^{i θ} - 1 ∣ \leq ∣ θ ∣$ with $θ : = k (R - r + n \cdot y)$ and $∣ θ ∣ \leq 2 k a^{2} / r$ . Integrating against $α$ with the bound of [F3], applied to the bounded continuous function $y \mapsto G_{k} (r n - y) - e^{i k r} e^{- i k n \cdot y} / (4 π r)$ on the compact support, gives $∣ u (r n) - e^{i k r} (4 π r)^{- 1} \int e^{- i k n \cdot y} d α (y) ∣ \leq ∥ α ∥_{T V} (π r)^{- 1} [a / r + k a^{2} / r]$ , uniformly in $n$ ; this is the displayed Fraunhofer estimate with a universal constant.

2.2step 1.3F1F2F3F6algebra

[Absolute field error of the Fresnel replacement] With the substitutions of step 1.3, for $y \in S$ : $∣ G_{k} (x - y) - e^{i k R_{F}} / (4 π z) ∣ \leq (1 / (4 π)) ∣ 1 / R - 1 / z ∣ + (1 / (4 π z)) ∣ e^{i k R} - e^{i k R_{F}} ∣ \leq (1 / (4 π)) d_{0}^{2} / (2 z^{3}) + (1 / (4 π z)) k d_{0}^{4} / (8 z^{3}) = (1 / (4 π z)) [d_{0}^{2} / (2 z^{2}) + k d_{0}^{4} / (8 z^{3})]$ , where the first summand used $∣ 1 / R - 1 / z ∣ = (1 / z) ∣ (1 + s)^{- 1 / 2} - 1 ∣ \leq d_{0}^{2} / (2 z^{3})$ and the second used $∣ e^{i k R} - e^{i k R_{F}} ∣ \leq k ∣ R - R_{F} ∣ \leq k d_{0}^{4} / (8 z^{3})$ from the phase estimate of step 1.3. Integrating against $α$ with [F3] gives $∣ u (x) - u_{F r e s n e l} (x) ∣ \leq ∥ α ∥_{T V} (4 π z)^{- 1} [d_{0}^{2} / (2 z^{2}) + k d_{0}^{4} / (8 z^{3})]$ , the displayed absolute field error with universal constants; the phase and relative amplitude errors of step 1.3 are those displayed in (b).

3.1step 1.2step 2.1F1F2F3algebra

[Outgoing residual] With the third bound of step 1.2 and the same integration against the total variation [F3], $r ∣ \partial_{r} u (r n) - i k u (r n) ∣ \leq r \int ∣ \partial_{r} G_{k} (r n - y) - i k G_{k} (r n - y) ∣ d ∣ α ∣ (y) \leq ∥ α ∥_{T V} [2 k a^{2} / (π r^{2}) + 1 / (π r)]$ , which is uniformly $O (∥ α ∥_{T V} r^{- 1})$ in direction; this and step 2.1 give clause (a).

3.2step 1.4step 2.1step 2.2F5algebra

[Intensity error transfer] Let $u_{a p p r o x}$ satisfy $∣ u_{a p p r o x} (x) - u (x) ∣ \leq δ (x)$ at the point $x$ . By the reverse triangle inequality of [F5], $∣ ∣ u_{a p p r o x} ∣ - ∣ u ∣ ∣ \leq ∣ u_{a p p r o x} - u ∣ \leq δ$ , and by $∣ z ∣^{2} = z \overline{z}$ and multiplicativity of the modulus [F5], $∣ ∣ u_{a p p r o x} ∣^{2} - ∣ u ∣^{2} ∣ = ∣ ∣ u_{a p p r o x} ∣ - ∣ u ∣ ∣ (∣ u_{a p p r o x} ∣ + ∣ u ∣) \leq δ (x) (∣ u (x) ∣ + ∣ u_{a p p r o x} (x) ∣)$ ; multiplying by $\frac{1}{2}$ and comparing with step 1.4 gives the first bound of (c). If additionally $u (x) \neq 0$ and $∣ u_{a p p r o x} - u ∣ \leq ϵ (x) ∣ u (x) ∣$ , then $∣ u_{a p p r o x} ∣ \leq (1 + ϵ (x)) ∣ u (x) ∣$ , so the absolute intensity error is at most $\frac{1}{2} ϵ (x) ∣ u (x) ∣ (2 + ϵ (x)) ∣ u (x) ∣ = (ϵ (x) + \frac{1}{2} ϵ (x)^{2}) ∣ u (x) ∣^{2} = (2 ϵ (x) + ϵ (x)^{2}) I (x)$ , which is the stated relative bound; applying these two estimates to the Fraunhofer and Fresnel field errors of steps 2.1, 3.1 and 2.2 gives the final sentence of (c) with no new approximation.

4.1step 3.1step 2.2step 3.2F1F2F4given∎

[Trace caveat, scope and choice accounting] The aperture distribution $α$ is the adopted datum of [F1] and is not identified with the exact Maxwell boundary data on the screen; by [F2] assigning both $u$ and $\partial_{n} u$ on the aperture independently is not a well-posed exact boundary problem, so no such identification or well-posedness is made. The construction never asserts that $u$ or $U$ satisfies the Maxwell equations; it ignores polarization and aperture back-reaction, and it makes no measured-intensity, apparatus, sampling or empirical claim, only model comparisons of $I$ between observation points; this is clause (d). The only choice principle used is the Countable Choice carried by the two suppliers of [F2] and used in their trace and kernel interfaces; the period average of step 1.4 uses only the choice-free period-average lemma, and all other steps are explicit estimates and finite computations, so no further choice is invoked.

Remarks

The two big-O statements of (a) are established by the proof in the quantitative form $∣ u (r n) - e^{i k r} (4 π r)^{- 1} \int e^{- i k n \cdot y} d α ∣ \leq ∥ α ∥_{T V} (π r)^{- 1} [a / r + k a^{2} / r]$ and $r ∣ \partial_{r} u - i k u ∣ \leq ∥ α ∥_{T V} (2 k a^{2} / (π r^{2}) + 1 / (π r))$ , both uniform in direction; the big-O form is the same statement with a universal constant.
The Fresnel error transfer is an error statement about the adopted model field, not about any measured optical intensity; the model comparison is between observation points of $I = \frac{1}{2} ∣ u ∣^{2}$ only.

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Monochromatic vector scattering by a smooth PEC obstacle: phasor Maxwell fields, the PEC trace condition and outgoing-correction uniqueness

Physical scope

Conditional vector scattering of a smooth frequency- $ω$ source-free vacuum phasor pair by a smooth bounded stationary perfect electric conductor with connected exterior: the scattered and total phasor pairs, the associated real fields solving the zero-source smooth vacuum Maxwell equations on the exterior, the ideal PEC boundary condition $n \times E_{t o t} = 0$ , and uniqueness of the outgoing scattered solution with the prescribed tangential trace. Excluded: lossy, impedance, nonsmooth or dispersive obstacles; determination of induced surface charge or current; radiation-zone amplitude asymptotics beyond the outgoing modal class; singular or point sources; finite-energy, coupled self-force and every empirical claim.

Statement

Assume the Axiom of Choice as carried by Existence and uniqueness for the exterior vector Helmholtz problem with prescribed tangential trace and outgoing radiation, whose variational, Sobolev-compactness and compact-spectral steps use it; the algebraic computations below use no further choice. Let $O \subset R^{3}$ be a bounded open set with $C^{\infty}$ boundary, $D : = R^{3} ∖ \overline{O}$ connected, $k > 0$ , $ω : = c k$ with $c$ the vacuum speed of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and $R > 0$ with $\overline{O} \subset B_{R} (0)$ ; let $V \subseteq R^{3}$ be open with $\overline{D} \subseteq V$ , and let $({\tilde{E}}_{i n c}, {\tilde{B}}_{i n c})$ be a $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair on $V$ (Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values). Plane-wave instances are provided by Dispersion, transversality and the two polarization conditions for plane-wave solutions: if $κ \cdot e = 0$ , $b = κ \times e / ω$ and $∣ κ ∣ = ω / c$ , then $({\tilde{E}}_{i n c}, {\tilde{B}}_{i n c}) (x) = (e e^{i κ \cdot x}, b e^{i κ \cdot x})$ satisfies the four phasor equations on $R^{3}$ (Real plane-wave fields encoded by complex amplitudes). Adopt the ideal perfectly conducting (PEC) boundary model: the tangential component of the total electric field vanishes on $\partial O$ . This is a declared idealization of the stationary interface conditions of Stationary macroscopic interface jump conditions for D, B, E and H with vanishing fields inside the conductor; it is not one of the finite-conductivity branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, and the induced surface charge and current are neither prescribed nor determined here. Let $e_{T}$ be the tangential projection of $- {\tilde{E}}_{i n c}$ on $\partial O$ , and let ${\tilde{E}}_{s c a t}$ be the unique $C^{\infty}$ field on $\overline{D}$ of Existence and uniqueness for the exterior vector Helmholtz problem with prescribed tangential trace and outgoing radiation with $(Δ + k^{2}) {\tilde{E}}_{s c a t} = 0$ and $div {\tilde{E}}_{s c a t} = 0$ on $D$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Divergence and curl of a $C^{1}$ vector field), tangential trace $e_{T}$ on $\partial O$ and that lemma's outgoing modal condition on $S_{R}$ . Set ${\tilde{B}}_{s c a t} : = (i ω)^{- 1} curl {\tilde{E}}_{s c a t}$ and, on $D$ , ${\tilde{E}}_{t o t} : = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ , ${\tilde{B}}_{t o t} : = {\tilde{B}}_{i n c} + {\tilde{B}}_{s c a t}$ . Then: (a) [Phasor Maxwell] $({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ and $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ are $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pairs on $D$ ; and for every $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair $(\tilde{E}, \tilde{B})$ on an open set $O^{'} \subseteq R^{3}$ , the associated real fields of Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding satisfy on $O^{'} \times R$ the smooth-branch vacuum Maxwell equations of The microscopic vacuum Maxwell equations: smooth and weak branches with $ρ \equiv 0$ and $J \equiv 0$ . Consequently the associated real total fields $E_{t o t}, B_{t o t}$ satisfy those source-free equations on $D \times R$ . (b) [PEC boundary condition] the tangential trace of ${\tilde{E}}_{t o t}$ on $\partial O$ vanishes; hence for every unit normal field $n$ of $\partial O$ the associated real total electric field $E_{t o t} (x, t) = Re ({\tilde{E}}_{t o t} (x) e^{- i ω t})$ (Real and imaginary parts, complex conjugation, and modulus) satisfies $n \times E_{t o t} = 0$ on $\partial O \times R$ . (c) [Uniqueness] if $({\tilde{E}}^{'}, {\tilde{B}}^{'})$ is a frequency- $ω$ source-free vacuum phasor pair on $D$ whose fields belong to $C^{\infty} (\overline{D}; C^{3})$ that is outgoing on $S_{R}$ in the modal sense of the vector Helmholtz lemma and has tangential trace $e_{T}$ on $\partial O$ , then $({\tilde{E}}^{'}, {\tilde{B}}^{'}) = ({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ on $D$ ; equivalently $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ is the unique total source-free phasor pair in the corresponding $C^{\infty}$ closure class with that prescribed incident phasor, vanishing tangential electric trace, and outgoing scattered correction ${\tilde{E}}_{t o t} - {\tilde{E}}_{i n c}$ in the lemma's modal class. The total field itself is not required to be outgoing; an incident plane wave is generally not outgoing. (d) [Scope] one positive frequency $ω = c k$ , one smooth bounded stationary PEC obstacle with connected exterior, and $C^{\infty}$ incident and scattered classes only; no lossy, impedance, nonsmooth or dispersive obstacle, no determination of the induced surface charge or current, no radiation-zone amplitude or far-field asymptotics beyond the lemma's outgoing modal class, no point or singular source, no finite-energy or empirical claim.

Facts & Assumptions

Given: The Axiom of Choice; the bounded $C^{\infty}$ obstacle $O$ with connected exterior $D$ ; $k > 0$ , $ω = c k$ , $R > 0$ with $\overline{O} \subset B_{R} (0)$ ; the open $V \supseteq \overline{D}$ and the $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair $({\tilde{E}}_{i n c}, {\tilde{B}}_{i n c})$ on $V$ ; the tangential projection $e_{T}$ of $- {\tilde{E}}_{i n c}$ on $\partial O$ ; the field ${\tilde{E}}_{s c a t}$ and ${\tilde{B}}_{s c a t} = (i ω)^{- 1} curl {\tilde{E}}_{s c a t}$ of the Statement; the totals ${\tilde{E}}_{t o t} = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ and ${\tilde{B}}_{t o t} = {\tilde{B}}_{i n c} + {\tilde{B}}_{s c a t}$ on $D$ .

[F1]

Exterior vector Helmholtz lemma: for the geometry, $k$ and $R$ above and every smooth $C$ -valued tangential field $e_{T}$ on $\partial O$ , there is exactly one $E \in C^{\infty} (\overline{D}; C^{3})$ with $(Δ + k^{2}) E = 0$ on $D$ , $div E = 0$ on $D$ , tangential trace $E_{t a n g e n t i a l} = e_{T}$ on $\partial O$ , and the modal outgoing condition on $S_{R}$ (its trace lies in $H^{1 / 2} (S_{R}; C^{3})$ and its normal derivative on $S_{R}$ equals the sphere Dirichlet-to-Neumann image of that trace componentwise in the weak trace sense); its normal boundary values satisfy the closure relation; the lemma assumes the Axiom of Choice (Existence and uniqueness for the exterior vector Helmholtz problem with prescribed tangential trace and outgoing radiation).

[F2]

Phasor conventions: a frequency- $ω$ source-free vacuum phasor pair on an open set is a $C^{\infty}$ pair $(\tilde{E}, \tilde{B})$ satisfying $div \tilde{E} = 0$ , $div \tilde{B} = 0$ , $curl \tilde{E} = i ω \tilde{B}$ and $curl \tilde{B} = - i ω c^{- 2} \tilde{E}$ there, with divergence and curl acting componentwise on the complex fields; the associated real fields are $E = Re (\tilde{E} e^{- i ω t})$ , $B = Re (\tilde{B} e^{- i ω t})$ (Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding, Real and imaginary parts, complex conjugation, and modulus).

[F3]

Phasor calculus and equivalence: for $C^{1}$ complex fields the displayed partial-derivative and divergence/curl formulas hold; the real-part evaluation is injective; and the phasor system holds on an open set if and only if the associated real fields satisfy $div E = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = c^{- 2} \partial_{t} E$ there (Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system).

[F4]

Smooth vacuum Maxwell branch at zero sources: for jointly $C^{1}$ fields with the additional derivatives required by each equation existing and continuous and continuous sources, the adopted vacuum Maxwell postulate consists of the four equations $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ ; with $ρ \equiv 0$ and $J \equiv 0$ this is exactly the source-free system of [F3] (The microscopic vacuum Maxwell equations: smooth and weak branches, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F5]

Plane-wave amplitude equations: for $k \in R^{3} ∖ {0}$ , $ω > 0$ and $e, b \in C^{3}$ , the plane-wave fields satisfy the source-free smooth Maxwell equations if and only if $κ \cdot e = 0$ , $κ \cdot b = 0$ , $κ \times e = ω b$ and $κ \times b = - (ω / c^{2}) e$ ; on the dispersion relation $∣ κ ∣ = ω / c$ the solutions are exactly the pairs with $κ \cdot e = 0$ and $b = κ \times e / ω$ (Dispersion, transversality and the two polarization conditions for plane-wave solutions, Real plane-wave fields encoded by complex amplitudes).

[F6]

Vector-calculus identities: for a $C^{2}$ real field on an open subset of $R^{3}$ , $curl curl F = \nabla div F - Δ F$ and $div (curl F) = 0$ ; divergence and curl are linear over the reals, and both identities and both operators extend to complex fields by applying them to real and imaginary parts componentwise (The curl of a curl is the gradient of the divergence minus the Laplacian, The divergence of the curl of a $C^{2}$ field vanishes, Divergence and curl are linear and satisfy the scalar product rules, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Divergence and curl of a $C^{1}$ vector field).

[F7]

Tangential projection and cross product: for a unit normal field $n$ of $\partial O$ and a field $w$ , the tangential projection is $w_{t a n g} : = w - (w \cdot n) n$ with the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , and $n \times w = n \times w_{t a n g}$ because $n \times n = 0$ ; the cross product is bilinear and alternating (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F8]

Material context: the PEC condition is the declared idealization of the stationary interface conditions of Stationary macroscopic interface jump conditions for D, B, E and H with vanishing fields inside the conductor; the material-closure branches of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch contain no $σ \to \infty$ ideal-conductor branch, and the fields of The electric and magnetic fields as fixed-time spatial vector fields with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values on the laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields are the physical reading adopted by Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding.

[F9]

Choice accounting: the Axiom of Choice of the Statement is used exactly through [F1]; all other computations are pointwise algebra and finite sums, and no further selection from infinite families is made (The Axiom of Choice).

Proof

technique · direct; instantiate the exterior vector Helmholtz lemma for the scattered field, verify the four phasor equations algebraically, transfer the real-field Maxwell statement through the phasor equivalence, read the PEC condition off the tangential traces, and reduce uniqueness of any candidate to the lemma's uniqueness clause

1.1givenF1F6F7

[Setting and existence of the scattered field] The hypotheses of [F1] are met by the geometry, $k$ and $R$ , and $e_{T}$ is a smooth $C$ -valued tangential field on $\partial O$ by [F7] applied to the $C^{\infty}$ incident field; hence [F1] supplies a unique ${\tilde{E}}_{s c a t} \in C^{\infty} (\overline{D}; C^{3})$ with $(Δ + k^{2}) {\tilde{E}}_{s c a t} = 0$ , $div {\tilde{E}}_{s c a t} = 0$ on $D$ , tangential trace $e_{T}$ on $\partial O$ , and the outgoing modal condition on $S_{R}$ . The field ${\tilde{B}}_{s c a t} : = (i ω)^{- 1} curl {\tilde{E}}_{s c a t}$ is $C^{\infty}$ on $\overline{D}$ by [F6] and the smooth closure regularity of ${\tilde{E}}_{s c a t}$ , and the incident pair together with $e_{T}$ and the geometry is the data of the Statement.

2.1step 1.1F1F2F6algebra

[The scattered pair is source-free] On $D$ : $div {\tilde{E}}_{s c a t} = 0$ by step 1.1; $curl {\tilde{E}}_{s c a t} = i ω {\tilde{B}}_{s c a t}$ by the definition of ${\tilde{B}}_{s c a t}$ ; $div {\tilde{B}}_{s c a t} = (i ω)^{- 1} div (curl {\tilde{E}}_{s c a t}) = 0$ by [F6]; and $curl {\tilde{B}}_{s c a t} = (i ω)^{- 1} curl curl {\tilde{E}}_{s c a t} = (i ω)^{- 1} (\nabla div {\tilde{E}}_{s c a t} - Δ {\tilde{E}}_{s c a t}) = (i ω)^{- 1} k^{2} {\tilde{E}}_{s c a t}$ , which equals $- i ω c^{- 2} {\tilde{E}}_{s c a t}$ because $k = ω / c$ gives $k^{2} / (i ω) = - i k^{2} / ω = - i ω c^{- 2}$ . All four equations of [F2] hold on the open set $D$ for a $C^{\infty}$ pair, so $({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ is a frequency- $ω$ source-free vacuum phasor pair on $D$ .

2.2step 1.1F1F2F6algebra

[Uniqueness of the outgoing scattered correction] Let $({\tilde{E}}^{'}, {\tilde{B}}^{'})$ be a frequency- $ω$ source-free vacuum phasor pair on $D$ whose fields belong to $C^{\infty} (\overline{D}; C^{3})$ , outgoing on $S_{R}$ in the modal sense of [F1], with tangential trace $e_{T}$ on $\partial O$ . From the phasor equations and [F6], $curl curl {\tilde{E}}^{'} = i ω curl {\tilde{B}}^{'} = i ω (- i ω c^{- 2} {\tilde{E}}^{'}) = ω^{2} c^{- 2} {\tilde{E}}^{'} = k^{2} {\tilde{E}}^{'}$ and $div {\tilde{E}}^{'} = 0$ , so [F6] gives $\nabla div {\tilde{E}}^{'} - Δ {\tilde{E}}^{'} = - Δ {\tilde{E}}^{'} = k^{2} {\tilde{E}}^{'}$ , that is $(Δ + k^{2}) {\tilde{E}}^{'} = 0$ on $D$ . The field ${\tilde{E}}^{'}$ is $C^{\infty}$ on $\overline{D}$ , has tangential trace $e_{T}$ , divergence zero and the same modal outgoing condition on $S_{R}$ ; all hypotheses of the uniqueness clause of [F1] are therefore satisfied, and ${\tilde{E}}^{'} = {\tilde{E}}_{s c a t}$ on $D$ . Then ${\tilde{B}}^{'} = (i ω)^{- 1} curl {\tilde{E}}^{'} = (i ω)^{- 1} curl {\tilde{E}}_{s c a t} = {\tilde{B}}_{s c a t}$ on $D$ by the third phasor equation and the definition of ${\tilde{B}}_{s c a t}$ .

3.1step 2.1F2F6algebra

[The total pair is source-free] On $D$ the incident pair satisfies the four equations of [F2] because $D \subseteq V$ and the pair is source-free on $V$ ; the scattered pair satisfies them by step 2.1; the totals ${\tilde{E}}_{t o t} = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ and ${\tilde{B}}_{t o t} = {\tilde{B}}_{i n c} + {\tilde{B}}_{s c a t}$ are $C^{\infty}$ on $D$ as sums, and each of the four equations is linear over the complex scalars by [F6] applied to real and imaginary parts, so they hold for the totals; hence $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ is as well a frequency- $ω$ source-free vacuum phasor pair on $D$ .

4.1step 2.1step 3.1F3F4

[Real-field Maxwell statement] Let $(\tilde{E}, \tilde{B})$ be any $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair on an open $O^{'} \subseteq R^{3}$ . By [F3] its associated real fields satisfy the real system of [F3] on $O^{'} \times R$ , and by [F4] that system with zero sources is exactly the smooth branch of the adopted vacuum Maxwell equations at $ρ \equiv 0$ , $J \equiv 0$ ; the required regularity is the joint $C^{1}$ (indeed $C^{\infty}$ ) calculus supplied by [F3]. Applying this to $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ on $O^{'} = D$ , which is admissible by step 3.1, gives the asserted source-free equations for $E_{t o t}, B_{t o t}$ on $D \times R$ . This proves the two statements of (a), the scattered and total clauses following from steps 2.1 and 3.1.

4.2step 3.1F2F7algebra

[PEC boundary condition] On $\partial O$ one has ${\tilde{E}}_{t o t, t a n g} = {\tilde{E}}_{i n c, t a n g} + {\tilde{E}}_{s c a t, t a n g} = {\tilde{E}}_{i n c, t a n g} + e_{T} = 0$ , because the tangential projection is the real-linear map of [F7] and $e_{T}$ is the tangential projection of $- {\tilde{E}}_{i n c}$ ; thus the tangential trace of ${\tilde{E}}_{t o t}$ vanishes. For the associated real field, the tangential projection of $E_{t o t} (x, t) = Re ({\tilde{E}}_{t o t} (x) e^{- i ω t})$ is $Re ({\tilde{E}}_{t o t, t a n g} (x) e^{- i ω t}) = 0$ by the real-linearity of the projection [F7] and of $Re$ ; hence $n \times E_{t o t} = n \times E_{t o t, t a n g} = 0$ on $\partial O \times R$ by [F7]. This is (b).

5.1step 3.1step 4.2step 2.2F6algebra

[Equivalent total formulation] Write a candidate total pair as $({\tilde{E}}_{i n c} + {\tilde{E}}^{''}, {\tilde{B}}_{i n c} + {\tilde{B}}^{''})$ on $D$ , with both fields smooth up to its boundary, where $({\tilde{E}}^{''}, {\tilde{B}}^{''})$ is the scattered correction, required to be a $C^{\infty}$ frequency- $ω$ source-free vacuum phasor pair and outgoing on $S_{R}$ in the modal sense, with the total tangential electric trace vanishing. By linearity [F6] the total tangential trace is ${\tilde{E}}_{i n c, t a n g} + {\tilde{E}}_{t a n g}^{''}$ , so the vanishing of the total trace is equivalent to ${\tilde{E}}_{t a n g}^{''} = e_{T}$ ; by step 2.2 applied to $({\tilde{E}}^{''}, {\tilde{B}}^{''})$ , the correction equals $({\tilde{E}}_{s c a t}, {\tilde{B}}_{s c a t})$ on $D$ , hence the total equals $({\tilde{E}}_{t o t}, {\tilde{B}}_{t o t})$ ; conversely the constructed total has these properties by steps 3.1, 4.2 and the outgoing property of ${\tilde{E}}_{s c a t}$ from step 1.1. This is the equivalent formulation of (c).

6.1step 4.2step 2.2step 5.1F5F8F9given∎

[Scope and choice accounting] Clause (d) records the exact hypotheses: one positive frequency, one smooth bounded stationary PEC obstacle with connected exterior, and $C^{\infty}$ incident and scattered classes; no lossy, impedance, nonsmooth or dispersive obstacle, no induced surface charge or current, no radiation-zone asymptotics beyond the modal class, no singular source and no empirical claim enters the argument. The PEC condition is adopted as an idealization by [F8] and is not derived from the material closures; the total field is not asserted to be outgoing, as an incident plane wave generally is not, and the uniqueness statement is exactly about the total-minus-incident correction. The plane-wave instance clause of the Statement is the specialization of the equivalence of [F5] to $∣ κ ∣ = ω / c$ and $b = κ \times e / ω$ , so it introduces no commitment beyond that cited statement. The only choice principle used is the Axiom of Choice assumed in the Statement through [F1], with the accounting of [F9]; all remaining steps are pointwise algebraic identities and finite sums.

Remarks

The phrase "outgoing" applies to the scattered correction only. The constructed total field ${\tilde{E}}_{t o t} = {\tilde{E}}_{i n c} + {\tilde{E}}_{s c a t}$ is source-free on $D$ but need not be outgoing, and no decay or far-field asymptotic beyond the lemma's modal class is claimed.
The ideal PEC condition is a boundary model, not a theorem about any particular conductor; the induced surface charge and current that would accompany it in a microscopic model are neither prescribed nor derived here.

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Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds

Physical scope

Conditional cross sections of the adopted ideal harmonic dipole model in a plane wave: the directional and total model cross sections in SI constants, the scalar-polarizability branch, and the explicit fidelity bounds comparing the model with an actual compact smooth source. Excluded: measured cross sections, frequency-domain response laws, magnetic-dipole and higher-multipole content, self-force and radiation reaction, material response, finite-size and every empirical claim.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ) as carried by Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source and Zonal and quadratic sphere moments for the polar surface measure on $S^{2}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the period-average and cross-section algebra itself uses no choice. Adopt the ideal harmonic electric-dipole radiation model of The ideal harmonic electric-dipole radiation model and its adopted status with frequency $ω > 0$ , wavenumber $k = ω / c$ and amplitude $p_{0} \in C^{3}$ ; let $E_{d}, B_{d}, S_{d}$ be its fields and model flux. Let $κ \in R^{3} ∖ {0}$ with $∣ κ ∣ = ω / c$ , $e \in C^{3} ∖ {0}$ with $κ \cdot e = 0$ , $b : = κ \times e / ω$ (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and let $(E_{i n c}, B_{i n c})$ be the corresponding plane-wave fields of Real plane-wave fields encoded by complex amplitudes on $R^{3} \times R$ ; by Dispersion, transversality and the two polarization conditions for plane-wave solutions they satisfy the smooth-branch source-free vacuum Maxwell equations, and by Period-averaged energy density, Poynting vector and momentum flux of a plane wave(b) their period-averaged energy flux has magnitude $I_{i n c} = \frac{ε_{0} c}{2} ∣ e ∣^{2} = \frac{∣ e ∣^{2}}{2 μ_{0} c}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values). Define the model differential scattering cross section by

$\frac{d σ}{d Ω} (n) : = \frac{r^{2} ⟨ S_{d} (r n, \cdot) \cdot n ⟩}{⟨ S_{i n c} \cdot \hat{κ} ⟩}, n \in S^{2}, r > 0,$

where $\hat{κ} : = κ / ∣ κ ∣$ , $S_{i n c} : = E_{i n c} \times B_{i n c} / μ_{0}$ (Vacuum electromagnetic energy density and energy flux (the Poynting vector)), and $⟨ f ⟩ : = \frac{1}{T} \int_{0}^{T} f (s) d s$ , $T = 2 π / ω$ , is the period average at fixed observation point. Then: (a) [Angular pattern] the ratio is independent of $r$ and of the averaging start, and for every $n \in S^{2}$

$\frac{d σ}{d Ω} (n) = \frac{ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ε_{0}^{2} c^{4} ∣ e ∣^{2}} = \frac{μ_{0}^{2} ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ∣ e ∣^{2}} .$

(b) [Total cross section] with $d σ_{S^{2}}$ the surface measure of the unit sphere, $σ : = \int_{S^{2}} \frac{d σ}{d Ω} d σ_{S^{2}}$ equals

$σ = \frac{ω^{4} ∣ p_{0} ∣^{2}}{6 π ε_{0}^{2} c^{4} ∣ e ∣^{2}} = \frac{μ_{0}^{2} ω^{4} ∣ p_{0} ∣^{2}}{6 π ∣ e ∣^{2}};$

if $p_{0} = α e$ for a complex scalar $α$ (the scalar polarizability, in $F m^{2}$ ), then $σ = \frac{ω^{4} ∣ α ∣^{2}}{6 π ε_{0}^{2} c^{4}}$ and $\frac{d σ}{d Ω} (n) = \frac{ω^{4} ∣ α ∣^{2}}{16 π^{2} ε_{0}^{2} c^{4}} ∣ n \times \hat{e} ∣^{2}$ with $\hat{e} : = e / ∣ e ∣$ . (c) [Fidelity for arbitrary actual sources and matched windows] Let an actual compact smooth conserved source satisfy the exact hypotheses of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source, with spatial support radius $a > 0$ , source-time interval $[τ_{0}, τ_{1}]$ , moment $p_{a c t} (u)$ , variation bound $D (u)$ and constants $Γ_{E}, Γ_{B}$ . These source data are independent of the prescribed model amplitude $p_{0}$ . Put

$p_{h} (u) : = Re (p_{0} e^{- i ω u}), M (u) : = ∣ p_{a c t}^{''} (u) - p_{h}^{''} (u) ∣ .$

For $n \in S^{2}$ , $u \in [τ_{0}, τ_{1}]$ , $r \geq 2 a$ , define

$H (u, r) : = \frac{Γ_{E}}{r^{2}} + \frac{μ_{0}}{4 π r} [D (u) + M (u)] .$

The actual retarded fields and the model fields satisfy, uniformly in $n$ ,

$∣ E_{a c t} (r n, u + r / c) - E_{d} (r n, u + r / c) ∣ \leq H (u, r),$

$∣ B_{a c t} (r n, u + r / c) - B_{d} (r n, u + r / c) ∣ \leq \frac{Γ_{B} + Γ_{E} / c}{r^{2}} + \frac{H (u, r)}{c} .$

Also retain the sharper direct source-leading-field consequence

$∣ B_{a c t} - B_{d} ∣ \leq \frac{Γ_{B}}{r^{2}} + \frac{μ_{0} [D (u) + M (u)]}{4 π c r}$

at the same events. If $p_{a c t}^{''} = p_{h}^{''}$ on a specified finite window, $M \equiv 0$ there and these give the full matched dipole bounds; in particular they imply the coarser electric bound $Γ_{E} / r^{2} + μ_{0} D / (2 π r)$ . Neither moment matching nor $D \equiv 0$ is automatic. A sound sufficient matched branch is $p_{a c t} = p_{h} + a_{0} + b_{0} u$ on an open neighbourhood of the window (any fixed real vectors $a_{0}, b_{0}$ ). Such a branch has actual compact realizations over any finite window: choose a smooth time cutoff $η$ equal to $1$ near the window, let $p_{η} : = η (p_{h} + a_{0} + b_{0} u)$ , choose a nonnegative smooth spatial bump $σ$ with support inside $B_{a}$ and integral $1$ , and set $ρ : = - p_{η} \cdot \nabla σ$ , $J : = p_{η}^{'} σ$ . Then continuity holds and the actual moment is $p_{η}$ ; outside the window it is tapered. Apply the actual-source theorem with the resulting $D$ and $Γ$ , retaining every finite-radius and source-variation error. No eternal harmonic moment of a compact-time source is asserted. (d) [Scope] conditional predictions of the adopted ideal model: no measured cross section, no frequency-domain response law (the amplitude $p_{0}$ is prescribed model data), no magnetic-dipole or higher-multipole content, no self-force, material response, finite-size or empirical claim; the scalar- $α$ clause records a declared proportional response, not a derived polarizability. In all complex-amplitude formulas $∣ v ∣^{2} = \sum_{i} ∣ v_{i} ∣^{2}$ is the Hermitian norm square; $n \times v$ is complex-linear for real $n$ . The $p_{0} = 0$ case gives zero scattered fields and cross sections while the nonzero incident intensity still defines the ratios. Actual-source fidelity assumes only the explicitly quantified mismatch and source hypotheses; it does not turn the model cross section into a measured or self-consistent scattering prediction.

Facts & Assumptions

Given: The model data of The ideal harmonic electric-dipole radiation model and its adopted status: $ω > 0$ , $k = ω / c$ , $p_{0} \in C^{3}$ , the model fields $E_{d}, B_{d}$ and flux $S_{d}$ on $(R^{3} ∖ {0}) \times R$ ; the plane-wave data $κ$ , $e \neq 0$ with $κ \cdot e = 0$ , $b = κ \times e / ω$ , and the fields $E_{i n c}, B_{i n c}$ , $S_{i n c}$ ; the constants $c, ε_{0}, μ_{0}$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values with $ε_{0} μ_{0} c^{2} = 1$ ; the period $T = 2 π / ω$ and the cross section of the Statement.

[F1]

Model interface: $E_{d} (x, t) = Re (w (x) e^{- i ω t})$ with $w (x) : = - \frac{μ_{0} ω^{2}}{4 π r} n \times (n \times p_{0}) e^{i k r}$ , $r = ∣ x ∣$ , $n = x / r$ ; $B_{d} = n \times E_{d} / c$ ; $S_{d} = E_{d} \times B_{d} / μ_{0}$ is the flux of Vacuum electromagnetic energy density and energy flux (the Poynting vector) evaluated at the model fields (The ideal harmonic electric-dipole radiation model and its adopted status).

[F2]

Incident plane wave: $(E_{i n c}, B_{i n c})$ are the plane-wave fields of Real plane-wave fields encoded by complex amplitudes with amplitude $e$ and $b = κ \times e / ω$ , $∣ κ ∣ = ω / c$ , satisfying the smooth source-free vacuum Maxwell equations by Dispersion, transversality and the two polarization conditions for plane-wave solutions; their period-averaged flux satisfies $⟨ S_{i n c} ⟩ = I_{i n c} \hat{κ}$ with $I_{i n c} = \frac{ε_{0} c}{2} ∣ e ∣^{2} = \frac{∣ e ∣^{2}}{2 μ_{0} c}$ (Period-averaged energy density, Poynting vector and momentum flux of a plane wave, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F3]

Period averages: for $a, d \in C$ , $\frac{1}{T} \int_{0}^{T} Re (a e^{- i ω t}) Re (d e^{- i ω t}) d t = \frac{1}{2} Re (a \overline{d})$ ; in particular the mean of the real square is $\frac{1}{2} ∣ a ∣^{2}$ (Period averages of products of real parts of complex exponentials, Real and imaginary parts, complex conjugation, and modulus).

[F4]

Cross product and norm identities: the cross product is bilinear and alternating; $∣ n \times (n \times v) ∣ = ∣ n \times v ∣ \leq ∣ v ∣$ for $∣ n ∣ = 1$ ; $n \times (u + i v) = n \times u + i n \times v$ for real $n$ ; $∣ n \times v ∣^{2} = \sum_{i} ∣ (n \times v)_{i} ∣^{2}$ is the Hermitian norm square (The cross product is bilinear, alternating, and orthogonal to both factors, The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Real and imaginary parts, complex conjugation, and modulus).

[F5]

Sphere moments: for the surface measure $σ_{S^{2}}$ of the unit sphere, $σ_{S^{2}} (S^{2}) = 4 π$ , $\int_{S^{2}} n d σ_{S^{2}} = 0$ , $\int_{S^{2}} n_{i} n_{j} d σ_{S^{2}} = (4 π / 3) δ_{i j}$ , and for every $a \in R^{3}$ , $\int_{S^{2}} (n \cdot a)^{2} d σ_{S^{2}} = \frac{4 π}{3} ∣ a ∣^{2}$ and $\int_{S^{2}} ∣ n \times a ∣^{2} d σ_{S^{2}} = \frac{8 π}{3} ∣ a ∣^{2}$ (Zonal and quadratic sphere moments for the polar surface measure on $S^{2}$ ).

[F6]

Actual-source theorem: under the hypotheses of Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source (jointly $C^{\infty}$ conserved sources vanishing outside ${\overline{B}}_{a} (0) \times [τ_{0}, τ_{1}]$ , $u \in [τ_{0}, τ_{1}]$ ), for $n \in S^{2}$ , $r \geq 2 a$ , $∣ E_{a c t} (r n, u + r / c) - \frac{μ_{0}}{4 π r} n \times (n \times p_{a c t}^{''} (u)) ∣ \leq \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D (u)}{4 π r}$ and $∣ B_{a c t} (r n, u + r / c) - \frac{n \times E_{a c t} (r n, u + r / c)}{c} ∣ \leq \frac{Γ_{B} + Γ_{E} / c}{r^{2}}$ , and the sharper bound $∣ B_{a c t} (r n, u + r / c) - \frac{n \times E_{l e a d} (n, u, r)}{c} ∣ \leq \frac{Γ_{B}}{r^{2}}$ with $E_{l e a d} = \frac{μ_{0}}{4 π r} n \times (n \times V_{a c t})$ ; the constants $Γ_{E}, Γ_{B}$ depend only on $a, c, μ_{0}$ and the source suprema and the theorem assumes ${A C}_{ω}$ (Controlled electric-dipole far field and retarded-time radiated power of a compact smooth source, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F7]

Window construction toolbox: smooth compactly supported cutoffs and localized bumps exist with the properties used below (Test function cutoffs and euclidean localization, The mollifier family generated by a unit-mass smooth bump); integration by parts holds on a bounding rectangle (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections), and the Riemann-Lebesgue comparison identifies the Lebesgue and Riemann integrals of the continuous integrands used there (Riemann–Lebesgue comparison for distribution test integrands); the first fundamental theorem gives the derivative of an integral over a moving endpoint (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive), and a continuous function on an interval with zero derivative is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant). Rectangles, cubes and their volumes are those of Axis-parallel rectangles in $R^{m}$ and their volume; continuous real functions on nondegenerate closed rectangles are Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable). The integral is monotone (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ), so a nonnegative continuous cutoff equal to $1$ near a nonempty cube has a strictly positive finite integral; dividing by that integral normalizes the spatial bump.

Proof

technique · direct; compute the model flux pointwise from the double cross product, evaluate its period average with the period-average lemma, divide by the incident flux, integrate the pattern against the sphere moments for the total, and obtain the actual-source fidelity bounds by adding the dipole theorem's errors to the moment mismatch through the norm inequalities of the cross product

1.1givenF1F4algebra

[Model flux along the observation direction] Fix $n \in S^{2}$ and $r > 0$ . By [F1], $B_{d} = n \times E_{d} / c$ and $n \cdot E_{d} = 0$ , since $n \times (n \times p_{0})$ is orthogonal to $n$ ; hence with the vector identity $E_{d} \times (n \times E_{d}) = n ∣ E_{d} ∣^{2} - E_{d} (n \cdot E_{d})$ of [F4], $S_{d} \cdot n = \frac{1}{μ_{0}} (E_{d} \times B_{d}) \cdot n = \frac{1}{μ_{0} c} (E_{d} \times (n \times E_{d})) \cdot n = \frac{∣ E_{d} ∣^{2}}{μ_{0} c}$ , because $n \cdot n = 1$ .

1.2givenF1F4algebra

[The model field is the leading field of the harmonic waveform] For $u = t - r / c$ one has $e^{i (k r - ω t)} = e^{- i ω u}$ ; since $p_{h}^{''} (u) = Re (- ω^{2} p_{0} e^{- i ω u})$ by [F1] of the harmonic lemma as recorded in The ideal harmonic electric-dipole radiation model and its adopted status, and $n \times (n \times \cdot)$ is real-linear by [F4], $E_{d} (r n, u + r / c) = \frac{μ_{0}}{4 π r} n \times (n \times p_{h}^{''} (u))$ and $B_{d} (r n, u + r / c) = \frac{1}{c} n \times E_{d} (r n, u + r / c) = - \frac{μ_{0}}{4 π c r} n \times p_{h}^{''} (u)$ , the last identity by the double-cross computation of [F4].

2.1step 1.1F1F3F4algebra

[Period average and the averaged radial power] At fixed $x = r n$ the amplitude $w = w (x)$ of [F1] is independent of $t$ , and $E_{d} (r n, t) = Re (w e^{- i ω t})$ ; by [F3] with $a = d = w$ , $⟨ ∣ E_{d} ∣^{2} ⟩ = \frac{1}{2} ∣ w ∣^{2}$ . Evaluating $∣ w ∣^{2}$ with [F4], $∣ w ∣^{2} = (\frac{μ_{0} ω^{2}}{4 π r})^{2} ∣ n \times (n \times p_{0}) ∣^{2} = \frac{μ_{0}^{2} ω^{4}}{16 π^{2} r^{2}} ∣ n \times p_{0} ∣^{2}$ , because $∣ n \times (n \times p_{0}) ∣ = ∣ n \times p_{0} ∣$ ; therefore $r^{2} ⟨ S_{d} \cdot n ⟩ = \frac{r^{2}}{μ_{0} c} ⟨ ∣ E_{d} ∣^{2} ⟩ = \frac{μ_{0} ω^{4}}{32 π^{2} c} ∣ n \times p_{0} ∣^{2}$ , independent of $r$ .

2.2step 1.2F4F6algebra

[Electric and magnetic fidelity bounds] Let $p_{a c t}$ and $D, Γ_{E}, Γ_{B}$ be the actual-source data of [F6]. Subtracting the two leading fields, $E_{a c t} (r n, u + r / c) - E_{d} (r n, u + r / c) = [E_{a c t} - \frac{μ_{0}}{4 π r} n \times (n \times p_{a c t}^{''})] + \frac{μ_{0}}{4 π r} n \times (n \times (p_{a c t}^{''} - p_{h}^{''}))$ , so taking norms and using $∣ n \times (n \times w) ∣ \leq ∣ w ∣$ of [F4] and $∣ p_{a c t}^{''} - p_{h}^{''} ∣ = M (u)$ , the first bound of [F6] gives $∣ E_{a c t} - E_{d} ∣ \leq \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D (u)}{4 π r} + \frac{μ_{0} M (u)}{4 π r} = H (u, r)$ , uniformly in $n$ at fixed $u$ . For the magnetic field, $∣ B_{a c t} - B_{d} ∣ \leq ∣ B_{a c t} - \frac{1}{c} n \times E_{a c t} ∣ + \frac{1}{c} ∣ n \times (E_{a c t} - E_{d}) ∣ \leq \frac{Γ_{B} + Γ_{E} / c}{r^{2}} + \frac{H (u, r)}{c}$ by the second bound of [F6], [F4] and the electric bound just proved. For the sharper direct estimate, [F6] gives $∣ B_{a c t} - \frac{1}{c} n \times E_{l e a d} (V_{a c t}) ∣ \leq \frac{Γ_{B}}{r^{2}}$ with $E_{l e a d} = \frac{μ_{0}}{4 π r} n \times (n \times V_{a c t})$ , and by the double-cross computation of [F4] applied to $V_{a c t}$ and $p_{h}^{''}$ , $\frac{1}{c} n \times E_{l e a d} (V_{a c t}) - B_{d} = - \frac{μ_{0}}{4 π c r} n \times (V_{a c t} - p_{h}^{''})$ ; hence $∣ B_{a c t} - B_{d} ∣ \leq \frac{Γ_{B}}{r^{2}} + \frac{μ_{0} (D (u) + M (u))}{4 π c r}$ by the triangle inequality and $∣ V_{a c t} - p_{h}^{''} ∣ \leq D + M$ from the replacement bound of [F6]. When $M \equiv 0$ on the window the bounds reduce to $H = \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D}{4 π r} \leq \frac{Γ_{E}}{r^{2}} + \frac{μ_{0} D}{2 π r}$ , and neither $M \equiv 0$ nor $D \equiv 0$ is automatic.

3.1step 2.1F1F7algebra

[Independence of the averaging start] For any real $s$ , define $G (s) : = \int_{s}^{s + T} ∣ E_{d} (r n, t) ∣^{2} d t$ . By [F7] the first fundamental theorem gives $G^{'} (s) = ∣ E_{d} (r n, s + T) ∣^{2} - ∣ E_{d} (r n, s) ∣^{2} = 0$ , because the exponential encoding of [F1] and the addition law give $e^{- i ω (t + T)} = e^{- i ω t} e^{- 2 π i} = e^{- i ω t}$ , so $t \mapsto E_{d} (r n, t)$ is $T$ -periodic; hence $G$ is constant by the zero-derivative corollary of [F7] and the mean $\frac{1}{T} \int_{s}^{s + T} ∣ E_{d} ∣^{2} = \frac{1}{T} \int_{0}^{T} ∣ E_{d} ∣^{2}$ is independent of the start, so the cross-section ratio of step 2.1 is as well.

3.2step 2.2F6F7algebra

[Matched compact-window realization] Let the window be a compact interval $W$ in the source-time line and let $a_{0}, b_{0} \in R^{3}$ . Choose a real $C^{\infty}$ cutoff $η$ in time, equal to $1$ on a neighbourhood of $W$ , supported in a compact interval $(t_{0}, t_{1})$ strictly containing $W$ , as supplied by [F7], and set $p_{η} : = η (p_{h} + a_{0} + b_{0} u)$ . Choose a nonempty closed cube $Q \subset B_{a}$ and a cutoff $χ \in C_{c}^{\infty} (B_{a})$ with $0 \leq χ \leq 1$ and $χ = 1$ near $Q$ , by [F7]. On a bounding rectangle, Fubini and integral monotonicity give $0 < vol (Q) \leq C : = \int χ < \infty$ ; set $σ : = χ / C$ , which is nonnegative, smooth, supported in $B_{a}$ and has $\int σ = 1$ and set $ρ (y, u) : = - p_{η} (u) \cdot \nabla σ (y)$ , $J (y, u) : = p_{η}^{'} (u) σ (y)$ ; both are $C^{\infty}$ and vanish outside ${\overline{B}}_{a} (0) \times [t_{0}, t_{1}]$ , and $\partial_{u} ρ + div J = - p_{η}^{'} \cdot \nabla σ + p_{η}^{'} \cdot \nabla σ = 0$ , so the conservation equation holds. Integrating by parts on a rectangle containing $B_{a}$ and using Fubini as in [F7], $\int_{B_{a}} ρ (y, u) d y = - p_{η} (u) \cdot \int_{B_{a}} \nabla σ = 0$ , so the total charge vanishes, and $\int_{B_{a}} y_{i} ρ (y, u) d y = - \sum_{j} p_{η, j} (u) \int_{B_{a}} y_{i} \partial_{j} σ (y) d y = p_{η, i} (u)$ because $\int_{B_{a}} y_{i} \partial_{j} σ = - δ_{i j} \int_{B_{a}} σ = - δ_{i j}$ by the same integration by parts; hence the actual moment of the source is $p_{a c t} = p_{η}$ . On $W$ one has $η \equiv 1$ , so $p_{a c t}^{''} = p_{h}^{''}$ there and the mismatch $M$ vanishes on the window; applying the fidelity bounds of step 2.2 to this source preserves every finite-radius and source-variation error term $D, Γ_{E}, Γ_{B}$ computed from it, and outside the window the moment is tapered, so no eternal harmonic moment is asserted.

4.1step 2.1step 3.1F2algebra

[Angular pattern] Dividing the averaged radial power of step 2.1 by $I_{i n c} = \frac{∣ e ∣^{2}}{2 μ_{0} c}$ of [F2] gives $\frac{d σ}{d Ω} (n) = \frac{μ_{0}^{2} ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ∣ e ∣^{2}}$ , and substituting $μ_{0}^{2} = \frac{1}{ε_{0}^{2} c^{4}}$ from Framework-local SI units, the vacuum constants and the exact/measured status of their values gives the displayed second form $\frac{ω^{4} ∣ n \times p_{0} ∣^{2}}{16 π^{2} ε_{0}^{2} c^{4} ∣ e ∣^{2}}$ ; with step 3.1 this is clause (a).

5.1step 4.1F4F5algebra

[Total cross section] Write $p_{0} = u + i v$ with $u, v \in R^{3}$ ; by [F4], $∣ n \times p_{0} ∣^{2} = ∣ n \times u ∣^{2} + ∣ n \times v ∣^{2}$ for every $n$ , so integrating the pattern of step 4.1 over $S^{2}$ and using $\int_{S^{2}} ∣ n \times a ∣^{2} d σ_{S^{2}} = \frac{8 π}{3} ∣ a ∣^{2}$ of [F5] for the real vectors $u$ and $v$ gives $\int_{S^{2}} ∣ n \times p_{0} ∣^{2} d σ_{S^{2}} = \frac{8 π}{3} (∣ u ∣^{2} + ∣ v ∣^{2}) = \frac{8 π}{3} ∣ p_{0} ∣^{2}$ ; hence $σ = \frac{μ_{0}^{2} ω^{4}}{16 π^{2} ∣ e ∣^{2}} \cdot \frac{8 π}{3} ∣ p_{0} ∣^{2} = \frac{μ_{0}^{2} ω^{4} ∣ p_{0} ∣^{2}}{6 π ∣ e ∣^{2}}$ , and the $ε_{0}$ form follows from $μ_{0}^{2} = \frac{1}{ε_{0}^{2} c^{4}}$ . For the scalar branch $p_{0} = α e$ , [F4] gives $∣ n \times p_{0} ∣^{2} = ∣ α ∣^{2} ∣ n \times e ∣^{2} = ∣ α ∣^{2} ∣ e ∣^{2} ∣ n \times \hat{e} ∣^{2}$ and $∣ p_{0} ∣^{2} = ∣ α ∣^{2} ∣ e ∣^{2}$ , so both displayed substitutions hold; this is clause (b).

6.1step 4.1step 5.1step 2.2step 3.2given∎

[Scope and choice accounting] Clause (d) is the exact scope: the cross sections are conditional predictions of the adopted model, $p_{0}$ is prescribed data rather than a derived response, and no measured, material, multipole, self-force or finite-size claim is made; the $p_{0} = 0$ case gives zero model fields and zero cross sections while the nonzero $I_{i n c}$ still normalizes the ratios. The only choice principle used is the Countable Choice carried by the actual-source theorem and the sphere-moment interface of [F5], both invoked in the finite-window and integration steps; the period averages, the double-cross algebra, the division by the incident flux and the integration of the pattern use only finite sums and the explicit suppliers [F3], [F4], so no further choice is made.

Remarks

The incident intensity used is the framework's own $I_{i n c} = ε_{0} c ∣ e ∣^{2} / 2$ from Period-averaged energy density, Poynting vector and momentum flux of a plane wave(b); no source's printed flux normalisation is used, and the two displayed $ε_{0}$ and $μ_{0}$ forms of the cross sections are related by $ε_{0} μ_{0} c^{2} = 1$ .
The fidelity clause compares the model with actual compact smooth sources through the explicit $D, Γ$ bounds plus the moment mismatch $M$ ; it is not a claim that an actual scatterer's measured cross section equals the model value, and the matched-window branch only shows that the mismatch term is genuinely removable for a realized source class.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Controlled weak-field first variation of the relativistic force law and the leading Thomson-model cross section

Physical scope

Conditional weak-field (small-amplitude) first variation of the adopted relativistic external-force law for a test charge in an ideal plane wave: the linearized periodic displacement, the induced harmonic dipole amplitude $p_{0} = - \frac{q^{2}}{m ω^{2}} e e^{i κ \cdot x_{0}}$ , and the resulting leading Thomson-model cross section $\frac{q^{4}}{6 π ε_{0}^{2} m^{2} c^{4}}$ independent of frequency. Excluded: exact Newtonian force laws, existence, persistence, stability or self-consistency of the weak-field family, self-force and radiation reaction, material media, quantum and atomic-resonance content, and every measured, statistical or empirical claim. The nonlinear family remains on any common open $I$ containing $0$ ; only its harmonic first-order ODE response is globally extended and periodically prepared. $x_{0}$ is a zeroth-order rest/reference position, with arbitrary first-order response mean $a$ ; no exact nonlinear-family mean or periodicity is claimed. All zero-charge and complex-polarization cases remain.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ) as carried by Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds; the linearization below and Periodic responses of $\ddot{z} = Re (w e^{- i ω t})$ and the affine classification of all solutions use no choice. Assume the setting and postulates of The four-force equation reproduces the coordinate Lorentz force and power for a test particle with charge parameter $q \in R$ (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current(a)) and externally supplied rest mass $m > 0$ , in the incident plane-wave field of Real plane-wave fields encoded by complex amplitudes with wavevector $κ$ , complex amplitude $e \neq 0$ , $κ \cdot e = 0$ , $∣ κ ∣ = ω / c$ and $b = κ \times e / ω$ as in Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds; write $E_{i n c} (x, t) = Re (e e^{i (κ \cdot x - ω t)})$ and $B_{i n c} = κ \times E_{i n c} / ω$ (Real and imaginary parts, complex conjugation, and modulus, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values). Assume that for some $η_{0} > 0$ and a common open interval $I ∋ 0$ , for every $η \in (- η_{0}, η_{0})$ there is a $C^{2}$ future timelike massive solution $r_{η} : I \to R^{3}$ of the adopted four-force law in the external field $η (E_{i n c}, B_{i n c})$ (no self-field, recoil or material force), and that the family admits a uniform first-order expansion: there are $r_{1} \in C^{2} (I; R^{3})$ and, for every compact $K \subset I$ , a constant $C_{K}$ with

$∣ r_{η} (t) - x_{0} - η r_{1} (t) ∣ + ∣ {\dot{r}}_{η} (t) - η {\dot{r}}_{1} (t) ∣ \leq C_{K} η^{2} (t \in K, ∣ η ∣ < η_{0})$

about a fixed position $x_{0} \in R^{3}$ . The family time interval $I$ is not required to contain a full period, and no solution $r_{η}$ outside $I$ is assumed. For the periodic-response conclusions (b), (c), impose the following explicit preparation (P): the unique global harmonic-ODE extension of $r_{1}$ supplied after (a) by Periodic responses of $\ddot{z} = Re (w e^{- i ω t})$ and the affine classification of all solutions(d), with its actual value and derivative at $0$ , is $T$ -periodic, $T = 2 π / ω$ . Equivalently its affine drift is zero, or $r_{1}^{'} (0) = Re (i w / ω)$ with $w = (q / m) e e^{i κ \cdot x_{0}}$ . The linearized equation and affine local/global classification in (a) hold for every stated family even without (P); overlap-only periodicity on a short $I$ is not substituted for (P). Then: (a) [Linearized equation] ${\ddot{r}}_{1} (t) = \frac{q}{m} E_{i n c} (x_{0}, t) = \frac{q}{m} Re (e e^{i (κ \cdot x_{0} - ω t)})$ on $I$ . (b) [Induced dipole under preparation (P)] $x_{0}$ is the zeroth-order rest/reference position. Let $a$ be the mean of the global periodic first-order response, so on $I$ one has $r_{1} = z_{p} + a$ with $z_{p} (t) = Re (- w ω^{- 2} e^{- i ω t})$ . The arbitrary constant $a$ is retained: no exact mean or periodicity of the full nonlinear family is asserted. The oscillating part of the first-order dipole coefficient $q (r_{1} - a)$ is $Re (p_{0} e^{- i ω t})$ with complex amplitude

$p_{0} = - \frac{q^{2}}{m ω^{2}} e e^{i κ \cdot x_{0}};$

equivalently $p_{0} = α e^{'}$ with $α = - \frac{q^{2}}{m ω^{2}}$ (units $F m^{2}$ ) and the unimodular factor $e^{'} = e e^{i κ \cdot x_{0}}$ , $∣ e^{'} ∣ = ∣ e ∣$ . (c) [Leading Thomson-model cross section] substituting this $p_{0}$ into Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds(b) gives

$σ_{T} = \frac{q^{4}}{6 π ε_{0}^{2} m^{2} c^{4}} = \frac{8 π}{3} (\frac{q^{2}}{4 π ε_{0} m c^{2}})^{2},$

independent of $ω$ and of the phase factor. (d) [Scope] this is a controlled conditional first variation of the adopted relativistic external-force law under the stated solution-family hypotheses. It is not the exact Newtonian force postulate, it does not assert existence, persistence, stability or self-consistency of the family, it excludes self-force, radiation reaction and material media, and it makes no empirical, measurement, atomic-physics or frequency-response claim. In (a), every local $r_{1}$ has the unique global harmonic extension $z_{p} + a + b t$ , with $b = r_{1}^{'} (0) - z_{p}^{'} (0)$ , $a = r_{1} (0) - z_{p} (0)$ . Thus the general affine-drift response is also retained; preparation (P) sets $b = 0$ for (b), (c) and leaves $a$ arbitrary. For a nonzero $η$ , the associated leading ideal harmonic model has incident amplitude $η e$ and induced dipole amplitude $η p_{0}$ . Their squared flux factors $η^{2}$ cancel in its cross-section ratio; $σ_{T}$ is this leading-model coefficient, not an assertion of the exact nonlinear particle's emitted field or cross section. At $q = 0$ the induced amplitude and $σ_{T}$ are zero without any division by $q$ . All $η$ estimates apply only to the open parameter domain on which the family was declared.

Facts & Assumptions

Given: The setting, family and preparation (P) of the Statement: charge $q \in R$ and mass $m > 0$ (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, Framework-local SI units, the vacuum constants and the exact/measured status of their values); the plane-wave data $κ, e, ω$ with $∣ κ ∣ = ω / c$ , $κ \cdot e = 0$ , $b = κ \times e / ω$ ; the family $r_{η}$ on the common open interval $I ∋ 0$ with the uniform first-order expansion about $x_{0}$ ; the first-order function $r_{1} \in C^{2} (I; R^{3})$ ; the harmonic data $w = (q / m) e e^{i κ \cdot x_{0}}$ and $z_{p} (t) = Re (- w ω^{- 2} e^{- i ω t})$ .

[F1]

Adopted force law (external supplier, whose carrier is the referenced in-run item): under its postulates, with the coordinate reading $r$ , velocity $v = r^{'}$ and $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ of a future timelike $C^{2}$ massive worldline in a prescribed exterior field $(E, B)$ , the coordinate momentum $p = m γ v$ satisfies $\frac{d p}{d t} = q (E + v \times B)$ and $\frac{d (γ m c^{2})}{d t} = q E \cdot v$ ; the norm $η (U, U)$ is constant and a solution with future timelike data remains future timelike; and for every $δ \in (0, 1)$ , wherever $∣ v ∣ / c \leq δ$ , $∣ γ - 1 ∣ \leq \frac{∣ v ∣^{2} / c^{2}}{2 (1 - δ^{2})^{3 / 2}}$ and $∣ p - m v ∣ \leq m ∣ v ∣ \frac{∣ v ∣^{2}}{2 c^{2} (1 - δ^{2})^{3 / 2}}$ , with constants depending only on $δ$ ; the statement is conditional on the adopted postulates and excludes self-force, radiation reaction and material media (The four-force equation reproduces the coordinate Lorentz force and power, Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, Framework-local SI units, the vacuum constants and the exact/measured status of their values, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields).

[F2]

Plane-wave fields: $E_{i n c} (x, t) = Re (e e^{i (κ \cdot x - ω t)})$ , $B_{i n c} = κ \times E_{i n c} / ω$ are jointly $C^{\infty}$ on $R^{3} \times R$ (Real plane-wave fields encoded by complex amplitudes, Real and imaginary parts, complex conjugation, and modulus, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ); in particular, on every compact subset of $R^{3} \times R$ they and their first derivatives are bounded, and on a compact interval $L \subset I$ the maps $t \mapsto E_{i n c} (x_{0}, t)$ , $t \mapsto B_{i n c} (x_{0}, t)$ are continuous (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion). On a compact spatial ball times $L$ , integrating each bounded spatial derivative along the segment from $x_{0}$ to $x$ with The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ and The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ gives $∣ E_{i n c} (x, t) - E_{i n c} (x_{0}, t) ∣ \leq C ∣ x - x_{0} ∣$ , uniformly for $t \in L$ ; the componentwise integral bounds are those of If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ .

[F3]

Harmonic response lemma: for $ω > 0$ , $T = 2 π / ω$ , $w \in C^{3}$ , the particular solution $z_{p} (t) = Re (- w ω^{- 2} e^{- i ω t})$ is $C^{\infty}$ with $z_{p}^{''} = Re (w e^{- i ω t})$ and is $T$ -periodic; every $C^{2}$ solution of that equation on $R$ is $z_{p} + a + b t$ with unique $a, b$ ; it is $T$ -periodic if and only if $b = 0$ ; on any nonempty open interval the local solutions have a unique global extension $Z$ ; with $b = z^{'} (t_{*}) - z_{p}^{'} (t_{*})$ , and for $b = 0$ the mean over every complete period is $a$ (Periodic responses of $\ddot{z} = Re (w e^{- i ω t})$ and the affine classification of all solutions).

[F4]

Cross-section theorem: for the ideal harmonic model with incident amplitude $e$ and dipole amplitude $p_{0}$ , the total model cross section is $σ = \frac{ω^{4} ∣ p_{0} ∣^{2}}{6 π ε_{0}^{2} c^{4} ∣ e ∣^{2}} = \frac{μ_{0}^{2} ω^{4} ∣ p_{0} ∣^{2}}{6 π ∣ e ∣^{2}}$ , and the scalar branch with $p_{0} = α e$ is $\frac{ω^{4} ∣ α ∣^{2}}{6 π ε_{0}^{2} c^{4}}$ ; the result assumes Countable Choice as carried by its suppliers (Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F5]

Elementary calculus and algebra: vector-valued differentiation and integration are componentwise (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral); the second fundamental theorem evaluates $\int_{a}^{b} G^{'} = G (b) - G (a)$ for integrable $G^{'}$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ), a continuous real function on a closed interval is Riemann integrable (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion), the first fundamental theorem differentiates the integral function (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive), and $m \leq f \leq M$ on $[a, b]$ gives $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ with the corresponding error bound (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ); sums, products and compositions of differentiable maps obey the algebra and chain rules (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ); the cross product is bilinear and alternating with $∣ u \times v ∣ \leq ∣ u ∣ ∣ v ∣$ (The cross product is bilinear, alternating, and orthogonal to both factors); and $∣ e^{i θ} ∣ = 1$ for real $θ$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

Proof

technique · direct; take the exact integrated coordinate momentum equation of the adopted force law for the family, pass the uniform first-order expansion through the finite-interval integral, differentiate the resulting identity, and then classify the first-order harmonic response with the affine/periodic lemma and substitute into the cross-section theorem

1.1givenF1algebra

[Uniform smallness of the velocity and of $γ - 1$ ] Let $K \subset I$ be a nonempty compact interval containing $0$ in its interior (any compact subinterval of $I$ containing $0$ will do); the expansion hypothesis gives $∣ v_{η} (t) ∣ \leq ∣ η ∣ ∣ {\dot{r}}_{1} (t) ∣ + C_{K} η^{2} \leq ∣ η ∣ (∥ {\dot{r}}_{1} ∥_{K} + C_{K})$ for $∣ η ∣ < \min {η_{0}, 1}$ and $t \in K$ , where $∥ {\dot{r}}_{1} ∥_{K}$ is finite by continuity on the compact $K$ . Choose $η_{1} \in (0, \min {η_{0}, 1})$ with $η_{1} (∥ {\dot{r}}_{1} ∥_{K} + C_{K}) \leq c / 2$ ; then for $0 < ∣ η ∣ < η_{1}$ every $t \in K$ has $∣ v_{η} (t) ∣ / c \leq 1 / 2$ , so the low-speed clause of [F1] with $δ = 1 / 2$ gives $∣ γ_{η} (t) - 1 ∣ \leq \frac{∣ v_{η} (t) ∣^{2}}{2 c^{2} (1 - 1 / 4)^{3 / 2}} \leq C_{1} η^{2}$ for a constant $C_{1}$ depending only on $K, η_{1}$ and the data, uniformly in $t \in K$ .

2.1step 1.1givenF1F2F5algebra

[Uniform convergence of the scaled momentum and of its derivative] With $γ_{η} = (1 - ∣ v_{η} ∣^{2} / c^{2})^{- 1 / 2}$ and $P_{η} : = m γ_{η} v_{η}$ , write $\frac{1}{η} P_{η} = m γ_{η} \frac{v_{η}}{η} = m \frac{v_{η}}{η} + m (γ_{η} - 1) \frac{v_{η}}{η}$ ; by the expansion hypothesis $\frac{v_{η}}{η} \to {\dot{r}}_{1}$ uniformly on $K$ with error $C_{K} ∣ η ∣$ , and by step 1.1 $∣ γ_{η} - 1 ∣ \leq C_{1} η^{2}$ while $∣ \frac{v_{η}}{η} ∣ \leq ∥ {\dot{r}}_{1} ∥_{K} + C_{K} ∣ η ∣$ is uniformly bounded, so $\frac{P_{η}}{η} \to m {\dot{r}}_{1}$ uniformly on $K$ as $η \to 0$ with $η \neq 0$ . Moreover, dividing the exact coordinate force law of [F1] by $η$ gives, for $t \in K$ and $η \neq 0$ , $\frac{1}{η} P_{η}^{'} (t) = q [E_{i n c} (r_{η} (t), t) + v_{η} (t) \times B_{i n c} (r_{η} (t), t)]$ , and this right-hand side converges uniformly on $K$ to $q E_{i n c} (x_{0}, t)$ : the spatial Lipschitz bound of [F2] on a compact tube around ${x_{0}} \times K$ gives $∣ E_{i n c} (r_{η} (t), t) - E_{i n c} (x_{0}, t) ∣ \leq C ∣ r_{η} (t) - x_{0} ∣ \to 0$ uniformly by the expansion, while $∣ v_{η} (t) \times B_{i n c} (r_{η} (t), t) ∣ \leq ∣ v_{η} (t) ∣ \sup ∣ B_{i n c} ∣ \leq C_{2} ∣ η ∣$ by [F2] and the bilinearity and norm bound of [F5].

3.1step 2.1F2F5algebra

[First-order equation] Fix $η \neq 0$ with $∣ η ∣ < η_{1}$ and fix a component index $i$ . The scalar map $t \mapsto (P_{η})_{i} (t)$ is $C^{1}$ on $K$ with continuous derivative $(P_{η}^{'})_{i} = q η [E_{i n c} + v_{η} \times B_{i n c}]_{i}$ , so $(P_{η}^{'})_{i}$ is Riemann integrable on every closed subinterval of $K$ by [F5]; splitting at $0$ and applying the second fundamental theorem [F5] to the intervals $[0, t]$ for $t \geq 0$ and $[t, 0]$ for $t \leq 0$ (with the orientation convention of the displayed one-sided differences) gives, for every $t \in K$ , $\frac{(P_{η})_{i} (t) - (P_{η})_{i} (0)}{η} = \int_{0}^{t} \frac{(P_{η}^{'})_{i} (s)}{η} d s = q \int_{0}^{t} [E_{i n c} (r_{η} (s), s) + v_{η} (s) \times B_{i n c} (r_{η} (s), s)]_{i} d s$ . By the uniform convergence of step 2.1 applied componentwise, the integrands converge uniformly on $K$ to $q E_{i n c} (x_{0}, \cdot)_{i}$ , so the integral bound of [F5] lets $η \to 0$ : the left side tends to $m [({\dot{r}}_{1})_{i} (t) - ({\dot{r}}_{1})_{i} (0)]$ and the right side to $q \int_{0}^{t} E_{i n c} (x_{0}, s)_{i} d s$ . The fundamental theorem [F5] differentiates the right side in $t$ to $q E_{i n c} (x_{0}, t)_{i}$ , and the left side is differentiable with derivative $m ({\ddot{r}}_{1})_{i} (t)$ ; hence $m {\ddot{r}}_{1} (t) = q E_{i n c} (x_{0}, t)$ componentwise on $K$ . Every point $t \in I$ lies in such a compact $K \subset I$ containing $0$ , so the identity holds on all of $I$ and is clause (a).

4.1step 3.1F2F3F5algebra

[Harmonic classification and the induced dipole] Equation (a) reads ${\ddot{r}}_{1} (t) = Re (w e^{- i ω t})$ on $I$ with $w = (q / m) e e^{i κ \cdot x_{0}}$ by [F2]. By the local-clause of the harmonic lemma [F3], on $I$ one has $r_{1} = z_{p} + a + b t$ for a unique pair $(a, b)$ , the same formula defines the unique global extension $Z$ of $r_{1}$ , and $b = 0$ exactly when $Z$ is $T$ -periodic, equivalently when $r_{1}^{'} (0) = z_{p}^{'} (0) = Re (i w / ω)$ (differentiating $z_{p}$ at $0$ with the rules of [F5]). Preparation (P) therefore says exactly $b = 0$ , so on $I$ we have $r_{1} = z_{p} + a$ with $a$ the mean of the periodic extension over every complete period by [F3]. Then $q (r_{1} - a) = q z_{p} (t) = Re (- q w ω^{- 2} e^{- i ω t}) = Re (p_{0} e^{- i ω t})$ with $p_{0} = - q w / ω^{2} = - \frac{q^{2}}{m ω^{2}} e e^{i κ \cdot x_{0}}$ , because $w = (q / m) e e^{i κ \cdot x_{0}}$ ; writing $e^{'} = e e^{i κ \cdot x_{0}}$ one has $∣ e^{'} ∣ = ∣ e ∣$ by [F5], so $p_{0} = α e^{'}$ with $α = - \frac{q^{2}}{m ω^{2}}$ . The arbitrary mean $a$ is retained and no mean or periodicity of the nonlinear family is asserted; this is clause (b).

5.1step 4.1F4F5algebra

[Thomson cross section] Apply the cross-section formula of [F4] with the incident amplitude $e$ and dipole amplitude $p_{0}$ of step 4.1: $σ = \frac{ω^{4} ∣ p_{0} ∣^{2}}{6 π ε_{0}^{2} c^{4} ∣ e ∣^{2}}$ , and $∣ p_{0} ∣^{2} = \frac{q^{4}}{m^{2} ω^{4}} ∣ e ∣^{2}$ because $∣ e^{i κ \cdot x_{0}} ∣ = 1$ ; substituting cancels $ω^{4}$ and $∣ e ∣^{2}$ and gives $σ_{T} = \frac{q^{4}}{6 π ε_{0}^{2} m^{2} c^{4}}$ . Multiplying out, $\frac{8 π}{3} (\frac{q^{2}}{4 π ε_{0} m c^{2}})^{2} = \frac{8 π}{3} \cdot \frac{q^{4}}{16 π^{2} ε_{0}^{2} m^{2} c^{4}} = \frac{q^{4}}{6 π ε_{0}^{2} m^{2} c^{4}}$ , the same value, and it depends neither on $ω$ nor on the phase factor beyond the unimodular $e^{'}$ ; this is clause (c).

6.1step 3.1step 4.1step 5.1F1F3F4given∎

[Scope and choice accounting] Clause (d) records the exact content: the result is a conditional first variation of the adopted external-force law under the stated family and preparation hypotheses, with the affine drift retained in general and set to zero only by (P) for (b) and (c); no existence, persistence, stability or self-consistency of the family, no self-force, radiation reaction or material content, and no empirical, atomic or frequency-response claim is made. The $η^{2}$ scaling statement is a remark on the leading model: for nonzero $η$ the incident amplitude $η e$ and the induced amplitude $η p_{0}$ scale together, so the ratio defining $σ_{T}$ is unchanged; at $q = 0$ the amplitude $p_{0} = - \frac{q^{2}}{m ω^{2}} e e^{i κ \cdot x_{0}}$ and $σ_{T}$ vanish by direct substitution without any division by $q$ . The only choice principle used is the Countable Choice carried by the cross-section theorem [F4]; the uniform-limit, integral and harmonic-classification arguments use only the explicit suppliers [F1], [F3], [F5] and finite compact intervals.

Source notes

Supplier reconciled. The supplier pthm-em-relativistic-lorentz-force is now authored (carrier on disk, level 18) and its exact clauses were checked against the uses here: clause (a) supplies the coordinate force law $d p / d t = q (E + v \times B)$ for $p = m γ v$ , clause (b) the constancy of $η (U, U)$ with the future-timelike branch, and clause (c) the controlled low-speed bounds $∣ γ - 1 ∣ \leq \frac{∣ v ∣^{2} / c^{2}}{2 (1 - δ^{2})^{3 / 2}}$ and $∣ p - m v ∣ \leq m ∣ v ∣ \frac{∣ v ∣^{2}}{2 c^{2} (1 - δ^{2})^{3 / 2}}$ on any interval where $∣ v ∣ / c \leq δ$ , with constants depending only on $δ$ . The constants agree with [F1], step 1.1 uses exactly the $δ = 1 / 2$ instance, and step 2.1 divides the exact coordinate ODE by $η$ ; the earlier open obligation is cleared with no change of formulation.
The identity $E_{i n c} (x_{0}, t) = Re (e e^{i (κ \cdot x_{0} - ω t)})$ and the phase factor $e^{i κ \cdot x_{0}}$ are inherited from the plane-wave definition; the phase is not absorbed into a redefinition of the incident amplitude in clause (b).

5 · Examples, experiments and counterexamples

None yet.

Vector PEC scattering, dipole limits and controlled scalar diffraction

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Frequency-ωω source-free vacuum phasor pairs and their real-field encoding

Definition

Well-definedness and conventions

Real parts of e−iωte−iωt phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system

Statement

Facts & Assumptions

Proof

Remarks

Periodic responses of z¨=Re⁡(we−iωt)z¨=Re(we−iωt) and the affine classification of all solutions

Statement

Facts & Assumptions

Proof

Remarks

The ideal harmonic electric-dipole radiation model and its adopted status

Definition

Adopted status

Well-definedness and conventions

The adopted scalar aperture-diffraction model

Postulate

Well-definedness and non-derivation notes

Controlled Fraunhofer and Fresnel estimates for the scalar aperture model, with intensity-error transfer

Statement

Facts & Assumptions

Proof

Remarks

Monochromatic vector scattering by a smooth PEC obstacle: phasor Maxwell fields, the PEC trace condition and outgoing-correction uniqueness

Statement

Facts & Assumptions

Proof

Remarks

Differential and total cross sections of the ideal harmonic dipole in a plane wave, with actual-source fidelity bounds

Statement

Facts & Assumptions

Proof

Remarks

Controlled weak-field first variation of the relativistic force law and the leading Thomson-model cross section

Statement

Facts & Assumptions

Proof

Source notes

5 · Examples, experiments and counterexamples

Sources

Frequency- $ω$ source-free vacuum phasor pairs and their real-field encoding

Real parts of $e^{- i ω t}$ phasors: calculus, injectivity and equivalence with the real Helmholtz-Maxwell system

Periodic responses of $\ddot{z} = Re (w e^{- i ω t})$ and the affine classification of all solutions