Alphabeta Math

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

Calculus of separated phasor fields: real-part derivative identities and amplitude injectivity

Statement

Let $Ω_{t} \subseteq R^{2}$ be open, let $β \in C$ , $ω > 0$ , and let $\tilde{F}, \tilde{G} : Ω_{t} \to C^{3}$ be $C^{1}$ maps (componentwise, in the sense of $C^{k}$ Euclidean maps and diffeomorphisms). Write $θ (x_{3}, t) : = β x_{3} - ω t$ and define $F : Ω_{t} \times R \times R \to R^{3}$ by $F (x_{1}, x_{2}, x_{3}, t) : = Re (\tilde{F} (x_{1}, x_{2}) e^{i θ (x_{3}, t)})$ componentwise, and $G$ likewise from $\tilde{G}$ ; here $e^{i θ}$ is the complex exponential of The complex exponential by its power series and $Re$ the real part of Real and imaginary parts, complex conjugation, and modulus. Then $F$ is $C^{1}$ and, for every point of the domain: (a) $\partial_{x_{j}} F = Re (\partial_{x_{j}} \tilde{F} e^{i θ})$ for $j = 1, 2$ , $\partial_{x_{3}} F = Re (i β \tilde{F} e^{i θ})$ , $\partial_{t} F = Re (- i ω \tilde{F} e^{i θ})$ , where the divergence and curl are those of Divergence and curl of a $C^{1}$ vector field and the partial derivatives are those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ; (b) $div F = Re ((\partial_{1} {\tilde{F}}_{1} + \partial_{2} {\tilde{F}}_{2} + i β {\tilde{F}}_{3}) e^{i θ})$ ; (c) $curl F = Re ((\partial_{2} {\tilde{F}}_{3} - i β {\tilde{F}}_{2}, i β {\tilde{F}}_{1} - \partial_{1} {\tilde{F}}_{3}, \partial_{1} {\tilde{F}}_{2} - \partial_{2} {\tilde{F}}_{1}) e^{i θ})$ ; (d) [amplitude injectivity] if $C \in C^{3}$ and $Re (C e^{i φ_{1}}) = Re (C e^{i φ_{2}}) = 0$ for two real phases with $φ_{1} - φ_{2} \notin π Z$ , then $C = 0$ ; consequently, if $F = G$ on a nonempty open subset $U$ of $Ω_{t} \times R \times R$ , then $\tilde{F} = \tilde{G}$ on the projection of $U$ to $Ω_{t}$ (a nonempty open subset of $Ω_{t}$ ).

Facts & Assumptions

Given: An open set $Ω_{t} \subseteq R^{2}$ ; complex numbers $β = p + i q$ ( $p, q \in R$ ) and $ω > 0$ ; $C^{1}$ maps $\tilde{F}, \tilde{G} : Ω_{t} \to C^{3}$ , with real functions $a_{l} : = Re {\tilde{F}}_{l}$ and $b_{l} : = Im {\tilde{F}}_{l}$ for $l = 1, 2, 3$ ; the phase $θ (x_{3}, t) = β x_{3} - ω t$ ; and the fields $F, G$ of the Statement.

[F1]

A map into $R^{n}$ is $C^{k}$ when each component is, and a $C^{1}$ component has continuous first partial derivatives; the $j$ th partial derivative $\partial_{j} f (a) = D_{e_{j}} f (a)$ is the derivative along the coordinate line $t \mapsto f (a + t e_{j})$ ( $C^{k}$ Euclidean maps and diffeomorphisms, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F2]

$C$ is a field with $i^{2} = - 1$ , every $z$ has the unique form $z = Re z + i Im z$ with real parts, and $Re$ is $R$ -linear; in particular $Re (i ζ) = - Im ζ$ , $Re (- i ζ) = Im ζ$ , and for real $a, b, φ$ the cartesian multiplication gives $(a + i b) (\cos φ + i \sin φ) = (a \cos φ - b \sin φ) + i (a \sin φ + b \cos φ)$ , so $Re ((a + i b) (\cos φ + i \sin φ)) = a \cos φ - b \sin φ$ (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Real and imaginary parts, complex conjugation, and modulus).

[F3]

For real $x, y$ , $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ); the complex exponential of The complex exponential by its power series is entire with $\exp^{'} = \exp$ (The complex exponential is entire and its complex derivative is itself).

[F4]

The real exponential is $C^{\infty}$ with $(\exp)^{'} = \exp$ (The exponential function is smooth and $(\exp)^{'} = \exp$ ) and $\exp x > 0$ for real $x$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ); $(\sin x)^{'} = \cos x$ and $(\cos x)^{'} = - \sin x$ (The derivatives of sine and cosine are cosine and minus sine); sums, scalar multiples and products of differentiable real functions satisfy the sum, scalar and product rules and composites satisfy the chain rule for total derivatives, which applies to the affine substitutions $x_{3} \mapsto \pm q x_{3}$ and $x_{3}, t \mapsto p x_{3} - ω t$ composed with the outer exponential, sine and cosine (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: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F5]

For all real $A, B$ , $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\cos (A + B) = \cos A \cos B - \sin A \sin B$ (The addition formulas for sine and cosine); $\sin (- A) = - \sin A$ , $\cos (- A) = \cos A$ and $\sin^{2} A + \cos^{2} A = 1$ (Parity and the Pythagorean identity for sine and cosine); finally $\sin x = 0$ if and only if $x \in π Z$ (The zero sets of sine and cosine and the least positive common period 2 pi).

[F6]

For a $C^{1}$ field $F = (F_{1}, F_{2}, F_{3})$ on an open subset of $R^{3}$ , $div F = \partial_{1} F_{1} + \partial_{2} F_{2} + \partial_{3} F_{3}$ and $curl F = (\partial_{2} F_{3} - \partial_{3} F_{2}, \partial_{3} F_{1} - \partial_{1} F_{3}, \partial_{1} F_{2} - \partial_{2} F_{1})$ (Divergence and curl of a $C^{1}$ vector field).

Proof

technique · direct: reduce the complex phase to real cartesian factors, differentiate the explicit components along coordinate lines with the real one-variable rules, assemble the divergence and curl, and settle the injectivity clause by a two-phase linear system

1.1givenF2F3algebra

[Reduction of the phase and of the components] Since $β = p + i q$ and $i^{2} = - 1$ , $i θ = i (β x_{3} - ω t) = i (p x_{3} - ω t) + i (i q x_{3}) = - q x_{3} + i (p x_{3} - ω t)$ ; with $φ : = p x_{3} - ω t$ the cartesian form [F3] gives $e^{i θ} = e^{- q x_{3}} (\cos φ + i \sin φ)$ . By [F2] the component of $\tilde{F} e^{i θ}$ with index $l$ is $e^{- q x_{3}} [(a_{l} \cos φ - b_{l} \sin φ) + i (a_{l} \sin φ + b_{l} \cos φ)]$ ; taking real parts, $F_{l} = e^{- q x_{3}} (a_{l} \cos φ - b_{l} \sin φ)$ .

1.2givenF2algebra

[The two-phase system] For clause (d), write $C = a + i b$ with $a, b \in R^{3}$ . By [F2], $Re (C e^{i φ_{j}}) = a \cos φ_{j} - b \sin φ_{j}$ componentwise, so $Re (C e^{i φ_{1}}) = Re (C e^{i φ_{2}}) = 0$ is the system $a_{l} \cos φ_{1} = b_{l} \sin φ_{1}$ and $a_{l} \cos φ_{2} = b_{l} \sin φ_{2}$ for every $l$ . Multiplying the two equations of index $l$ by $\cos φ_{2}$ and $\cos φ_{1}$ and subtracting gives $b_{l} (\sin φ_{1} \cos φ_{2} - \cos φ_{1} \sin φ_{2}) = 0$ , and multiplying them by $\sin φ_{2}$ and $\sin φ_{1}$ and subtracting gives $a_{l} (\cos φ_{1} \sin φ_{2} - \cos φ_{2} \sin φ_{1}) = 0$ .

2.1step 1.1F2F4

[The phase factor and its derivatives] By step 1.1 the real and imaginary parts of $(x_{3}, t) \mapsto e^{i θ}$ are $e^{- q x_{3}} \cos φ$ and $e^{- q x_{3}} \sin φ$ , which are $C^{\infty}$ on $R^{2}$ by [F4] (the chain rule applied to the affine substitutions $x_{3} \mapsto - q x_{3}$ and $x_{3}, t \mapsto p x_{3} - ω t$ composed with the outer exponential, sine and cosine, together with the sum, scalar and product rules), with $\partial_{x_{3}} e^{- q x_{3}} = - q e^{- q x_{3}}$ , $\partial_{t} e^{- q x_{3}} = 0$ , $\partial_{x_{3}} \cos φ = - p \sin φ$ , $\partial_{x_{3}} \sin φ = p \cos φ$ , $\partial_{t} \cos φ = ω \sin φ$ and $\partial_{t} \sin φ = - ω \cos φ$ . Pairing these, $\partial_{x_{3}} e^{i θ} = (- q + i p) e^{i θ} = i β e^{i θ}$ and $\partial_{t} e^{i θ} = - i ω e^{i θ}$ by [F2], while $e^{i θ}$ does not depend on $x_{1}, x_{2}$ , so $\partial_{x_{j}} e^{i θ} = 0$ for $j = 1, 2$ .

2.2step 1.1F1F2F4

[Transverse derivatives] Fix $j \in {1, 2}$ and $l \in {1, 2, 3}$ . Along the $x_{j}$ -coordinate line the factors $e^{- q x_{3}}$ , $\cos φ$ and $\sin φ$ of step 1.1 are constant, so the derivative of the partial application is $e^{- q x_{3}} [(\partial_{j} a_{l}) \cos φ - (\partial_{j} b_{l}) \sin φ]$ by the sum and scalar rules [F4], the derivatives $\partial_{j} a_{l}, \partial_{j} b_{l}$ existing by [F1]; by the expansion of step 1.1 applied to $\partial_{j} {\tilde{F}}_{l}$ in place of ${\tilde{F}}_{l}$ , this number is $Re ((\partial_{j} {\tilde{F}}_{l}) e^{i θ})$ . Hence $\partial_{x_{j}} F = Re (\partial_{x_{j}} \tilde{F} e^{i θ})$ componentwise for $j = 1, 2$ .

2.3step 1.2F5

[Injectivity] By the addition and parity formulas [F5], $\sin (φ_{1} - φ_{2}) = \sin φ_{1} \cos φ_{2} - \cos φ_{1} \sin φ_{2}$ and $- \sin (φ_{1} - φ_{2}) = \cos φ_{1} \sin φ_{2} - \cos φ_{2} \sin φ_{1}$ . Since $φ_{1} - φ_{2} \notin π Z$ , [F5] gives $\sin (φ_{1} - φ_{2}) \neq 0$ , so the two equations of step 1.2 force $b_{l} = 0$ and $a_{l} = 0$ for every $l$ , that is, $C = 0$ .

3.1step 1.1step 2.1F2F4algebra

[Longitudinal and time derivatives] Fix $l$ . Along the $x_{3}$ -line the amplitudes $a_{l}, b_{l}$ are constant, so the product and scalar rules [F4] and the derivatives of step 2.1 give $\partial_{x_{3}} F_{l} = - q e^{- q x_{3}} (a_{l} \cos φ - b_{l} \sin φ) + e^{- q x_{3}} (- p a_{l} \sin φ - p b_{l} \cos φ)$ ; by [F2], $i β (a_{l} + i b_{l}) = (- q a_{l} - p b_{l}) + i (p a_{l} - q b_{l})$ , so $Re (i β {\tilde{F}}_{l} e^{i θ}) = e^{- q x_{3}} [(- q a_{l} - p b_{l}) \cos φ - (p a_{l} - q b_{l}) \sin φ]$ , which is the displayed expression. Along the $t$ -line the amplitudes and $e^{- q x_{3}}$ are constant, so $\partial_{t} F_{l} = e^{- q x_{3}} (a_{l} ω \sin φ + b_{l} ω \cos φ)$ ; by [F2], $- i ω (a_{l} + i b_{l}) = ω b_{l} - i ω a_{l}$ , so $Re (- i ω {\tilde{F}}_{l} e^{i θ}) = e^{- q x_{3}} (ω b_{l} \cos φ + ω a_{l} \sin φ)$ , the displayed expression. Assembling the components gives $\partial_{x_{3}} F = Re (i β \tilde{F} e^{i θ})$ and $\partial_{t} F = Re (- i ω \tilde{F} e^{i θ})$ .

4.1step 2.2step 3.1F1F4

[Regularity] Every partial derivative displayed in steps 2.2 and 3.1 is continuous on $Ω_{t} \times R \times R$ : the partial derivatives $\partial_{j} a_{l}, \partial_{j} b_{l}$ ( $j = 1, 2$ ) are continuous by [F1], the factors $e^{- q x_{3}}$ , $\cos φ$ , $\sin φ$ and their displayed partial derivatives are smooth by step 2.1, and each displayed formula is a finite sum of products of such continuous functions. Thus all first partial derivatives of $F$ exist and are continuous, so $F$ is $C^{1}$ .

4.2step 2.2step 3.1F2F6algebra

[Divergence] By [F6] and steps 2.2 and 3.1, $div F = \partial_{1} F_{1} + \partial_{2} F_{2} + \partial_{3} F_{3} = Re ((\partial_{1} {\tilde{F}}_{1} + \partial_{2} {\tilde{F}}_{2} + i β {\tilde{F}}_{3}) e^{i θ})$ : indeed, for complex $ζ_{l}$ the $R$ -linearity of $Re$ and distributivity in $C$ [F2] give $Re (ζ_{1} e^{i θ}) + Re (ζ_{2} e^{i θ}) + Re (ζ_{3} e^{i θ}) = Re ((ζ_{1} + ζ_{2} + ζ_{3}) e^{i θ})$ .

4.3step 2.2step 3.1F2F6algebra

[Curl] Similarly, [F6] together with steps 2.2 and 3.1 gives the three components of $curl F$ as $\partial_{2} F_{3} - \partial_{3} F_{2} = Re ((\partial_{2} {\tilde{F}}_{3} - i β {\tilde{F}}_{2}) e^{i θ})$ , $\partial_{3} F_{1} - \partial_{1} F_{3} = Re ((i β {\tilde{F}}_{1} - \partial_{1} {\tilde{F}}_{3}) e^{i θ})$ and $\partial_{1} F_{2} - \partial_{2} F_{1} = Re ((\partial_{1} {\tilde{F}}_{2} - \partial_{2} {\tilde{F}}_{1}) e^{i θ})$ , since [F2] gives $Re (ζ e^{i θ}) - Re (η e^{i θ}) = Re ((ζ - η) e^{i θ})$ for complex $ζ, η$ . This is clause (c).

5.1step 2.3F2F4F5algebra∎

[Open-set consequence] Let $U \subseteq Ω_{t} \times R \times R$ be nonempty open with $F = G$ on $U$ . Its projection $π (U)$ to $Ω_{t}$ is nonempty and open. Fix $x_{t} \in π (U)$ ; the fibre ${(x_{3}, t) : (x_{t}, x_{3}, t) \in U}$ is a nonempty open subset of $R^{2}$ , so there are $x_{3} \in R$ and an open interval $J$ of $t$ -values on which $(x_{t}, x_{3}, t) \in U$ ; choose $t_{1}, t_{2} \in J$ with $0 < ω ∣ t_{1} - t_{2} ∣ < π$ . Put $C : = (\tilde{F} (x_{t}) - \tilde{G} (x_{t})) e^{i β x_{3}}$ . For $j = 1, 2$ , the $R$ -linearity of $Re$ [F2] and $F = G$ on $U$ give $Re (C e^{- i ω t_{j}}) = Re ((\tilde{F} - \tilde{G}) (x_{t}) e^{i θ (x_{3}, t_{j})}) = F (x_{t}, x_{3}, t_{j}) - G (x_{t}, x_{3}, t_{j}) = 0$ . The phases $φ_{j} : = - ω t_{j}$ are real and $φ_{1} - φ_{2} = - ω (t_{1} - t_{2}) \notin π Z$ , so step 2.3 gives $C = 0$ . Finally $e^{i β x_{3}} = e^{- q x_{3}} (\cos (p x_{3}) + i \sin (p x_{3}))$ by step 1.1, where $e^{- q x_{3}} > 0$ by [F4] and $\cos (p x_{3}) + i \sin (p x_{3}) \neq 0$ because its real and imaginary parts have $\cos^{2} (p x_{3}) + \sin^{2} (p x_{3}) = 1 \neq 0$ [F5]; since a product of nonzero elements of the field $C$ is nonzero [F2], $\tilde{F} (x_{t}) = \tilde{G} (x_{t})$ . As $x_{t} \in π (U)$ was arbitrary and $π (U)$ is nonempty open, this is the stated consequence.

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

Transverse vector-calculus identities for the splitting $R^{3} = R^{2} \oplus R e_{z}$

Statement

Let $e_{z}$ be a unit vector of $R^{3}$ and identify $R^{3} = R^{2} \oplus R e_{z}$ with transverse coordinates $(x_{1}, x_{2})$ ; write $\nabla_{t} h = (\partial_{1} h, \partial_{2} h)$ , $Δ_{t} h = \partial_{1}^{2} h + \partial_{2}^{2} h$ and, for a transverse $C^{1}$ map $u_{t}$ with components $(u_{1}, u_{2})$ , ${div}_{t} u_{t} = \partial_{1} u_{1} + \partial_{2} u_{2}$ and ${curl}_{2} u_{t} = \partial_{1} u_{2} - \partial_{2} u_{1}$ ; regard $h$ , $\nabla_{t} h$ , $u_{t}$ and $e_{z} \times u_{t}$ as $z$ -independent fields on $Ω_{t} \times R$ , with $e_{z}$ the third basis vector. Let $h : Ω_{t} \to R$ be $C^{2}$ and $u_{t} : Ω_{t} \to R^{2}$ be $C^{1}$ . Then, with the divergence, curl, cross product and inner product of Divergence and curl of a $C^{1}$ vector field, The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : (a) $div (e_{z} \times \nabla_{t} h) = 0$ ; $curl (e_{z} \times \nabla_{t} h) = e_{z} Δ_{t} h$ ; $e_{z} \times (e_{z} \times \nabla_{t} h) = - \nabla_{t} h$ ; $\nabla_{t} h \cdot (e_{z} \times \nabla_{t} h) = 0$ ; $e_{z} \cdot (e_{z} \times \nabla_{t} h) = 0$ ; $∣ e_{z} \times \nabla_{t} h ∣ = ∣ \nabla_{t} h ∣$ ; (b) $curl (e_{z} h) = - e_{z} \times \nabla_{t} h$ and $div (e_{z} h) = \partial_{3} h$ (which vanishes for $z$ -independent $h$ ); (c) $curl (u_{t}) = e_{z} {curl}_{2} u_{t}$ , $div (u_{t}) = {div}_{t} u_{t}$ , and $e_{z} \times u_{t}$ is transverse.

Facts & Assumptions

Given: A unit vector $e_{z} \in R^{3}$ , the orthogonal splitting $R^{3} = R^{2} \oplus R e_{z}$ with transverse coordinates $(x_{1}, x_{2})$ and $e_{z} = e_{3}$ the third standard basis vector; an open set $Ω_{t} \subseteq R^{2}$ ; a $C^{2}$ function $h : Ω_{t} \to R$ and a $C^{1}$ map $u_{t} : Ω_{t} \to R^{2}$ with components $(u_{1}, u_{2})$ ; and the fields $h$ , $\nabla_{t} h$ , $u_{t}$ , $e_{z} \times u_{t}$ regarded as functions on $Ω_{t} \times R$ independent of the third coordinate $x_{3}$ .

[F1]

$\partial_{j} f$ is the derivative of $f$ along the $j$ th coordinate line, and a $C^{1}$ map has continuous first partial derivatives (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F2]

For a $C^{1}$ field $F = (F_{1}, F_{2}, F_{3})$ on an open subset of $R^{3}$ , $div F = \partial_{1} F_{1} + \partial_{2} F_{2} + \partial_{3} F_{3}$ and $curl F = (\partial_{2} F_{3} - \partial_{3} F_{2}, \partial_{3} F_{1} - \partial_{1} F_{3}, \partial_{1} F_{2} - \partial_{2} F_{1})$ (Divergence and curl of a $C^{1}$ vector field).

[F3]

The cross product on $R^{3}$ is given by its coordinate formula, it is bilinear and alternating, it is orthogonal to both factors, and the Euclidean inner product is the coordinate dot product $⟨ v, w ⟩ = \sum_{j} v_{j} w_{j}$ (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

The Laplacian of a $C^{2}$ scalar field on an open subset of $R^{n}$ is $Δ f = \partial_{1} \partial_{1} f + \dots + \partial_{n} \partial_{n} f$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field); in particular $Δ_{t} f = \partial_{1}^{2} f + \partial_{2}^{2} f$ for $f$ on $Ω_{t} \subseteq R^{2}$ .

[F5]

If $f$ is $C^{2}$ on an open subset of $R^{m}$ , then $\partial_{i} \partial_{j} f = \partial_{j} \partial_{i} f$ for all coordinate indices (Clairaut--Schwarz theorem for continuous second partial derivatives).

[F6]

Derivatives of sums, scalar multiples and products of differentiable real functions obey the sum, scalar and product 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 $g (c) \neq 0$ ); in particular a constant factor and $- \partial_{j} h$ may be moved through $\partial_{i}$ .

Proof

technique · direct: write every object in the orthonormal basis $(e_1,e_2,e_z)$, use the coordinate formulas for the cross product, divergence and curl, and evaluate the coordinate-derivative expressions with the mixed-partials theorem

1.1givenF1F2F3

[Component forms] In the orthonormal basis $(e_{1}, e_{2}, e_{z})$ a transverse vector has the form $v = (v_{1}, v_{2}, 0)$ , and the coordinate formula for the cross product [F3] gives $e_{z} \times v = (e_{z} \times v_{1} e_{1}) + (e_{z} \times v_{2} e_{2}) = v_{1} e_{2} - v_{2} e_{1} = (- v_{2}, v_{1}, 0)$ . Hence $e_{z} \times \nabla_{t} h = (- \partial_{2} h, \partial_{1} h, 0)$ , $e_{z} \times u_{t} = (- u_{2}, u_{1}, 0)$ , and $e_{z} h$ is the field $(0, 0, h)$ ; every field displayed in the Statement is independent of $x_{3}$ , so its $\partial_{3}$ -derivative vanishes wherever the derivative is taken.

2.1step 1.1F2F5F6algebra

[Divergence of $e_{z} \times \nabla_{t} h$ ] By [F2] and step 1.1, $div (e_{z} \times \nabla_{t} h) = \partial_{1} (- \partial_{2} h) + \partial_{2} (\partial_{1} h) + \partial_{3} 0 = - \partial_{1} \partial_{2} h + \partial_{2} \partial_{1} h$ , which is $0$ by [F5] because $h$ is $C^{2}$ ; this is the first identity of clause (a).

2.2step 1.1F2F4F6

[Curl of $e_{z} \times \nabla_{t} h$ ] By [F2] and step 1.1, again with the third-coordinate derivative vanishing, $curl (e_{z} \times \nabla_{t} h) = (\partial_{2} 0 - \partial_{3} \partial_{1} h, \partial_{3} (- \partial_{2} h) - \partial_{1} 0, \partial_{1} \partial_{1} h - \partial_{2} (- \partial_{2} h)) = (0, 0, \partial_{1}^{2} h + \partial_{2}^{2} h) = e_{z} Δ_{t} h$ , which is the second identity of clause (a) by [F4].

2.3step 1.1F3algebra

[Triple product, orthogonality and norm] For an arbitrary transverse $v = (v_{1}, v_{2}, 0)$ , step 1.1 gives $e_{z} \times (e_{z} \times v) = e_{z} \times (- v_{2}, v_{1}, 0) = (- v_{1}, - v_{2}, 0) = - v$ ; the coordinate formulas of [F3] give $v \cdot (e_{z} \times v) = v_{1} (- v_{2}) + v_{2} v_{1} + 0 \cdot 0 = 0$ , $e_{z} \cdot (e_{z} \times v) = 0$ as the third component of $e_{z} \times v$ is $0$ , and $∣ e_{z} \times v ∣^{2} = (- v_{2})^{2} + v_{1}^{2} + 0 = v_{1}^{2} + v_{2}^{2} = ∣ v ∣^{2}$ , both sides nonnegative, so $∣ e_{z} \times v ∣ = ∣ v ∣$ . At $v = \nabla_{t} h$ these are the remaining four identities of clause (a); the case $v = 0$ is included since then both sides vanish.

2.4step 1.1F1F2F6

[Clause (b)] By [F2] and step 1.1, $curl (e_{z} h) = (\partial_{2} h - \partial_{3} 0, \partial_{3} 0 - \partial_{1} h, \partial_{1} 0 - \partial_{2} 0) = (\partial_{2} h, - \partial_{1} h, 0) = - e_{z} \times \nabla_{t} h$ , and $div (e_{z} h) = \partial_{1} 0 + \partial_{2} 0 + \partial_{3} h = \partial_{3} h$ , which is $0$ because $h$ is independent of $x_{3}$ .

3.1step 1.1F1F2∎

[Clause (c)] For $u_{t} = (u_{1}, u_{2}, 0)$ with $\partial_{3} u_{j} = 0$ , [F2] gives $curl (u_{t}) = (\partial_{2} 0 - \partial_{3} u_{2}, \partial_{3} u_{1} - \partial_{1} 0, \partial_{1} u_{2} - \partial_{2} u_{1}) = (0, 0, {curl}_{2} u_{t}) = e_{z} {curl}_{2} u_{t}$ and $div (u_{t}) = \partial_{1} u_{1} + \partial_{2} u_{2} = {div}_{t} u_{t}$ ; and $e_{z} \times u_{t} = (- u_{2}, u_{1}, 0)$ has third component $0$ , so it is transverse.

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

The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals

Statement

Let $L_{1}, L_{2} > 0$ and $R : = (0, L_{1}) \times (0, L_{2})$ . For integers $m, n \geq 0$ put $h_{m n} (x_{1}, x_{2}) : = \cos (\frac{m π x_{1}}{L_{1}}) \cos (\frac{n π x_{2}}{L_{2}})$ and $k_{m n}^{2} : = (\frac{m π}{L_{1}})^{2} + (\frac{n π}{L_{2}})^{2}$ ; for integers $m, n \geq 1$ put $e_{m n} (x_{1}, x_{2}) : = \sin (\frac{m π x_{1}}{L_{1}}) \sin (\frac{n π x_{2}}{L_{2}})$ . Then, with $Δ = \partial_{1}^{2} + \partial_{2}^{2}$ , $\nabla = (\partial_{1}, \partial_{2})$ and the Riemann integrals over the rectangle of Axis-parallel rectangles in $R^{m}$ and their volume: (a) $Δ h_{m n} = - k_{m n}^{2} h_{m n}$ on $R$ and $\partial_{1} h_{m n} = 0$ on both faces $x_{1} \in {0, L_{1}}$ , $\partial_{2} h_{m n} = 0$ on both faces $x_{2} \in {0, L_{2}}$ ; hence $\partial_{n} h_{m n} = 0$ on every open face of $\partial R$ ; (b) $Δ e_{m n} = - k_{m n}^{2} e_{m n}$ on $R$ and $e_{m n} = 0$ on $\partial R$ for $m, n \geq 1$ ; (c) [orthogonality and norms] $\int_{R} h_{m n} h_{m^{'} n^{'}} = 0$ unless $(m, n) = (m^{'}, n^{'})$ , and $\int_{R} h_{m n}^{2} = \frac{L_{1} L_{2}}{4} (1 + δ_{m 0}) (1 + δ_{n 0})$ ; $\int_{R} e_{m n} e_{m^{'} n^{'}} = 0$ unless $(m, n) = (m^{'}, n^{'})$ , and $\int_{R} e_{m n}^{2} = \frac{L_{1} L_{2}}{4}$ for $m, n \geq 1$ ; the mixed integrals are instead $\int_{R} h_{m n} e_{p q} = J_{m p} (L_{1}) J_{n q} (L_{2}), J_{m p} (L) : = \int_{0}^{L} \cos (m π x / L) \sin (p π x / L) d x,$ for $m, n \geq 0$ , $p, q \geq 1$ . Here $J_{m p} (L) = 0$ when $m, p$ have matching parity (including $p = m$ ), and $J_{m p} (L) = 2 p L / [π (p^{2} - m^{2})]$ when their parities differ. In particular $\int_{R} h_{00} e_{11} = 4 L_{1} L_{2} / π^{2}$ , so the sine and cosine systems are not mutually orthogonal; (d) [Dirichlet-type identities] $\int_{R} ∣ \nabla h_{m n} ∣^{2} = k_{m n}^{2} \int_{R} h_{m n}^{2}$ for all $m, n \geq 0$ and $\int_{R} ∣ \nabla e_{m n} ∣^{2} = k_{m n}^{2} \int_{R} e_{m n}^{2}$ for $m, n \geq 1$ .

Facts & Assumptions

Given: Positive reals $L_{1}, L_{2}$ , the rectangle $R = (0, L_{1}) \times (0, L_{2})$ , and the profiles $h_{m n}$ ( $m, n \geq 0$ ), $e_{m n}$ ( $m, n \geq 1$ ) of the Statement.

[F1]

For an open $U \subseteq R^{2}$ and $f : U \to R$ , $\partial_{j} f (a) = D_{e_{j}} f (a)$ is the $j$ th partial derivative, computed by restricting to coordinate lines (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ), and for $f \in C^{2}$ the Laplacian is $Δ f = div \nabla f = \partial_{1} \partial_{1} f + \partial_{2} \partial_{2} f$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F2]

$[a, b] = {x : a_{j} \leq x_{j} \leq b_{j}}$ with $vol [a, b] = \prod_{j} (b_{j} - a_{j})$ , and $[a, b]$ is nondegenerate when every $a_{j} < b_{j}$ (Axis-parallel rectangles in $R^{m}$ and their volume); every continuous real function on a closed nondegenerate rectangle is Riemann integrable (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable); and for $f$ Riemann integrable on a product $A \times B$ , the section-integral functions are integrable and $\int_{A \times B} f = \int_{A} \int_{B} f_{x}$ whenever the inner sections are integrable (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections).

[F3]

If $G : [a, b] \to R$ is differentiable on $[a, b]$ and $G^{'}$ is integrable there, then $\int_{a}^{b} G^{'} = G (b) - G (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F4]

$(\sin x)^{'} = \cos x$ and $(\cos x)^{'} = - \sin x$ on $R$ , with $\sin 0 = 0$ , $\cos 0 = 1$ (The derivatives of sine and cosine are cosine and minus sine); derivatives of sums, scalar multiples and products obey the sum, scalar and product 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 $g (c) \neq 0$ ); and composites of totally differentiable maps obey the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ), which applies to the affine maps $x_{j} \mapsto \frac{m π x_{j}}{L_{j}}$ and $x_{j} \mapsto \frac{n π x_{j}}{L_{j}}$ .

[F5]

$\sin (k π) = 0$ and $\cos (k π) = (- 1)^{k}$ for every integer $k$ , and $\sin 0 = 0$ , $\cos 0 = 1$ : the zero sets, period and special values of sine and cosine (The zero sets of sine and cosine and the least positive common period 2 pi, Quarter-turn values and shifts by pi/2 and pi).

[F6]

For all real $A, B$ , $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\cos (A + B) = \cos A \cos B - \sin A \sin B$ (The addition formulas for sine and cosine), hence the product-to-sum identities $\cos A \cos B = \frac{1}{2} [\cos (A - B) + \cos (A + B)], \sin A \sin B = \frac{1}{2} [\cos (A - B) - \cos (A + B)], \sin A \cos B = \frac{1}{2} [\sin (A + B) + \sin (A - B)] .$

Proof

technique · direct: differentiate the products, then integrate the resulting continuous one-variable functions by product-to-sum, the fundamental theorem and Fubini

1.1givenF1F4F5algebra

[First derivatives and traces] Let $m, n \geq 0$ and $x \in R$ . The partial derivatives of the cosine products are $\partial_{1} h_{m n} (x) = - \frac{m π}{L_{1}} \sin (\frac{m π x_{1}}{L_{1}}) \cos (\frac{n π x_{2}}{L_{2}})$ and $\partial_{2} h_{m n} (x) = - \frac{n π}{L_{2}} \cos (\frac{m π x_{1}}{L_{1}}) \sin (\frac{n π x_{2}}{L_{2}})$ : fix the other coordinate and apply the chain rule to $x_{j} \mapsto \cos (\frac{m π x_{j}}{L_{j}})$ composed with the affine map, i.e. the derivative $- \sin \cdot \frac{m π}{L_{j}}$ , together with the product rule. Evaluating with [F5] gives $\partial_{1} h_{m n} (0, x_{2}) = \partial_{1} h_{m n} (L_{1}, x_{2}) = 0$ and $\partial_{2} h_{m n} (x_{1}, 0) = \partial_{2} h_{m n} (x_{1}, L_{2}) = 0$ . Similarly, for $m, n \geq 1$ , $\partial_{1} e_{m n} (x) = \frac{m π}{L_{1}} \cos (\frac{m π x_{1}}{L_{1}}) \sin (\frac{n π x_{2}}{L_{2}})$ , $\partial_{2} e_{m n} (x) = \frac{n π}{L_{2}} \sin (\frac{m π x_{1}}{L_{1}}) \cos (\frac{n π x_{2}}{L_{2}})$ , and $e_{m n} = 0$ on $\partial R$ because a sine factor vanishes whenever $x_{1} \in {0, L_{1}}$ or $x_{2} \in {0, L_{2}}$ .

1.2F3F5F6algebra

[One-dimensional orthogonality and norms] Let $m, m^{'} \geq 0$ and $p, p^{'} \geq 1$ be integers. By [F6], $\cos (m π x / L) \cos (m^{'} π x / L) = \frac{1}{2} [\cos (\frac{(m - m^{'}) π x}{L}) + \cos (\frac{(m + m^{'}) π x}{L})]$ , so the second fundamental theorem [F3] applied to the continuous antiderivatives, together with the integer values [F5], gives $\int_{0}^{L} \cos \frac{m π x}{L} \cos \frac{m^{'} π x}{L} d x = 0$ for $m \neq m^{'}$ , $= \frac{L}{2}$ for $m = m^{'} \geq 1$ and $= L$ for $m = m^{'} = 0$ ; uniformly $\int_{0}^{L} \cos \frac{m π x}{L} \cos \frac{m^{'} π x}{L} d x = δ_{m m^{'}} \frac{L}{2} (1 + δ_{m 0})$ . Likewise $\int_{0}^{L} \sin \frac{p π x}{L} \sin \frac{p^{'} π x}{L} d x = \frac{L}{2} δ_{p p^{'}}$ , because $\sin \frac{p π x}{L} \sin \frac{p^{'} π x}{L} = \frac{1}{2} [\cos \frac{(p - p^{'}) π x}{L} - \cos \frac{(p + p^{'}) π x}{L}]$ and every nonzero integer multiple of $π$ is a zero of sine (and the case $p = p^{'}$ leaves the constant $1$ in the first term).

2.1step 1.1F1F4algebra

[Eigen-identities] Differentiating the formulas of step 1.1 once more in the same coordinate gives $\partial_{1}^{2} h_{m n} = - (\frac{m π}{L_{1}})^{2} h_{m n}$ and $\partial_{2}^{2} h_{m n} = - (\frac{n π}{L_{2}})^{2} h_{m n}$ on $R$ , so $Δ h_{m n} = - k_{m n}^{2} h_{m n}$ by [F1]; the same computation gives $\partial_{1}^{2} e_{m n} = - (\frac{m π}{L_{1}})^{2} e_{m n}$ , $\partial_{2}^{2} e_{m n} = - (\frac{n π}{L_{2}})^{2} e_{m n}$ and $Δ e_{m n} = - k_{m n}^{2} e_{m n}$ . Since each open face of $\partial R$ carries one of the normals $\pm e_{1}, \pm e_{2}$ and the corresponding normal derivative $\partial_{1} h_{m n}$ or $\partial_{2} h_{m n}$ vanishes on it by step 1.1, clause (a) and clause (b) hold.

2.2step 1.2F2algebra

[Rectangle orthogonality, norms and mixed integrals] Each of the functions $h_{m n} h_{m^{'} n^{'}}$ , $e_{m n} e_{m^{'} n^{'}}$ and $h_{m n} e_{p q}$ is continuous on the closed rectangle $\overline{R} = [0, L_{1}] \times [0, L_{2}]$ , hence Riemann integrable there by [F2]; their sections in each coordinate are continuous, hence integrable. Fubini's theorem [F2] therefore factorizes each integral into a product of one-dimensional integrals. By step 1.2, $\int_{R} h_{m n} h_{m^{'} n^{'}} = \frac{L_{1} L_{2}}{4} (1 + δ_{m 0}) (1 + δ_{n 0})$ if $(m, n) = (m^{'}, n^{'})$ and $= 0$ otherwise, and $\int_{R} e_{m n} e_{m^{'} n^{'}} = \frac{L_{1} L_{2}}{4}$ if $(m, n) = (m^{'}, n^{'})$ and $= 0$ otherwise. For the mixed integral, $\int_{R} h_{m n} e_{p q} = J_{m p} (L_{1}) J_{n q} (L_{2})$ with the stated $J_{m p}$ .

3.1step 2.2F3F5F6algebra

[Evaluation of $J$ ] Let $m \geq 0$ , $p \geq 1$ and $L > 0$ . By [F6], $\cos \frac{m π x}{L} \sin \frac{p π x}{L} = \frac{1}{2} [\sin \frac{(p + m) π x}{L} + \sin \frac{(p - m) π x}{L}]$ . If $p = m$ , the second sine vanishes identically and the integral of the first over $[0, L]$ is $\frac{L}{2 π m} (1 - \cos (2 m π)) = 0$ by the fundamental theorem [F3] and [F5] (here $m = p \geq 1$ ); thus $J_{m m} = 0$ . If $p \neq m$ , the fundamental theorem gives $J_{m p} = \frac{L}{2 π} [\frac{1 - \cos ((p + m) π)}{p + m} + \frac{1 - \cos ((p - m) π)}{p - m}] = \frac{L}{2 π} (1 - (- 1)^{p + m}) [\frac{1}{p + m} + \frac{1}{p - m}] = \frac{p L}{π (p^{2} - m^{2})} (1 - (- 1)^{p + m})$ , using $\cos (k π) = (- 1)^{k}$ and $\frac{1}{p + m} + \frac{1}{p - m} = \frac{2 p}{p^{2} - m^{2}}$ . Hence $J_{m p} = 0$ when $m, p$ have matching parity (the case $p = m$ included) and $J_{m p} = \frac{2 p L}{π (p^{2} - m^{2})}$ when their parities differ. In particular $J_{01} (L) = \frac{2 L}{π}$ for $L = L_{1}, L_{2}$ , so $\int_{R} h_{00} e_{11} = J_{01} (L_{1}) J_{01} (L_{2}) = \frac{4 L_{1} L_{2}}{π^{2}}$ .

4.1step 1.1step 2.1F2F3F4algebra∎

[Dirichlet-type identities] Fix $x_{2} \in (0, L_{2})$ and put $G_{1} (x_{1}) : = h_{m n} (x_{1}, x_{2}) \partial_{1} h_{m n} (x_{1}, x_{2})$ on $[0, L_{1}]$ . By the product rule [F4], $G_{1}$ is differentiable on $[0, L_{1}]$ with $G_{1}^{'} = (\partial_{1} h_{m n})^{2} + h_{m n} \partial_{1}^{2} h_{m n}$ , a continuous hence integrable function, and $G_{1} (0) = G_{1} (L_{1}) = 0$ by step 1.1; the fundamental theorem [F3] gives $\int_{0}^{L_{1}} [(\partial_{1} h_{m n})^{2} + h_{m n} \partial_{1}^{2} h_{m n}] d x_{1} = 0$ . The same argument in the second coordinate, and step 2.1 (respectively step 1.1 for the vanishing trace), give $\int_{0}^{L_{2}} [(\partial_{2} h_{m n})^{2} + h_{m n} \partial_{2}^{2} h_{m n}] d x_{2} = 0$ , and the identical argument with $e_{m n}$ in place of $h_{m n}$ gives $\int_{0}^{L_{1}} [(\partial_{1} e_{m n})^{2} + e_{m n} \partial_{1}^{2} e_{m n}] d x_{1} = 0$ and its second-coordinate counterpart, the boundary terms vanishing because $e_{m n}$ is zero on $\partial R$ . Integrating the resulting pointwise identities over the rectangle against the other coordinate and applying Fubini [F2] (all integrands are continuous on $\overline{R}$ ), then substituting $\partial_{1}^{2} h_{m n} = - (\frac{m π}{L_{1}})^{2} h_{m n}$ , $\partial_{2}^{2} h_{m n} = - (\frac{n π}{L_{2}})^{2} h_{m n}$ and the analogous two identities for $e_{m n}$ from step 2.1, yields both statements of clause (d).

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The sine and cosine systems are complete orthogonal systems of L^2 on an interval and on a rectangle

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $L > 0$ and $R = (0, L_{1}) \times (0, L_{2})$ with $L_{1}, L_{2} > 0$ . (a) [One interval] The families ${\cos (m π x / L)}_{m \geq 0}$ and ${\sin (m π x / L)}_{m \geq 1}$ are orthogonal complete systems in the real Hilbert space $L^{2} (0, L)$ (The space $L^{p} (μ)$ as the quotient by null functions, $L^{2}$ with the integral pairing is a Hilbert space, Orthonormal families, complete orthonormal systems and Hilbert bases): each is orthogonal, and its closed linear span is all of $L^{2} (0, L)$ . (b) [Rectangle] With $m, n$ ranging over $N_{0}$ in cosine slots and over $N_{\geq 1}$ in sine slots, each of the four product families ${\cos \frac{m π x_{1}}{L_{1}} \cos \frac{n π x_{2}}{L_{2}}}$ , ${\cos \frac{m π x_{1}}{L_{1}} \sin \frac{n π x_{2}}{L_{2}}}$ , ${\sin \frac{m π x_{1}}{L_{1}} \cos \frac{n π x_{2}}{L_{2}}}$ and ${\sin \frac{m π x_{1}}{L_{1}} \sin \frac{n π x_{2}}{L_{2}}}$ is an orthogonal complete system in $L^{2} (R)$ ; in particular the Neumann family ${h_{m n}}_{m, n \geq 0}$ and the Dirichlet family ${e_{m n}}_{m, n \geq 1}$ of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals are orthogonal complete systems in $L^{2} (R)$ .

Facts & Assumptions

Given: The Axiom of Countable Choice; a length $L > 0$ , rectangle dimensions $L_{1}, L_{2} > 0$ ; the real Hilbert space $L^{2} (0, L)$ (similarly $L^{2} (R)$ ) of The space $L^{p} (μ)$ as the quotient by null functions with inner product $⟨ f, g ⟩ = \int f g$ and norm $∥ \cdot ∥_{2}$ ( $L^{2}$ with the integral pairing is a Hilbert space); and the one-interval families $c_{m} (x) : = \cos (m π x / L)$ ( $m \geq 0$ ), $s_{p} (x) : = \sin (p π x / L)$ ( $p \geq 1$ ).

[F1]

On real $L^{2} (μ)$ the formula $⟨ f, g ⟩ = \int f g d μ$ is an inner product with $⟨ f, f ⟩ = ∥ f ∥_{2}^{2}$ , $L^{2} (μ)$ is complete for the induced norm, and $∣ \int f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ for $f, g \in L^{2}$ ( $L^{2}$ with the integral pairing is a Hilbert space, Cauchy-Schwarz inequality for $L^{2}$ , The space $L^{p} (μ)$ as the quotient by null functions). A family is orthogonal when its distinct members pair to $0$ and complete when its closed linear span is the whole space (Orthonormal families, complete orthonormal systems and Hilbert bases).

[F2]

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the torus $T = R / Z$ carries the normalized Haar integral $m_{T}$ and the characters $e_{k} ([t]) = \exp (2 π i k t)$ , and the characters form an orthonormal basis of the complex Hilbert space $L^{2} (T; C)$ (The one-dimensional torus and its normalized Haar integral, The trigonometric system is complete in $L^{2}$ of the torus); for real $φ$ , $\exp (i φ) = \cos φ + i \sin φ$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F3]

Change of variables: invertible linear maps $T$ scale Lebesgue measure by $∣ \det T ∣$ (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not), and a $C^{1}$ diffeomorphism $T : U \to V$ substitutes in $L^{1}$ integrands by $\int_{V} f = \int_{U} f \circ T ∣ \det D T ∣$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions); both hold under ${A C}_{ω}$ . In particular the affine substitutions $t \mapsto 2 L t$ and $x \mapsto - x$ below preserve integrability and transform the displayed integrals.

[F4]

Fubini for $L^{1}$ : for sigma-finite spaces and $h \in L^{1} (μ \times ν)$ , the sections are a.e. integrable, the section-integral functions are integrable, and the iterated and double integrals agree (Fubini's theorem for L^1 functions on a sigma-finite product).

[F5]

One-variable calculus: $(\sin x)^{'} = \cos x$ , $(\cos x)^{'} = - \sin x$ (The derivatives of sine and cosine are cosine and minus sine); sums, scalar multiples and products obey the sum, scalar and product rules and composites obey the chain rule, applicable to affine substitutions (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: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ); a differentiable $G$ on $[a, b]$ with integrable $G^{'}$ satisfies $\int_{a}^{b} G^{'} = G (b) - G (a)$ (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 continuous function on $[a, b]$ is Riemann integrable (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion) and a bounded Riemann integrable function is Lebesgue measurable with the same integral (A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral).

[F6]

$\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\cos (A + B) = \cos A \cos B - \sin A \sin B$ for real $A, B$ (The addition formulas for sine and cosine); sine is odd, cosine is even, and $\sin^{2} + \cos^{2} = 1$ (Parity and the Pythagorean identity for sine and cosine).

[F7]

Every $f \in L^{2} (R)$ satisfies $\int_{R} ∣ f ∣^{2} < + \infty$ ; bounded measurable factors on the finite-measure rectangle preserve $L^{1}$ and $L^{2}$ membership, and $\int_{R} ∣ f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ is the Cauchy–Schwarz estimate (Cauchy-Schwarz inequality for $L^{2}$ , The space $L^{p} (μ)$ as the quotient by null functions, $L^{2}$ with the integral pairing is a Hilbert space, Fubini's theorem for L^1 functions on a sigma-finite product).

[F8]

Assume ${A C}_{ω}$ ; if $(e_{i})_{i \in I}$ is a complete orthonormal family in a real or complex Hilbert space and $x$ is in the space, then the finite-subset net of partial sums $\sum_{i \in F} ⟨ x, e_{i} ⟩ e_{i}$ converges to $x$ (Fourier expansion in a Hilbert space).

[F9]

The profiles of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals satisfy $h_{m n} = c_{m}^{(1)} \otimes c_{n}^{(2)}$ and $e_{m n} = s_{m}^{(1)} \otimes s_{n}^{(2)}$ , where the superscripts select the $x_{1}$ - and $x_{2}$ -families, and the lemma states their rectangle orthogonality and norms. The four families of clause (b) are the products $c^{(i)} \otimes d^{(j)}$ of one cosine or sine family on each factor.

Proof

technique · direct: compute the one-dimensional inner products with the product-to-sum identities and the fundamental theorem, transfer completeness of the trigonometric system of the torus to the interval through even and odd periodic extensions, and obtain the rectangle families by Fubini and one completeness argument applied on each factor

1.1F5F6algebra

[One-interval inner products] Let $m, m^{'} \geq 0$ and $p, p^{'} \geq 1$ be integers. By [F6], $\cos \frac{m π x}{L} \cos \frac{m^{'} π x}{L} = \frac{1}{2} [\cos \frac{(m - m^{'}) π x}{L} + \cos \frac{(m + m^{'}) π x}{L}]$ and $\sin \frac{p π x}{L} \sin \frac{p^{'} π x}{L} = \frac{1}{2} [\cos \frac{(p - p^{'}) π x}{L} - \cos \frac{(p + p^{'}) π x}{L}]$ ; for every nonzero integer $k$ the function $x \mapsto \frac{L}{k π} \sin \frac{k π x}{L}$ has derivative $\cos \frac{k π x}{L}$ on $[0, L]$ by [F5], so $\int_{0}^{L} \cos \frac{k π x}{L} d x = \frac{L}{k π} [\sin (k π) - \sin 0] = 0$ by [F5], [F6] (the case $k = 0$ , i.e. $m = m^{'}$ or $p = p^{'}$ , leaves the constant $1$ or the difference $0$ ). The integrands are continuous, hence Riemann integrable [F5], and their Riemann and Lebesgue integrals agree [F5]. Therefore $⟨ c_{m}, c_{m^{'}} ⟩ = 0$ for $m \neq m^{'}$ , $⟨ s_{p}, s_{p^{'}} ⟩ = 0$ for $p \neq p^{'}$ , while $⟨ c_{m}, c_{m} ⟩ = \frac{L}{2} (1 + δ_{m 0})$ and $⟨ s_{p}, s_{p} ⟩ = \frac{L}{2}$ are nonzero; each displayed member is nonzero because it equals $1$ at some point of $[0, L]$ . Hence both families of clause (a) are orthogonal in the sense of [F1].

1.2F1F2F3F6algebra

[Even extensions have vanishing torus coefficients] Let $f \in L^{2} (0, L)$ satisfy $⟨ f, c_{m} ⟩ = 0$ for every $m \geq 0$ . Define $F (x) : = f (∣ x ∣)$ on $(- L, L)$ and let $G$ be the $2 L$ -periodic extension of $F$ to $R$ ; define $g ([t]) : = G (2 L t)$ on $T$ . Then $g$ is a real-valued element of $L^{2} (T; C)$ : by [F1], [F3] the change of variables $x \mapsto 2 L t$ and the periodicity give $\int_{T} ∣ g ∣^{2} d m_{T} = \int_{0}^{1} ∣ G (2 L t) ∣^{2} d t = \frac{1}{2 L} \int_{0}^{2 L} ∣ G ∣^{2} = \frac{1}{2 L} \int_{- L}^{L} ∣ F ∣^{2} = \frac{1}{L} ∥ f ∥_{2}^{2} < \infty$ . Splitting the integral at $0$ , substituting $x \mapsto - x$ on $(- L, 0)$ by [F3] and using $F (- x) = F (x)$ , its character coefficients are $\hat{g} (k) = \int_{0}^{1} G (2 L t) e^{- 2 π i k t} d t = \frac{1}{2 L} \int_{- L}^{L} F (x) e^{- i π k x / L} d x = \frac{1}{2 L} \int_{0}^{L} f (x) (e^{- i π k x / L} + e^{i π k x / L}) d x = \frac{1}{L} \int_{0}^{L} f (x) \cos \frac{k π x}{L} d x = \frac{1}{L} ⟨ f, c_{∣ k ∣} ⟩ = 0$ for every $k \in Z$ , using the affine substitutions, the periodicity of $G$ , the cartesian form of [F2] and the parity of [F6] for the identification with $\cos \frac{k π x}{L} = \cos \frac{∣ k ∣ π x}{L}$ .

1.3F1F2algebra

[Trivial orthogonal complement in $L^{2}$ ] We record the Hilbert-space step used below. Let $H$ be a real or complex Hilbert space, let $V \subseteq H$ be the closed linear span of a family $(v_{i})_{i \in I}$ , and suppose the only $w \in H$ with $⟨ w, v_{i} ⟩ = 0$ for every $i$ is $w = 0$ . Then $V = H$ : for $u \in H$ put $d : = \inf_{v \in V} ∥ u - v ∥$ ; under [F2] (countable choice) choose $v_{n} \in V$ with $∥ u - v_{n} ∥ \to d$ , and the parallelogram identity $∥ v_{n} - v_{m} ∥^{2} = 2 ∥ u - v_{n} ∥^{2} + 2 ∥ u - v_{m} ∥^{2} - 4 ∥ u - \frac{v_{n} + v_{m}}{2} ∥^{2} \leq 2 ∥ u - v_{n} ∥^{2} + 2 ∥ u - v_{m} ∥^{2} - 4 d^{2}$ of [F1] makes $(v_{n})$ Cauchy, so $v_{n} \to v_{*} \in V$ by closedness and completeness; if $d > 0$ then $z : = u - v_{*} \neq 0$ and for every $w \in V$ and real $ε$ one has $d^{2} \leq ∥ z - ε w ∥^{2} = d^{2} - 2 ε Re ⟨ z, w ⟩ + ε^{2} ∥ w ∥^{2}$ , so taking $ε = Re ⟨ z, w ⟩ / ∥ w ∥^{2}$ for $w \neq 0$ forces $Re ⟨ z, w ⟩ = 0$ ; when $H$ is complex, applying that conclusion to $i w$ in place of $w$ gives $Im ⟨ z, w ⟩ = 0$ as well, so $⟨ z, w ⟩ = 0$ in either case; in particular $⟨ z, v_{i} ⟩ = 0$ for all $i$ , contradicting the hypothesis, so $d = 0$ and $u \in V$ .

1.4F1F2

[Torus elements with all coefficients zero vanish] If $h \in L^{2} (T; C)$ has $\hat{h} (k) = 0$ for every $k$ , then $h = 0$ : by [F2] the characters are complete orthonormal, so Cauchy–Schwarz [F1] makes $⟨ h, φ ⟩ = 0$ for every $φ$ in their closed span, which is all of $L^{2} (T; C)$ , and $⟨ h, h ⟩ = 0$ gives $h = 0$ .

2.1step 1.2F1F2F3F6algebra

[Odd extensions have vanishing torus coefficients] Let $f \in L^{2} (0, L)$ satisfy $⟨ f, s_{p} ⟩ = 0$ for every $p \geq 1$ . Define $F (x) : = sgn (x) f (∣ x ∣)$ (so $F$ is odd and $F = f$ on $(0, L)$ ) and define $G, g$ from $F$ as in step 1.2. The same change of variables and periodicity give $∥ g ∥^{2} = \frac{1}{L} ∥ f ∥_{2}^{2} < \infty$ , and now the reflection substitution yields $\hat{g} (k) = \frac{1}{2 L} \int_{- L}^{L} F (x) e^{- i π k x / L} d x = \frac{1}{2 L} \int_{0}^{L} f (x) (e^{- i π k x / L} - e^{i π k x / L}) d x = - \frac{i}{L} \int_{0}^{L} f (x) \sin \frac{k π x}{L} d x$ ; by the parity of sine this is $- \frac{i sgn (k)}{L} ⟨ f, s_{∣ k ∣} ⟩ = 0$ for $k \neq 0$ and $0$ for $k = 0$ .

2.2step 1.1step 1.2step 1.3step 1.4F3

[Interval completeness, cosine case] Let $f \in L^{2} (0, L)$ be orthogonal to every $c_{m}$ . Step 1.2 produces $g \in L^{2} (T; C)$ whose character coefficients all vanish, so $g = 0$ by step 1.4, and hence $G (2 L t) = 0$ for almost every $t \in [0, 1)$ . By [F3] (the linear map $t \mapsto 2 L t$ scales Lebesgue measure by $2 L$ ) the set of $x \in (0, 2 L)$ with $G (x) \neq 0$ is null, so $F$ , hence $f$ , vanishes almost everywhere on $(0, L)$ . Thus the only element of $L^{2} (0, L)$ orthogonal to the cosine family is $0$ ; by step 1.3 with the family $(c_{m})$ the closed span of the cosine family is $L^{2} (0, L)$ , and step 1.1 gives orthogonality.

2.3step 1.1F4F7algebra

[Rectangle orthogonality] Let $c$ denote either of the two $x_{1}$ -families and $d$ either of the two $x_{2}$ -families of clause (b); all members are continuous, hence bounded and measurable on the closed rectangle. For members $c_{m} \otimes d_{n}$ and $c_{m^{'}} \otimes d_{n^{'}}$ , the product $(c_{m} c_{m^{'}}) (d_{n} d_{n^{'}})$ is bounded and measurable on the finite-measure rectangle, hence in $L^{1}$ , and Fubini [F4] factors the inner product as $⟨ c_{m} \otimes d_{n}, c_{m^{'}} \otimes d_{n^{'}} ⟩ = ⟨ c_{m}, c_{m^{'}} ⟩ ⟨ d_{n}, d_{n^{'}} ⟩$ ; this vanishes whenever $(m, n) \neq (m^{'}, n^{'})$ by step 1.1 (a member with a matching index has nonzero squared norm $\frac{L_{1}}{2} (1 + δ_{m 0})$ or $\frac{L_{1}}{2}$ , or the analogous $x_{2}$ -norm). Hence each of the four product families is orthogonal in $L^{2} (R)$ .

3.1step 1.1step 1.3step 1.4step 2.1F3

[Interval completeness, sine case] The same argument with step 2.1 in place of step 1.2 shows that the only element of $L^{2} (0, L)$ orthogonal to the sine family is $0$ , so its closed span is $L^{2} (0, L)$ by step 1.3, and it is orthogonal by step 1.1; this proves clause (a) together with step 2.2.

4.1step 1.3step 2.2step 2.3step 3.1F1F4F7F9choose

[Rectangle completeness] Fix one of the four product families $c \otimes d$ in $L^{2} (R)$ and let $f \in L^{2} (R)$ be orthogonal to every $c_{m} \otimes d_{n}$ . By Fubini applied to the integrable function $∣ f ∣^{2}$ [F4], the section $f (\cdot, x_{2})$ lies in $L^{2} (0, L_{1})$ for almost every $x_{2}$ , and the identities of [F4] applied to the integrable function $f c_{m}$ define measurable functions $φ_{m} (x_{2}) : = \int_{0}^{L_{1}} f (x_{1}, x_{2}) c_{m} (x_{1}) d x_{1}$ almost everywhere, with $\int_{0}^{L_{2}} ∣ φ_{m} ∣^{2} \leq ∥ c_{m} ∥_{2}^{2} ∥ f ∥_{2}^{2} < \infty$ by Cauchy–Schwarz [F7], so $φ_{m} \in L^{2} (0, L_{2})$ ; and Fubini applied to $f c_{m} d_{n}$ gives $⟨ φ_{m}, d_{n} ⟩ = ⟨ f, c_{m} \otimes d_{n} ⟩ = 0$ for every $n$ . Since the $d$ -family is complete in $L^{2} (0, L_{2})$ by step 2.2 or step 3.1, Cauchy–Schwarz [F1] makes $φ_{m}$ orthogonal to the whole closed span, so $φ_{m} = 0$ almost everywhere. There are countably many $m$ , so [F2] (countable choice) selects the countably many exceptional null sets whose union $N$ is null, and for every $x_{2} \notin N$ the section $f (\cdot, x_{2}) \in L^{2} (0, L_{1})$ is orthogonal to every $c_{m}$ , hence vanishes almost everywhere in $x_{1}$ by step 2.2 or step 3.1. Fubini applied to $∣ f ∣^{2}$ [F4] now gives $∥ f ∥_{2}^{2} = 0$ , so $f = 0$ . By step 1.3 the closed span of the product family is $L^{2} (R)$ ; with step 2.3 this proves the first claim of clause (b) for all four parity choices, and the identification of [F9] gives the claim for the Neumann family ${h_{m n}}$ and the Dirichlet family ${e_{m n}}$ .

5.1step 1.1step 2.2step 2.3step 3.1step 4.1F1F8algebra∎

[Expansions] For each system $(u_{i})$ of clauses (a) and (b), put $v_{i} : = u_{i} / ∥ u_{i} ∥_{2}$ ; the norms are nonzero and $(v_{i})$ is a complete orthonormal system of [F1]. By [F8], every $f$ in the space is the norm limit of the finite-subset net $\sum_{i \in F} ⟨ f, v_{i} ⟩ v_{i}$ . Since $⟨ f, v_{i} ⟩ v_{i} = (⟨ f, u_{i} ⟩ / ∥ u_{i} ∥_{2}^{2}) u_{i}$ , the corresponding coefficients in the original orthogonal system are $⟨ f, u_{i} ⟩ / ∥ u_{i} ∥_{2}^{2}$ , obtained by dividing the normalized coefficients by $∥ u_{i} ∥_{2}$ .

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Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM

Physical scope

The ideal PEC boundary model of the straight rectangular guide and cavity (and of straight guides with a bounded connected C1 cross-section) in the declared laboratory chart, with lossless and explicitly declared constant scalar Ohmic fillings distinguished by their consuming theorems, together with the zero-normal-B reflection class and the TE/TM/TEM classification. The boundary model is an idealization of an interface with a perfectly conducting medium on the open walls; the definition asserts no field equations, constitutive laws or mode completeness.

Definition

Let the laboratory region $U ≅ I \times Ω$ , its frame basis $(e_{1}, e_{2}, e_{3})$ , its SI units and its fields $E, B$ be those of 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 and The electric and magnetic fields as fixed-time spatial vector fields. Fix $a, b > 0$ , write $x_{t} = (x_{1}, x_{2})$ , $e_{z} : = e_{3}$ , and define $\nabla_{t} f = (\partial_{1} f, \partial_{2} f, 0)$ , $Δ_{t} f = \partial_{1}^{2} f + \partial_{2}^{2} f$ and ${div}_{t} u_{t} = \partial_{1} u_{1} + \partial_{2} u_{2}$ for the stated regularity, using Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Divergence and curl of a $C^{1}$ vector field and The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field. (a) [Straight guide] A straight guide region is $Ω_{A} : = A \times R$ where $A$ is either a bounded connected $C^{1}$ plane domain (Bounded C^k domains and boundary charts) or the explicit rectangular cross-section; the rectangle has corners and is treated by its open faces, not asserted to be a $C^{1}$ boundary domain, with longitudinal axis $e_{z}$ ; for a rectangular cross-section $A = R_{a b} : = (0, a) \times (0, b)$ the region is called the rectangular guide and its walls are the four open faces ${x_{1} = 0}, {x_{1} = a}, {x_{2} = 0}, {x_{2} = b}$ . A rectangular cavity region is $Ω_{c} : = (0, a) \times (0, b) \times (0, d)$ for fixed $d > 0$ , with the six open faces as its walls. The separate smooth-cavity spectral branch uses an explicitly declared bounded $C^{\infty}$ domain in $R^{3}$ with its smooth PEC boundary; it is not the rectangular reflection class. Spatial derivatives are taken at fixed $t$ ; the fields are defined on $Ω \times I$ with the regularity declared by each theorem. (b) [PEC boundary model] On every open wall (a wall minus its edges) the ideal perfectly conducting boundary model declares the two conditions $n \times E = 0$ and $n \cdot B = 0$ , where $n$ is the outward unit normal of that wall ( $- e_{1}, + e_{1}, - e_{2}, + e_{2}$ on the four rectangular walls, and $\pm e_{j}$ on the six cavity walls). Its physical reading uses the stationary interface conditions of Stationary macroscopic interface jump conditions for D, B, E and H with $E = B = 0$ on the conductor side: tangential- $E$ and normal- $B$ continuity then give the two filling-side conditions. For that reading write $D_{f i l l}, H_{f i l l}$ and $D_{c o n d}, H_{c o n d}$ for the filling- and conductor-side traces of electric displacement (in $C m^{- 2}$ ) and magnetic intensity (in $A m^{- 1}$ ). Since $n$ points from the filling into the conductor, the general sheet readings are $σ_{f} = n \cdot (D_{c o n d} - D_{f i l l})$ and $K_{f} = n \times (H_{c o n d} - H_{f i l l})$ , in $C m^{- 2}$ and $A m^{- 1}$ respectively. Only with the additional conductor-side assumptions $D_{c o n d} = H_{c o n d} = 0$ do they reduce to $σ_{f} = - n \cdot D_{f i l l}$ and $K_{f} = - n \times H_{f i l l}$ ; $E = B = 0$ alone does not imply those additional assumptions. No sheet density or conductor constitutive relation is prescribed here. The supplier assumes ${A C}_{ω}$ , a compact embedded $C^{2}$ stationary hypersurface, joint one-sided spacetime $C^{1}$ extensions and distributional macroscopic equations with the stated thin sheet sources for its pointwise branch; it covers that smooth interface model, not a deduction of the $σ \to \infty$ idealization nor the box edges. The guide/cavity conditions on the open walls are therefore declared as an idealization of this definition, and no pointwise statement on the edges is made. (c) [Zero-normal-B reflection class] For the rectangular box $Q = \prod_{j = 1}^{3} (0, L_{j})$ with $L = (a, b, d)$ , the reflection class is the class of locally integrable fields $E, B$ on $Q$ whose extensions to the rectangular torus $\prod_{j} R / (2 L_{j} Z)$ , defined by extending $E_{1}$ evenly across the $x_{1}$ -walls and oddly across the $x_{2}$ -, $x_{3}$ -walls and cyclically for $E_{2}, E_{3}$ , and extending $B_{1}$ oddly across $x_{1}$ and evenly across $x_{2}, x_{3}$ and cyclically for $B_{2}, B_{3}$ , are $L^{2}$ fields, together with the distributional constraints that $div E$ and $div B$ vanish on the torus for the extensions. This is the declared weak PEC model used for the cavity initial-value problem; when the reflected extensions are continuous across the walls, their odd components vanish there, giving $n \times E = 0$ and $n \cdot B = 0$ on the open walls. Smoothness only inside $Q$ does not imply this extension continuity. Initial data not in this class (for example data with trapped normal $B$ ) are excluded, as are the general $H (curl)$ boundary classes, which are defined by the mathematical suppliers and are not identified here with the classical conditions. (d) [TE, TM, TEM] For a guide or cavity field $(E, B)$ , the longitudinal components are $E_{z} : = E \cdot e_{z}$ and $B_{z} : = B \cdot e_{z}$ ; the field is called TE (transverse electric) if $E_{z} \equiv 0$ , TM (transverse magnetic) if $B_{z} \equiv 0$ , and TEM if both vanish, always relative to the named axis $e_{z}$ . (e) [Separated phasor convention] For $ω > 0$ and $β \in C$ , a $β$ - $ω$ separated phasor field on the guide is a real field of the form $F (x, t) = Re (\tilde{F} (x_{1}, x_{2}) e^{i (β x_{3} - ω t)})$ with a complex amplitude $\tilde{F}$ , componentwise; $ω / (2 π)$ is the frequency and $β$ the longitudinal propagation constant; when $β$ is real, $β > 0$ describes propagation in the $+ e_{z}$ direction under this phase convention, while a complex $β$ gives the spatial factor $e^{- Im β x_{3}}$ and may grow in the opposite direction; each consuming branch states its selection. (f) [Scope] The definition fixes the geometry, the PEC boundary model, the weak reflection class and the mode terminology; it asserts no field equation, no constitutive law and no existence or completeness of modes, Rectangular TE/TM candidates and the rectangular reflection class, declared smooth bounded PEC cavity spectral classes, and smooth straight-guide TEM potential branches (including the coaxial annulus) are within this geometry. No general non-rectangular TE/TM guide theory, circular-guide spectral classification, open-end or junction scattering, or arbitrary lossy model is supplied. The same stationary PEC geometry also hosts the explicitly declared constant scalar Ohmic cavity class and the full longitudinal finite-energy reflection class of the straight rectangular guide; constitutive and evolution claims belong to their consuming theorems, not this definition.

Conventions fixed by this definition. The definition records objects and conditions only: the guide and cavity regions, their open walls and outward normals, the ideal PEC boundary conditions with their interface reading and excluded idealization, the zero-normal- $B$ reflection class with its distributional divergence constraints, the axis-relative TE/TM/TEM terminology, and the separated phasor encoding of real fields. It asserts no field equation, no constitutive relation, no existence, uniqueness or completeness of modes, and no dissipation or energy claim; those belong to the consuming theorems, each of which restates the branch of the material model it uses. The rectangular cross-section is treated through its four open faces and is not asserted to be a $C^{1}$ boundary domain, and no pointwise condition on box edges is made.

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The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff

Physical scope

The verified separated TE and TM candidate fields of the lossless isotropic scalar constant branch in the straight rectangular perfectly conducting guide with cross-section (0,a)x(0,b), at a declared frequency and longitudinal wave number, including the index exclusions and the cutoff relation. Conditional on the adopted material branch and PEC boundary model; no completeness, lossy or empirical claim.

Statement

Assume the elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in its lossless isotropic scalar case at rest in the laboratory frame: constants $ε, μ > 0$ , $σ = 0$ , $J_{e x t} = 0$ , the constitutive relations $D = ε E$ , $B = μ H$ of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch clause (b), read with the objects $D, H$ of Polarization, magnetization, bound and free sources, and the fields D and H, and the macroscopic equations (b) of The macroscopic Maxwell equations in terms of D, H and the free sources with $ρ_{f} = 0$ , $J_{f} = 0$ on the guide region; suppose also $ε μ \geq c^{- 2}$ whenever the broadband interpretation of that branch is used. Let the rectangular guide, its PEC boundary model, the axis $e_{z}$ , the TE/TM/TEM terminology and the separated phasor convention be those of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM with cross-section $(0, a) \times (0, b)$ , and let $ω > 0$ , $β \in C$ , $k_{c}^{2} : = ω^{2} ε μ - β^{2} \in R_{> 0}$ , $H : = B / μ$ and $θ : = β x_{3} - ω t$ . Write $\tilde{E}, \tilde{H}$ for the complex amplitudes of the separated real fields $E = Re (\tilde{E} e^{i θ})$ , $H = Re (\tilde{H} e^{i θ})$ , so that $B = μ H$ . All fields use SI units: $a, b, x_{j}$ carry $m$ , $t$ carries $s$ , $ω$ carries $s^{- 1}$ , $β, k_{c}, k_{m n}$ carry $m^{- 1}$ , $ε$ carries $F m^{- 1}$ , $μ$ carries $H m^{- 1}$ , $\tilde{E}$ carries $V m^{- 1}$ and $\tilde{H}$ carries $A m^{- 1}$ . The scalar profiles $h_{m n}, e_{m n}$ below are dimensionless. Fix positive real constant longitudinal amplitude quantities $H_{*}$ in $A m^{- 1}$ and $E_{*}$ in $V m^{- 1}$ ; the unit-amplitude choices are $H_{*} = 1 A m^{- 1}$ for TE and $E_{*} = 1 V m^{- 1}$ for TM. (a) [TE family] For integers $m, n \geq 0$ with $(m, n) \neq (0, 0)$ put $h_{m n} : = \cos \frac{m π x_{1}}{a} \cos \frac{n π x_{2}}{b}$ , $k_{m n}^{2} : = (\frac{m π}{a})^{2} + (\frac{n π}{b})^{2}$ , and require $β^{2} = ω^{2} ε μ - k_{m n}^{2}$ ; define the complex amplitudes $\tilde{E} : = - H_{*} \frac{i ω μ}{k_{m n}^{2}} e_{z} \times \nabla_{t} h_{m n}, \tilde{H} : = H_{*} (\frac{i β}{k_{m n}^{2}} \nabla_{t} h_{m n} + e_{z} h_{m n}) .$ Then the real fields $E = Re (\tilde{E} e^{i θ})$ , $B = μ Re (\tilde{H} e^{i θ})$ are $C^{\infty}$ on the guide region, have $E_{z} = 0$ (TE), satisfy the four source-free macroscopic equations $div D = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl H = \partial_{t} D$ , and satisfy $n \times E = 0$ and $n \cdot B = 0$ on every open wall of the guide. (b) [TM family] For integers $m, n \geq 1$ put $e_{m n} : = \sin \frac{m π x_{1}}{a} \sin \frac{n π x_{2}}{b}$ , $k_{m n}^{2} : = (\frac{m π}{a})^{2} + (\frac{n π}{b})^{2}$ , require $β^{2} = ω^{2} ε μ - k_{m n}^{2}$ , and define $\tilde{E} : = E_{*} (\frac{i β}{k_{m n}^{2}} \nabla_{t} e_{m n} + e_{z} e_{m n}), \tilde{H} : = E_{*} \frac{i ω ε}{k_{m n}^{2}} e_{z} \times \nabla_{t} e_{m n} .$ Then the corresponding real fields are $C^{\infty}$ , have $B_{z} = 0$ (TM), satisfy the same four equations and the same boundary conditions on the open walls. (c) [Index restrictions] For $m = n = 0$ one has $\nabla_{t} h_{00} = 0$ and $k_{00}^{2} = 0$ makes the displayed divided reconstruction formulas undefined. A purely longitudinal constant-profile magnetic amplitude with zero electric amplitude cannot be a nonzero monochromatic Maxwell candidate: Faraday gives $0 = i ω μ \tilde{H}$ , hence $\tilde{H} = 0$ because $ω > 0$ . At $β = 0$ a constant $H_{z}$ is divergence-free, so divergence alone does not exclude it. Thus $(0, 0)$ is excluded by the full field equations and the reconstruction denominator, not by a false divergence claim; for $m = 0$ or $n = 0$ the TM profile $e_{m n}$ vanishes identically, so TM indices are restricted to $m, n \geq 1$ ; conversely for $(m, n) \neq (0, 0)$ (resp. $m, n \geq 1$ ) the displayed profiles and fields are nonzero. (d) [Dispersion and cutoff] On the declared family, $k_{c}^{2} = ω^{2} ε μ - β^{2}$ defines the cutoff $ω_{c} : = k_{c} / \sqrt{ε μ} > 0$ ; real longitudinal wave numbers with $β^{2} = ω^{2} ε μ - k_{c}^{2}$ exist exactly for $ω \geq ω_{c}$ , with $β \neq 0$ exactly for $ω > ω_{c}$ and $β = 0$ at $ω = ω_{c}$ ; for $ω < ω_{c}$ the two algebraic roots are $β = \pm i \sqrt{k_{c}^{2} - ε μ ω^{2}}$ , and the same verified separated fields have the factor $e^{- Im β x_{3}}$ ; a declared half-guide may select the decaying sign, and no propagating-power claim is made there. (e) [Scope] These are separated candidate modes of the declared lossless model; the theorem asserts no completeness, no uniqueness among fields with the same data, and no lossy, dispersive or open-end behaviour, and it assumes the PEC model of the definition, not the physical infinite-conductivity limit.

Facts & Assumptions

Given: The elementary inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch in its isotropic lossless scalar case at rest in the laboratory frame: constants $ε, μ > 0$ , $σ = 0$ , $J_{e x t} = 0$ , the relations $D = ε E$ , $B = μ H$ of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch clause (b), read with the objects $D, H$ of Polarization, magnetization, bound and free sources, and the fields D and H, and the macroscopic equations (b) of The macroscopic Maxwell equations in terms of D, H and the free sources with $ρ_{f} = 0$ , $J_{f} = 0$ on the guide region; the rectangular guide, its PEC model, its axis $e_{z}$ , the TE/TM/TEM terminology and the separated phasor convention of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM with cross-section $(0, a) \times (0, b)$ ; $ω > 0$ , $β \in C$ , $k_{c}^{2} = ω^{2} ε μ - β^{2} > 0$ , $θ (x_{3}, t) = β x_{3} - ω t$ , $H = B / μ$ , the constant longitudinal amplitudes $H_{*}, E_{*} > 0$ with the SI units of the Statement; and separated real fields $E = Re (\tilde{E} e^{i θ})$ , $H = Re (\tilde{H} e^{i θ})$ ; for clause (a), integers $m, n \geq 0$ with $(m, n) \neq (0, 0)$ and $β^{2} = ω^{2} ε μ - k_{m n}^{2}$ ; for clause (b), integers $m, n \geq 1$ and the same relation; the real and imaginary parts of complex numbers are those of Real and imaginary parts, complex conjugation, and modulus and complex arithmetic is that of The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ .

[F1]

In the stated branch, $D = ε E$ and $B = μ H$ with constant $ε, μ > 0$ , and with $ρ_{f} = 0$ , $J_{f} = 0$ the four equations $div D = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl H = \partial_{t} D$ hold on the guide region under the stated regularity (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch clause (b), The macroscopic Maxwell equations in terms of D, H and the free sources clause (b), Polarization, magnetization, bound and free sources, and the fields D and H); the isotropic broadband condition $ε μ \geq c^{- 2}$ is required only when the broadband interpretation of the branch is used.

[F2]

The guide region is $Ω_{A} = (0, a) \times (0, b) \times R$ with axis $e_{z} = e_{3}$ ; its open walls are ${x_{1} = 0}, {x_{1} = a}$ with outward normals $- e_{1}, + e_{1}$ and ${x_{2} = 0}, {x_{2} = b}$ with outward normals $- e_{2}, + e_{2}$ , and the ideal PEC model declares $n \times E = 0$ and $n \cdot B = 0$ on every open wall; a $β$ - $ω$ separated field is $F = Re (\tilde{F} e^{i (β x_{3} - ω t)})$ with complex amplitude $\tilde{F}$ , TE means $E_{z} \equiv 0$ , TM means $B_{z} \equiv 0$ (Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM clauses (a),(b),(d),(e)).

[F3]

Under the separated convention, Calculus of separated phasor fields: real-part derivative identities and amplitude injectivity gives, componentwise and with $Re$ the real part: (a) $\partial_{x_{j}} F = Re (\partial_{j} \tilde{F} e^{i θ})$ , $\partial_{x_{3}} F = Re (i β \tilde{F} e^{i θ})$ , $\partial_{t} F = Re (- i ω \tilde{F} e^{i θ})$ ; (b) $div F = Re ((\partial_{1} {\tilde{F}}_{1} + \partial_{2} {\tilde{F}}_{2} + i β {\tilde{F}}_{3}) e^{i θ})$ ; (c) $curl F = Re ((\partial_{2} {\tilde{F}}_{3} - i β {\tilde{F}}_{2}, i β {\tilde{F}}_{1} - \partial_{1} {\tilde{F}}_{3}, \partial_{1} {\tilde{F}}_{2} - \partial_{2} {\tilde{F}}_{1}) e^{i θ})$ ; (d) if $Re (C e^{i φ_{1}}) = Re (C e^{i φ_{2}}) = 0$ for a complex vector $C$ and real phases with $φ_{1} - φ_{2} \notin π Z$ , then $C = 0$ , and consequently two separated fields that agree on a nonempty open subset have equal amplitudes on its projection to $Ω_{t}$ . In the sequel write $\overset{}{div} (v) : = \partial_{1} v_{1} + \partial_{2} v_{2} + i β v_{3}$ and $\overset{}{curl} (v) : = (\partial_{2} v_{3} - i β v_{2}, i β v_{1} - \partial_{1} v_{3}, \partial_{1} v_{2} - \partial_{2} v_{1})$ for a $C^{1}$ complex amplitude $v$ .

[F4]

Transverse vector-calculus identities for the splitting $R^{3} = R^{2} \oplus R e_{z}$ gives for a $C^{2}$ transverse profile $h$ and a $C^{1}$ transverse map $u_{t}$ , regarded as $z$ -independent fields on $Ω_{t} \times R$ : $div (e_{z} \times \nabla_{t} h) = 0$ , $curl (e_{z} \times \nabla_{t} h) = e_{z} Δ_{t} h$ , $e_{z} \times (e_{z} \times \nabla_{t} h) = - \nabla_{t} h$ , $v \cdot (e_{z} \times v) = 0$ , $e_{z} \cdot (e_{z} \times v) = 0$ , $∣ e_{z} \times v ∣ = ∣ v ∣$ for $v = \nabla_{t} h$ , $curl (e_{z} h) = - e_{z} \times \nabla_{t} h$ , $div (e_{z} h) = \partial_{3} h$ , $curl (u_{t}) = e_{z} {curl}_{2} u_{t}$ , $div (u_{t}) = {div}_{t} u_{t}$ , and $e_{z} \times u_{t}$ is transverse.

[F5]

The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals gives for $h_{m n} = \cos \frac{m π x_{1}}{a} \cos \frac{n π x_{2}}{b}$ and $e_{m n} = \sin \frac{m π x_{1}}{a} \sin \frac{n π x_{2}}{b}$ , with $k_{m n}^{2} = (\frac{m π}{a})^{2} + (\frac{n π}{b})^{2}$ : $Δ_{t} h_{m n} = - k_{m n}^{2} h_{m n}$ , $\partial_{1} h_{m n} = 0$ on $x_{1} \in {0, a}$ , $\partial_{2} h_{m n} = 0$ on $x_{2} \in {0, b}$ ; and for $m, n \geq 1$ , $Δ_{t} e_{m n} = - k_{m n}^{2} e_{m n}$ and $e_{m n} = 0$ on $\partial R$ ; in particular at the corners the vanishing statements hold and $k_{m n}^{2} > 0$ whenever $(m, n) \neq (0, 0)$ . By linearity, the eigen-identities and zero traces hold equally for the dimensionful profiles $H_{*} h_{m n}$ and $E_{*} e_{m n}$ .

[F6]

Smoothness rules: $(\sin x)^{'} = \cos x$ , $(\cos x)^{'} = - \sin x$ , the real exponential is smooth with $(\exp)^{'} = \exp$ , sums, scalar multiples and products obey the sum, scalar and product rules, composites obey the chain rule, and partial derivatives are one-variable derivatives along coordinate lines (The derivatives of sine and cosine are cosine and minus sine, The exponential function is smooth and $(\exp)^{'} = \exp$ , 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: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ); for real $x, y$ one has $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , so with $β = p + i q$ and $φ = p x_{3} - ω t$ one has $e^{i θ} = e^{- q x_{3}} (\cos φ + i \sin φ)$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[F7]

For every real $X \geq 0$ there is a unique $Y \geq 0$ with $Y^{2} = X$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ); and $i γ$ with real $γ$ satisfies $(i γ)^{2} = - γ^{2}$ .

Proof

technique · direct: translate the real source-free equations into complex amplitude equations with the separated phasor calculus, verify the amplitude equations and wall traces by the transverse and profile identities, then read off the index restrictions, the dispersion relation and the scope

1.1F1F3

[Equivalence of the real and the amplitude equations] Let $E, H$ be separated fields with amplitudes $\tilde{E}, \tilde{H}$ on the guide region, and use the combinations of [F3]. Then the real equations $div D = 0$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl H = \partial_{t} D$ of [F1] hold if and only if the four amplitude equations $\overset{}{div} (\tilde{E}) = 0$ , $\overset{}{div} (\tilde{H}) = 0$ , $\overset{}{curl} (\tilde{E}) = i ω μ \tilde{H}$ , $\overset{}{curl} (\tilde{H}) = - i ω ε \tilde{E}$ hold on $(0, a) \times (0, b)$ : indeed, since $ε, μ$ are constants, F3,(c) give $div D = ε Re (\overset{}{div} (\tilde{E}) e^{i θ})$ , $div B = μ Re (\overset{}{div} (\tilde{H}) e^{i θ})$ , $curl E = Re (\overset{}{curl} (\tilde{E}) e^{i θ})$ , $curl H = Re (\overset{}{curl} (\tilde{H}) e^{i θ})$ and F3 gives $\partial_{t} B = μ Re (- i ω \tilde{H} e^{i θ})$ , $\partial_{t} D = ε Re (- i ω \tilde{E} e^{i θ})$ , so substituting the amplitude equations and taking real parts reproduces the real equations; conversely, the four actual real residuals are $Re (ε \overset{}{div} (\tilde{E}) e^{i θ})$ , $Re (μ \overset{}{div} (\tilde{H}) e^{i θ})$ , $Re ((\overset{}{curl} (\tilde{E}) - i ω μ \tilde{H}) e^{i θ})$ and $Re ((\overset{}{curl} (\tilde{H}) + i ω ε \tilde{E}) e^{i θ})$ . If the real equations hold these residuals vanish for all times in a nonempty open interval. At each fixed spatial point, varying time supplies two real phases whose difference is not in $π Z$ because $ω > 0$ ; the nonzero spatial factor $e^{- Im β x_{3}}$ may be cancelled. By F3 each amplitude residual is therefore zero, proving the converse. All steps are complex-linear algebra in the amplitudes.

1.2givenF3F5F6algebra

[Smoothness of the constructed fields] In either family the amplitudes $\tilde{E}, \tilde{H}$ are finite sums of constant multiples of the components of $\nabla_{t} h$ or $\nabla_{t} e$ and of $h$ or $e$ ; the profiles $h, e$ are finite products of functions $\cos \frac{m π x_{1}}{a}$ , $\cos \frac{n π x_{2}}{b}$ , $\sin \frac{m π x_{1}}{a}$ , $\sin \frac{n π x_{2}}{b}$ composed with affine functions, hence are $C^{\infty}$ by [F6] (chain rule for the affine substitutions composed with the sine and cosine derivatives, and the product rule); and $e^{i θ}$ is $C^{\infty}$ by [F6] (the cartesian form exhibits it as $e^{- q x_{3}} (\cos φ + i \sin φ)$ with affine $φ$ , a finite combination of products of smooth functions). Therefore $E = Re (\tilde{E} e^{i θ})$ and $B = μ Re (\tilde{H} e^{i θ})$ , as finite sums of products of $C^{\infty}$ complex functions and of their real parts, have all iterated partial derivatives existing and continuous: differentiating a product once in a coordinate lowers no regularity by [F6], so induction on the order gives $C^{\infty}$ , and in particular every derivative used below exists and is continuous. The factor $(β / k^{2}) \nabla_{t}$ is dimensionless, $(ω μ / k^{2}) \nabla_{t}$ carries impedance $V A^{- 1}$ , and $(ω ε / k^{2}) \nabla_{t}$ carries admittance $A V^{- 1}$ ; multiplication by $H_{*}$ or $E_{*}$ therefore gives the stated electric and magnetic amplitude units.

1.3F3F6F7

[Dispersion relation and cutoff] Let $k_{c}^{2} = ω^{2} ε μ - β^{2} > 0$ and put $ω_{c} : = k_{c} / \sqrt{ε μ} > 0$ ; the number $ω_{c}$ is well defined because $ε μ > 0$ and $k_{c}^{2} > 0$ [F7]. Real $β$ with $β^{2} = ω^{2} ε μ - k_{c}^{2}$ exist exactly when $ω^{2} ε μ - k_{c}^{2} \geq 0$ , i.e. $ω^{2} \geq k_{c}^{2} / (ε μ) = ω_{c}^{2}$ , i.e. $ω \geq ω_{c}$ (both $ω$ and $ω_{c}$ are positive); then the real solutions are $β = \pm \sqrt{ω^{2} ε μ - k_{c}^{2}}$ [F7], and $β \neq 0$ exactly when $ω > ω_{c}$ , while $β = 0$ occurs exactly at $ω = ω_{c}$ . If $ω < ω_{c}$ the right side is negative and the two solutions of $β^{2} = - (⁣ k_{c}^{2} - ω^{2} ε μ)$ are $β = \pm i \sqrt{k_{c}^{2} - ω^{2} ε μ}$ because $(i γ)^{2} = - γ^{2}$ for real $γ$ [F7]. For any such root the phase factor obeys $e^{i θ} = e^{i β x_{3}} e^{- i ω t}$ , and $e^{i β x_{3}} = e^{- Im β x_{3}} (\cos (Re β x_{3}) + i \sin (Re β x_{3}))$ by [F6], so the field carries the real modulation factor $e^{- Im β x_{3}}$ in the declared phase convention; on a declared half-guide $x_{3} \geq 0$ with $Im β > 0$ this factor decays, and no propagating-power claim is made. The verification of clauses (a) and (b) below uses only $β^{2} = ω^{2} ε μ - k^{2}$ (equivalently $k^{2} + β^{2} = ω^{2} ε μ$ ), so it covers imaginary and complex $β$ as well.

2.1step 1.1step 1.2F1F2F3F4F5F6algebra

[The TE family] Fix $m, n \geq 0$ with $(m, n) \neq (0, 0)$ , put $h : = H_{*} h_{m n}$ , $k^{2} : = k_{m n}^{2}$ (so $k^{2} > 0$ by [F5] and $k^{2} = ω^{2} ε μ - β^{2} = k_{c}^{2}$ by the required dispersion relation), and let $\tilde{E} : = - \frac{i ω μ}{k^{2}} e_{z} \times \nabla_{t} h$ , $\tilde{H} : = \frac{i β}{k^{2}} \nabla_{t} h + e_{z} h$ , $E = Re (\tilde{E} e^{i θ})$ , $B = μ Re (\tilde{H} e^{i θ})$ . First, $e_{z} \times \nabla_{t} h$ is transverse by [F4], so ${\tilde{E}}_{3} = 0$ and $E_{z} = Re (0) = 0$ : the field is TE. Second, the amplitude equations of step 1.1 hold: $\overset{}{div} (\tilde{E}) = - \frac{i ω μ}{k^{2}} (\partial_{1} (- \partial_{2} h) + \partial_{2} (\partial_{1} h) + i β \cdot 0) = - \frac{i ω μ}{k^{2}} div (e_{z} \times \nabla_{t} h) = 0$ by [F4]; $\overset{}{div} (\tilde{H}) = \frac{i β}{k^{2}} (\partial_{1}^{2} h + \partial_{2}^{2} h) + i β h = \frac{i β}{k^{2}} Δ_{t} h + i β h = \frac{i β}{k^{2}} (- k^{2} h) + i β h = 0$ by [F5]; $\overset{}{curl} (\tilde{E}) = (- i β {\tilde{E}}_{2}, i β {\tilde{E}}_{1}, \partial_{1} {\tilde{E}}_{2} - \partial_{2} {\tilde{E}}_{1}) = (- \frac{β ω μ}{k^{2}} \partial_{1} h, - \frac{β ω μ}{k^{2}} \partial_{2} h, - \frac{i ω μ}{k^{2}} (\partial_{1}^{2} h + \partial_{2}^{2} h)) = (\frac{i β}{k^{2}} i ω μ \partial_{1} h, \frac{i β}{k^{2}} i ω μ \partial_{2} h, i ω μ h) = i ω μ \tilde{H}$ , using $i β {\tilde{E}}_{1} = i β \frac{i ω μ}{k^{2}} \partial_{2} h = - \frac{β ω μ}{k^{2}} \partial_{2} h$ and $- \frac{i ω μ}{k^{2}} (- k^{2} h) = i ω μ h$ ; and $\overset{}{curl} (\tilde{H}) = (\partial_{2} h - i β \frac{i β}{k^{2}} \partial_{2} h, i β \frac{i β}{k^{2}} \partial_{1} h - \partial_{1} h, \frac{i β}{k^{2}} (\partial_{1} \partial_{2} h - \partial_{2} \partial_{1} h)) = (1 + \frac{β^{2}}{k^{2}}) (\partial_{2} h, - \partial_{1} h, 0) = \frac{k^{2} + β^{2}}{k^{2}} (\partial_{2} h, - \partial_{1} h, 0) = \frac{ω^{2} ε μ}{k^{2}} (\partial_{2} h, - \partial_{1} h, 0) = - i ω ε \tilde{E}$ , where the last mixed-partial difference vanishes by the identity $div (e_{z} \times \nabla_{t} h) = 0$ of [F4] and the final equality uses $\tilde{E} = - \frac{i ω μ}{k^{2}} (- \partial_{2} h, \partial_{1} h, 0)$ . Third, the open-wall conditions: on ${x_{1} = 0}$ and ${x_{1} = a}$ (normals $\mp e_{1}$ ) the tangential components of $E$ are $E_{2} = Re (- \frac{i ω μ}{k^{2}} \partial_{1} h e^{i θ}) = - \frac{ω μ}{k^{2}} Re (i \partial_{1} h e^{i θ}) = 0$ and $E_{3} = 0$ , because $\partial_{1} h = 0$ there by [F5], and $n \cdot B = \mp B_{1} = \mp μ Re (\frac{i β}{k^{2}} \partial_{1} h e^{i θ}) = 0$ ; on ${x_{2} = 0}$ and ${x_{2} = b}$ (normals $\mp e_{2}$ ) the tangential components are $E_{1} = Re (\frac{i ω μ}{k^{2}} \partial_{2} h e^{i θ}) = 0$ , since $\partial_{2} h = 0$ there by [F5], and $E_{3} = 0$ , while $n \cdot B = \mp B_{2} = \mp μ Re (\frac{i β}{k^{2}} \partial_{2} h e^{i θ}) = 0$ . Hence $n \times E = 0$ and $n \cdot B = 0$ on every open wall, and clause (a) holds.

2.2step 1.1step 1.2F1F2F3F4F5F6algebra

[The TM family] Fix $m, n \geq 1$ , put $e : = E_{*} e_{m n}$ and $k^{2} : = k_{m n}^{2} = k_{c}^{2} > 0$ , and let $\tilde{E} : = \frac{i β}{k^{2}} \nabla_{t} e + e_{z} e$ , $\tilde{H} : = \frac{i ω ε}{k^{2}} e_{z} \times \nabla_{t} e$ , $E = Re (\tilde{E} e^{i θ})$ , $B = μ Re (\tilde{H} e^{i θ})$ . Here $e_{z} \times \nabla_{t} e$ is transverse by [F4], so ${\tilde{H}}_{3} = 0$ and $B_{z} = μ Re (0) = 0$ : the field is TM. The amplitude equations hold: $\overset{}{div} (\tilde{E}) = \frac{i β}{k^{2}} (\partial_{1}^{2} e + \partial_{2}^{2} e) + i β e = \frac{i β}{k^{2}} (- k^{2} e) + i β e = 0$ by [F5]; $\overset{}{div} (\tilde{H}) = \frac{i ω ε}{k^{2}} (\partial_{1} (- \partial_{2} e) + \partial_{2} (\partial_{1} e)) + i β \cdot 0 = 0$ by the identity $div (e_{z} \times \nabla_{t} e) = 0$ of [F4]; $\overset{}{curl} (\tilde{E}) = (\partial_{2} e - i β \frac{i β}{k^{2}} \partial_{2} e, i β \frac{i β}{k^{2}} \partial_{1} e - \partial_{1} e, \frac{i β}{k^{2}} (\partial_{1} \partial_{2} e - \partial_{2} \partial_{1} e)) = \frac{k^{2} + β^{2}}{k^{2}} (\partial_{2} e, - \partial_{1} e, 0) = \frac{ω^{2} ε μ}{k^{2}} (\partial_{2} e, - \partial_{1} e, 0) = i ω μ \tilde{H}$ (the mixed difference vanishes by [F4] and the final equality uses $\tilde{H} = \frac{i ω ε}{k^{2}} (- \partial_{2} e, \partial_{1} e, 0)$ ); and $\overset{}{curl} (\tilde{H}) = (- i β {\tilde{H}}_{2}, i β {\tilde{H}}_{1}, \partial_{1} {\tilde{H}}_{2} - \partial_{2} {\tilde{H}}_{1}) = (\frac{β ω ε}{k^{2}} \partial_{1} e, \frac{β ω ε}{k^{2}} \partial_{2} e, \frac{i ω ε}{k^{2}} (\partial_{1}^{2} e + \partial_{2}^{2} e)) = \frac{ω ε β}{k^{2}} \nabla_{t} e - i ω ε e_{z} e = - i ω ε \tilde{E}$ , using $\frac{i ω ε}{k^{2}} (- k^{2} e) = - i ω ε e$ , in agreement with $\tilde{E} = \frac{i β}{k^{2}} \nabla_{t} e + e_{z} e$ . The open-wall conditions: $e = 0$ on $\partial R$ by [F5], so $E_{3} = Re (e e^{i θ}) = 0$ on the walls; on ${x_{1} = 0}$ and ${x_{1} = a}$ (normals $\mp e_{1}$ ) the tangential components are $E_{2} = Re (\frac{i β}{k^{2}} \partial_{2} e e^{i θ}) = 0$ , since $\partial_{2} e$ carries the factor $\sin \frac{m π x_{1}}{a}$ vanishing there, and $E_{3} = 0$ , while $n \cdot B = \mp B_{1} = \mp μ Re (\frac{i ω ε}{k^{2}} (- \partial_{2} e) e^{i θ}) = 0$ ; on ${x_{2} = 0}$ and ${x_{2} = b}$ (normals $\mp e_{2}$ ) the tangential components are $E_{1} = Re (\frac{i β}{k^{2}} \partial_{1} e e^{i θ}) = 0$ , since $\partial_{1} e$ carries the factor $\sin \frac{n π x_{2}}{b}$ vanishing there, and $E_{3} = 0$ , while $n \cdot B = \mp B_{2} = \mp μ Re (\frac{i ω ε}{k^{2}} \partial_{1} e e^{i θ}) = 0$ . Hence the wall conditions hold and clause (b) follows.

2.3step 1.1F3F5F6algebra

[Zero indices and nonvanishing] For $m = n = 0$ one has $h_{00} \equiv 1$ , so $\nabla_{t} h_{00} = 0$ and $k_{00}^{2} = 0$ : the divided reconstruction formulas of the TE family are undefined, and this exclusion is by the reconstruction denominator. A purely longitudinal constant candidate has $\tilde{E} = 0$ and $\tilde{H} = c e_{z}$ ; the Faraday amplitude equation of step 1.1 then reads $\overset{}{curl} (0) = (0, 0, 0) = i ω μ c e_{z}$ , so $c = 0$ because $ω μ > 0$ : it is excluded by the full field equations, not by divergence. Consistently $\overset{}{div} (c e_{z}) = i β c$ , which vanishes at $β = 0$ , so the divergence equation alone would not exclude a constant longitudinal amplitude at $β = 0$ ; the exclusion uses Faraday as displayed. For $m = 0$ or $n = 0$ the TM profile vanishes identically, $e_{m n} \equiv 0$ , so $\tilde{E} = e_{z} e = 0$ and $\tilde{H} = 0$ ; hence the TM indices are restricted to $m, n \geq 1$ . Conversely, for $(m, n) \neq (0, 0)$ one has $\nabla_{t} h_{m n} \neq 0$ : if $m \neq 0$ then $\partial_{1} h_{m n} = - \frac{m π}{a} \sin \frac{m π x_{1}}{a} \cos \frac{n π x_{2}}{b}$ is nonzero at $x_{1} = \frac{a}{2 m}$ and an interior $x_{2} \in (0, b)$ with $\cos (n π x_{2} / b) \neq 0$ (take $x_{2} = b / (4 n)$ when $n > 0$ , and $x_{2} = b / 2$ when $n = 0$ ), and if $m = 0$ then $n \neq 0$ and the analogous $\partial_{2} h_{m n}$ is nonzero; and for $m, n \geq 1$ one has $e_{m n} \neq 0$ , for instance at $x_{1} = \frac{a}{2 m}, x_{2} = \frac{b}{2 n}$ . Since $H_{*}, E_{*} > 0$ , the TE electric amplitude is a nonzero multiple of $e_{z} \times \nabla_{t} h_{m n}$ and its longitudinal magnetic amplitude is $H_{*} h_{m n} \neq 0$ ; the TM longitudinal electric amplitude is $E_{*} e_{m n} \neq 0$ and its magnetic amplitude is a nonzero multiple of $e_{z} \times \nabla_{t} e_{m n}$ . Thus both displayed fields are nonzero for all admissible indices, even at $β = 0$ where the other transverse amplitude vanishes, and by the consequence clause of F3 the real fields $E = Re (\tilde{E} e^{i θ})$ , $B = μ Re (\tilde{H} e^{i θ})$ are not identically zero.

3.1step 1.1step 1.3step 2.1step 2.2step 2.3∎

[Conclusion and scope] Clauses (a) and (b) are steps 2.1 and 2.2: for every admissible index pair the displayed fields are $C^{\infty}$ by step 1.2, satisfy the four source-free equations by steps 1.1 and the amplitude verifications, have their stated TE or TM property, and satisfy the PEC wall conditions on every open wall by the trace identities of [F5]; clause (c) is step 2.3; clause (d) is step 1.3, whose algebra covers real, zero and imaginary roots alike with the factor $e^{- Im β x_{3}}$ and the declared half-guide reading; and the argument used only the declared lossless isotropic scalar branch, the PEC model of the definition, the four macroscopic equations with $ρ_{f} = J_{f} = 0$ and the separated phasor calculus, with no completeness, uniqueness, lossy, dispersive, open-end or empirical assertion and no propagating-power claim below cutoff. This is clause (e).

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The propagating candidate branch: cutoff, phase and group speeds, cross-section energy and flux, and energy transport at the group velocity

Physical scope

The dispersion relation, cutoff branch, phase and group speeds and the cross-section period-averaged energy and Poynting flux of the separated TE/TM candidate modes of the lossless rectangular guide, including the identity of the energy transport ratio with the group speed. Conditional on the adopted lossless material branch and PEC model; no signal-speed or general-dispersion claim.

Statement

In the setting of The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff let a TE or TM candidate of part (a) or (b) with $β > 0$ and $ω > ω_{c}$ be fixed (so $β = \sqrt{ω^{2} ε μ - k_{c}^{2}}$ is real and nonzero), retain its positive longitudinal amplitude $H_{*}$ (A m $^{- 1}$ ) in the TE case or $E_{*}$ (V m $^{- 1}$ ) in the TM case, let $T : = 2 π / ω$ , and let $u : = \frac{1}{2} (E \cdot D + B \cdot H)$ and $S : = E \times H$ be the stored-energy density and energy-flux vector of Energy balance and Ohmic dissipation for the constant nondispersive linear model, with the Euclidean inner and cross products of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and The cross product in $R^{3}$ . Define the period-averaged stored energy per unit length and the period-averaged longitudinal flux across the cross-section $U : = \frac{1}{T} \int_{0}^{T} ⁣ ⁣ \int_{R_{a b}} u d x_{1} d x_{2} d t, Φ : = \frac{1}{T} \int_{0}^{T} ⁣ ⁣ \int_{R_{a b}} S \cdot e_{z} d x_{1} d x_{2} d t,$ both independent of $z$ and of the starting time. Then: (a) [Cross-section identities] $\int_{R_{a b}} ∣ \nabla_{t} h_{m n} ∣^{2} = k_{c}^{2} \int_{R_{a b}} h_{m n}^{2}$ and $\int_{R_{a b}} ∣ \nabla_{t} e_{m n} ∣^{2} = k_{c}^{2} \int_{R_{a b}} e_{m n}^{2}$ , with the boundary terms of the integration by parts vanishing; (b) [Energy and flux] for the TE candidate $U = H_{*}^{2} \frac{μ^{2} ε ω^{2}}{2 k_{c}^{2}} \int_{R_{a b}} h_{m n}^{2}, Φ = H_{*}^{2} \frac{ω μ β}{2 k_{c}^{2}} \int_{R_{a b}} h_{m n}^{2},$ and for the TM candidate $U = E_{*}^{2} \frac{ε^{2} μ ω^{2}}{2 k_{c}^{2}} \int_{R_{a b}} e_{m n}^{2}, Φ = E_{*}^{2} \frac{ω ε β}{2 k_{c}^{2}} \int_{R_{a b}} e_{m n}^{2},$ with the explicit norms of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals(c); (c) [Energy transport and speed relations] $Φ = U β / (ε μ ω) = U v_{g}$ , where $v_{g} : = d ω / d β = β / (ε μ ω)$ is the group speed of the branch $ω (β) = \sqrt{(β^{2} + k_{c}^{2}) / (ε μ)}$ ; moreover the phase speed $v_{p} : = ω / β$ satisfies $v_{p} v_{g} = 1 / (ε μ)$ , $v_{p} > 1 / \sqrt{ε μ}$ and $0 < v_{g} < 1 / \sqrt{ε μ}$ for the declared indices; (d) [Finite propagating set] if a candidate with indices $(m, n)$ is propagating at frequency $ω$ (that is, $β^{2} = ω^{2} ε μ - k_{m n}^{2} > 0$ ), then $m < ω a \sqrt{ε μ} / π$ and $n < ω b \sqrt{ε μ} / π$ , so the propagating indices lie in the explicitly bounded index box ${m, n \geq 0 : m < ω a \sqrt{ε μ} / π, n < ω b \sqrt{ε μ} / π}$ ; (e) [Scope] This is a lossless separated-guide computation: the period average is over time at fixed position, the flux is through one cross-section, $v_{p}$ is not asserted to be a signal speed and no bound on information or energy speeds in general dispersive media follows; the material branch and the PEC model are those of the cited theorems.

Facts & Assumptions

Given: A TE or TM candidate of The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff with $β > 0$ and $ω > ω_{c}$ , so that $β = \sqrt{ω^{2} ε μ - k_{c}^{2}}$ is real and nonzero and $k_{c}^{2} = ω^{2} ε μ - β^{2}$ ; $T : = 2 π / ω$ ; the stored-energy density $u = \frac{1}{2} (E \cdot D + B \cdot H)$ and energy-flux vector $S = E \times H$ of Energy balance and Ohmic dissipation for the constant nondispersive linear model, with the Euclidean inner and cross products of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and The cross product in $R^{3}$ ; the guide, TEM and phasor conventions of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM and the SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

[F1]

The candidate amplitudes are those of The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff: for the TE family, $h = H_{*} h_{m n}$ with $H_{*} > 0$ in A m $^{- 1}$ , $k_{c}^{2} = k_{m n}^{2}$ , $\tilde{E} = - \frac{i ω μ}{k_{c}^{2}} e_{z} \times \nabla_{t} h$ , $\tilde{H} = \frac{i β}{k_{c}^{2}} \nabla_{t} h + e_{z} h$ ; for the TM family, $e = E_{*} e_{m n}$ with $E_{*} > 0$ in V m $^{- 1}$ , $\tilde{E} = \frac{i β}{k_{c}^{2}} \nabla_{t} e + e_{z} e$ , $\tilde{H} = \frac{i ω ε}{k_{c}^{2}} e_{z} \times \nabla_{t} e$ ; the real fields are $E = Re (\tilde{E} e^{i (β x_{3} - ω t)})$ , $H = Re (\tilde{H} e^{i (β x_{3} - ω t)})$ , $B = μ H$ , $D = ε E$ , and the four source-free macroscopic equations hold by that theorem ((The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff), Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM). The stored energy and flux are $u = \frac{1}{2} (ε ∣ E ∣^{2} + μ ∣ H ∣^{2})$ and $S = E \times H$ in this branch, with $\partial_{t} u + div S = 0$ and $u \geq 0$ (Energy balance and Ohmic dissipation for the constant nondispersive linear model clause (a), lossless case $J_{f} = 0$ ).

[F2]

Period averages: for $ω > 0$ , $T = 2 π / ω$ and $a, d \in C^{3}$ , the componentwise averages over one period are $\frac{1}{T} \int_{0}^{T} ⟨ Re (a e^{- i ω t}), Re (d e^{- i ω t}) ⟩ d t = \frac{1}{2} Re ⟨ a, \overline{d} ⟩$ and $\frac{1}{T} \int_{0}^{T} Re (a e^{- i ω t}) \times Re (d e^{- i ω t}) d t = \frac{1}{2} Re (a \times \overline{d})$ , with the complex-bilinear extensions of the Euclidean inner and cross products, and the scalar form for $a, d \in C$ (Period averages of products of real parts of complex exponentials). The integrands are $T$ -periodic because $e^{- i ω (t + T)} = e^{- i ω t} e^{- 2 π i} = e^{- i ω t}$ , so the average over any interval $[t_{0}, t_{0} + T]$ agrees with the average over $[0, T]$ .

[F3]

Cross-section identities and norms: for the profiles of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals, $\int_{R_{a b}} ∣ \nabla_{t} h_{m n} ∣^{2} = k_{m n}^{2} \int_{R_{a b}} h_{m n}^{2}$ and $\int_{R_{a b}} ∣ \nabla_{t} e_{m n} ∣^{2} = k_{m n}^{2} \int_{R_{a b}} e_{m n}^{2}$ , with all boundary terms of the integration by parts vanishing (The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals (d)); and the transverse identities $∣ e_{z} \times \nabla_{t} h ∣ = ∣ \nabla_{t} h ∣$ , $\nabla_{t} h \cdot (e_{z} \times \nabla_{t} h) = 0$ and $e_{z} \times (e_{z} \times \nabla_{t} h) = - \nabla_{t} h$ hold, together with their analogues for $e_{m n}$ (Transverse vector-calculus identities for the splitting $R^{3} = R^{2} \oplus R e_{z}$ (a)). The additional cross identities follow from The cross product in $R^{3}$ : for $g = (g_{1}, g_{2}, 0)$ , $w = e_{z} \times g = (- g_{2}, g_{1}, 0)$ gives $w \times g = - e_{z} (g_{1}^{2} + g_{2}^{2})$ , $g \times w = e_{z} ∣ g ∣^{2}$ and $w \times e_{z} = g$ .

[F4]

The branch function $ω (β) = \sqrt{(β^{2} + k_{c}^{2}) / (ε μ)}$ is well defined and positive for real $β$ by Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ; limits of real functions are algebraically computed by Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero, and the derivative is that of 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; sums, scalar multiples and composites obey the one-variable 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: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ .

Proof

technique · direct: insert the explicit candidate amplitudes into the real quadratic densities, average over one period with the recorded period-average identities, evaluate the cross-section integrals with the profile and transverse identities, and differentiate the real branch function

1.1F1F2algebra

[Period-averaged quadratic densities] Fix a point $(x_{1}, x_{2}, x_{3})$ and put $a : = \tilde{E} e^{i β x_{3}}$ , $d : = \tilde{H} e^{i β x_{3}}$ , so that $E = Re (a e^{- i ω t})$ and $H = Re (d e^{- i ω t})$ at that point. By [F2], $\frac{1}{T} \int_{0}^{T} ∣ E ∣^{2} d t = \frac{1}{2} ∣ \tilde{E} ∣^{2}$ , $\frac{1}{T} \int_{0}^{T} ∣ H ∣^{2} d t = \frac{1}{2} ∣ \tilde{H} ∣^{2}$ and $\frac{1}{T} \int_{0}^{T} (E \times H) d t = \frac{1}{2} Re (\tilde{E} \times \overline{\tilde{H}})$ , because the phase factors $e^{\pm i β x_{3}}$ are unimodular and cancel in the inner and cross products. Hence the period averages $⟨ u ⟩ : = \frac{1}{T} \int_{0}^{T} u d t = \frac{1}{4} (ε ∣ \tilde{E} ∣^{2} + μ ∣ \tilde{H} ∣^{2})$ and $⟨ S ⟩ : = \frac{1}{T} \int_{0}^{T} S d t = \frac{1}{2} Re (\tilde{E} \times \overline{\tilde{H}})$ depend only on $(x_{1}, x_{2})$ and not on $z$ or on the starting time, and therefore $U = \int_{R_{a b}} ⟨ u ⟩ d x_{1} d x_{2}$ and $Φ = \int_{R_{a b}} ⟨ S ⟩ \cdot e_{z} d x_{1} d x_{2}$ are the stated averages, independent of $z$ and of the starting time; consistency with the local balance of [F1] means $\partial_{t} ⟨ u ⟩ = 0$ and $\partial_{z} (⟨ S ⟩ \cdot e_{z}) = 0$ , which the displayed formulas show directly.

1.2F1F3algebra

[Cross-section identities] For the scaled profiles $h = H_{*} h_{m n}$ and $e = E_{*} e_{m n}$ of [F1], multiplying the two gradient-integral identities of [F3] by $H_{*}^{2}$ and $E_{*}^{2}$ , respectively, gives $\int_{R_{a b}} ∣ \nabla_{t} h ∣^{2} = k_{c}^{2} \int_{R_{a b}} h^{2}$ and $\int_{R_{a b}} ∣ \nabla_{t} e ∣^{2} = k_{c}^{2} \int_{R_{a b}} e^{2}$ with vanishing boundary terms (the traces $\partial_{n} h = 0$ and $e = 0$ on $\partial R_{a b}$ of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals (a),(b)), and the pointwise identities $∣ e_{z} \times \nabla_{t} h ∣^{2} = ∣ \nabla_{t} h ∣^{2}$ and $(e_{z} \times \nabla_{t} h) \times \nabla_{t} h = - e_{z} ∣ \nabla_{t} h ∣^{2}$ hold, with $\nabla_{t} h \cdot (e_{z} \times \nabla_{t} h) = 0$ ; the same holds with $e$ in place of $h$ .

2.1step 1.1step 1.2F1F3algebra

[TE energy and flux] Let $h : = H_{*} h_{m n}$ and $w : = e_{z} \times \nabla_{t} h$ . With the TE amplitudes of [F1], $∣ \tilde{E} ∣^{2} = \frac{ω^{2} μ^{2}}{k_{c}^{4}} ∣ w ∣^{2} = \frac{ω^{2} μ^{2}}{k_{c}^{4}} ∣ \nabla_{t} h ∣^{2}$ and $∣ \tilde{H} ∣^{2} = \frac{β^{2}}{k_{c}^{4}} ∣ \nabla_{t} h ∣^{2} + h^{2}$ (the cross term $\nabla_{t} h \cdot e_{z}$ vanishes), so $⟨ u ⟩ = \frac{1}{4} (\frac{ε ω^{2} μ^{2} + μ β^{2}}{k_{c}^{4}} ∣ \nabla_{t} h ∣^{2} + μ h^{2})$ ; integrating with step 1.2, $\int (ε ∣ \tilde{E} ∣^{2}) = \frac{ε ω^{2} μ^{2}}{k_{c}^{2}} \int h^{2}$ , $\int (μ ∣ \tilde{H} ∣^{2}) = \frac{μ β^{2}}{k_{c}^{2}} \int h^{2} + μ \int h^{2}$ , and $μ β^{2} + μ k_{c}^{2} = μ ω^{2} ε μ$ , so $\int (ε ∣ \tilde{E} ∣^{2} + μ ∣ \tilde{H} ∣^{2}) = \frac{2 ε μ^{2} ω^{2}}{k_{c}^{2}} \int h^{2}$ and $U = H_{*}^{2} \frac{μ^{2} ε ω^{2}}{2 k_{c}^{2}} \int_{R_{a b}} h_{m n}^{2}$ as claimed. For the flux, $\overline{\tilde{H}} = - \frac{i β}{k_{c}^{2}} \nabla_{t} h + e_{z} h$ (the profile $h$ and the parameters $β, ω, μ$ are real), so $\tilde{E} \times \overline{\tilde{H}} = - \frac{i ω μ}{k_{c}^{2}} [- \frac{i β}{k_{c}^{2}} w \times \nabla_{t} h + h w \times e_{z}] = \frac{ω μ β}{k_{c}^{4}} ∣ \nabla_{t} h ∣^{2} e_{z} - i \frac{ω μ}{k_{c}^{2}} h \nabla_{t} h$ using $w \times \nabla_{t} h = - e_{z} ∣ \nabla_{t} h ∣^{2}$ and $w \times e_{z} = \nabla_{t} h$ from [F3]; hence $Re (\tilde{E} \times \overline{\tilde{H}}) = \frac{ω μ β}{k_{c}^{4}} ∣ \nabla_{t} h ∣^{2} e_{z}$ , and step 1.2 gives $Φ = \frac{1}{2} \frac{ω μ β}{k_{c}^{4}} k_{c}^{2} \int h^{2} = H_{*}^{2} \frac{ω μ β}{2 k_{c}^{2}} \int_{R_{a b}} h_{m n}^{2}$ and the explicit norms of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals (c).

2.2step 1.1step 1.2F1F3algebra

[TM energy and flux] Let $e : = E_{*} e_{m n}$ and $w : = e_{z} \times \nabla_{t} e$ . With the TM amplitudes of [F1], $∣ \tilde{E} ∣^{2} = \frac{β^{2}}{k_{c}^{4}} ∣ \nabla_{t} e ∣^{2} + e^{2}$ and $∣ \tilde{H} ∣^{2} = \frac{ω^{2} ε^{2}}{k_{c}^{4}} ∣ \nabla_{t} e ∣^{2}$ , so $⟨ u ⟩ = \frac{1}{4} (\frac{ε β^{2} + μ ω^{2} ε^{2}}{k_{c}^{4}} ∣ \nabla_{t} e ∣^{2} + ε e^{2})$ ; integrating with step 1.2, $\int (ε ∣ \tilde{E} ∣^{2} + μ ∣ \tilde{H} ∣^{2}) = \frac{ε β^{2} + μ ω^{2} ε^{2} + ε k_{c}^{2}}{k_{c}^{2}} \int e^{2} = \frac{2 ε^{2} μ ω^{2}}{k_{c}^{2}} \int e^{2}$ because $ε β^{2} + ε k_{c}^{2} = ε ω^{2} ε μ$ , giving $U = E_{*}^{2} \frac{ε^{2} μ ω^{2}}{2 k_{c}^{2}} \int_{R_{a b}} e_{m n}^{2}$ . For the flux, $\overline{\tilde{H}} = - \frac{i ω ε}{k_{c}^{2}} w$ , so $\tilde{E} \times \overline{\tilde{H}} = (\frac{i β}{k_{c}^{2}} \nabla_{t} e + e_{z} e) \times (- \frac{i ω ε}{k_{c}^{2}} w) = \frac{ω ε β}{k_{c}^{4}} ∣ \nabla_{t} e ∣^{2} e_{z} + i \frac{ω ε}{k_{c}^{2}} e \nabla_{t} e$ , using $\nabla_{t} e \times w = e_{z} ∣ \nabla_{t} e ∣^{2}$ and $e_{z} \times w = - \nabla_{t} e$ from [F3]; hence $Re (\tilde{E} \times \overline{\tilde{H}}) = \frac{ω ε β}{k_{c}^{4}} ∣ \nabla_{t} e ∣^{2} e_{z}$ and $Φ = E_{*}^{2} \frac{ω ε β}{2 k_{c}^{2}} \int_{R_{a b}} e_{m n}^{2}$ . This proves clause (b) in both families with the norms recorded in the profile lemma.

2.3step 1.2F3F4algebra

[Bounded propagating index set] Let a candidate with indices $(m, n)$ be propagating at frequency $ω$ , that is $β^{2} = ω^{2} ε μ - k_{m n}^{2} > 0$ . Then $k_{m n}^{2} < ε μ ω^{2}$ ; since $k_{m n}^{2} = (\frac{m π}{a})^{2} + (\frac{n π}{b})^{2} \geq (\frac{m π}{a})^{2}$ and $\geq (\frac{n π}{b})^{2}$ by [F3] and the definitions of the profiles, $(\frac{m π}{a})^{2} < ε μ ω^{2}$ and $(\frac{n π}{b})^{2} < ε μ ω^{2}$ ; taking positive square roots [F4] gives $m < ω a \sqrt{ε μ} / π$ and $n < ω b \sqrt{ε μ} / π$ . Hence the set of propagating indices is contained in the displayed bounded box.

3.1step 2.1step 2.2F4algebra

[Energy transport and the speed relations] From steps 2.1 and 2.2, cancelling the common positive amplitude square $H_{*}^{2}$ or $E_{*}^{2}$ gives $Φ = \frac{β}{ε μ ω} U$ in both families, since $\frac{μ^{2} ε ω^{2}}{2 k_{c}^{2}} \frac{β}{ε μ ω} = \frac{ω μ β}{2 k_{c}^{2}}$ and $\frac{ε^{2} μ ω^{2}}{2 k_{c}^{2}} \frac{β}{ε μ ω} = \frac{ω ε β}{2 k_{c}^{2}}$ . The branch function $ω (β) = \sqrt{(β^{2} + k_{c}^{2}) / (ε μ)}$ of [F4] is differentiable with $v_{g} : = ω^{'} (β) = \frac{β}{ε μ ω (β)}$ : indeed for real $h \neq 0$ the difference quotient is $\frac{2 β + h}{ε μ (ω (β + h) + ω (β))}$ (rationalizing the difference of square roots), and its limit as $h \to 0$ is $\frac{2 β}{2 ε μ ω (β)}$ by [F4]; differentiating the identity $ε μ ω (β)^{2} = β^{2} + k_{c}^{2}$ with the chain rule gives the same value. Hence $Φ = U v_{g}$ . Moreover $v_{p} : = ω / β$ satisfies $v_{p} v_{g} = \frac{ω}{β} \frac{β}{ε μ ω} = \frac{1}{ε μ}$ ; $v_{p} = \frac{\sqrt{β^{2} + k_{c}^{2}}}{\sqrt{ε μ} β} = \frac{1}{\sqrt{ε μ}} \sqrt{1 + \frac{k_{c}^{2}}{β^{2}}} > \frac{1}{\sqrt{ε μ}}$ because $k_{c}^{2} > 0$ and $β > 0$ ; and $0 < v_{g} = \frac{1}{\sqrt{ε μ}} \frac{β}{\sqrt{β^{2} + k_{c}^{2}}} < \frac{1}{\sqrt{ε μ}}$ . This proves clause (c).

4.1step 1.2step 2.1step 2.2step 2.3step 3.1F1∎

[Scope] Clauses (a) and (b) are steps 1.2, 2.1 and 2.2, with the cross-section integrals evaluated by the profile and transverse identities and the explicit norms; clause (c) is step 3.1, with the transport relation $Φ = U v_{g}$ and the speed inequalities; clause (d) is step 2.3. Every step used the lossless separated candidate of the declared model, the ideal PEC boundary model and the choice-free local energy reading of the material-energy theorem: the period average is over time at a fixed position, the flux is through one cross-section, no signal-speed interpretation of $v_{p}$ is asserted, no bound on information or energy speeds in general dispersive media is claimed, and no empirical premise is used. This is clause (e).

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Constant-Ohmic PEC modal decay and exact energy-class completeness

Physical scope

The explicitly declared constant scalar Ohmic PEC energy classes of K10, including the lossless sigma=0 branch, with exact modal/Fourier completeness and energy balance. No arbitrary dispersive, nonlinear, hysteretic, nonsmooth-wall, open-end or junction-scattering claim; no empirical claim.

Statement

Assume the constant isotropic inertial-rest material branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch with $ε, μ > 0$ , $σ \geq 0$ , $D = ε E$ , $H = B / μ$ , $J_{e x t} = 0$ . The exact broadband physical reading additionally requires $ε μ \geq c^{- 2}$ as in that closure; otherwise the positive coefficients define its finite-band effective phase model without a universal front-speed claim. Put $v^{2} = (ε μ)^{- 1}$ and $γ = σ / ε$ . The smooth equations of The macroscopic Maxwell equations in terms of D, H and the free sources with initially zero volume charge give $E_{t} = v^{2} curl B - γ E$ , $B_{t} = - curl E$ and zero divergences; in the finite-energy classes below these first-order equations with the declared PEC weak/variational boundary realization are adopted explicitly, rather than deduced from a smooth-only postulate. The quadratic stored energy is $U (t) = \frac{1}{2} (ε ∥ E (t) ∥_{2}^{2} + μ^{- 1} ∥ B (t) ∥_{2}^{2})$ , consistent with Energy balance and Ohmic dissipation for the constant nondispersive linear model. (a) [Rectangular cavity] In the exact zero-normal- $B$ reflection-parity and divergence class of Rectangular PEC reflection class: Fourier completeness, weak solutions and energy conservation and Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM(c), every initial pair has a unique $C (R; L^{2})$ distributional solution of this damped system in the same class. Finite Fourier mode sums converge in energy norm uniformly on compact time intervals and are dense at every time. The Fourier coefficient equations at $k_{j} = π n_{j} / L_{j}$ are ${\hat{E}}_{t} = i v^{2} k \times \hat{B} - γ \hat{E}$ , ${\hat{B}}_{t} = - i k \times \hat{E}$ , with both transverse constraints and the original parities retained. (b) [Smooth closed PEC cavity] Assume the full Axiom of Choice (The Axiom of Choice) inherited from the smooth spectral and elliptic suppliers. On a bounded smooth cavity $Ω$ , let $X, V, H$ and its smooth orthonormal eigenfields $u_{j}$ , $λ_{j} \geq 0$ , be exactly Spectral completeness of the curl form in a smooth PEC cavity and Spectral completeness of the declared PEC waveguide and cavity classes. Define the full weak normal-zero divergence-free magnetic space $M : = {B \in L^{2} (Ω; R^{3}) : \int_{Ω} B \cdot \nabla ψ = 0 for every smooth scalar ψ on \overline{Ω}} .$ This is the distributional divergence constraint together with weak zero normal flux; for smooth fields it is $div B = 0$ , $n \cdot B = 0$ . For every positive eigenvalue put $w_{j} = curl u_{j} / \sqrt{λ_{j}}$ and put $K_{M} = {B \in M : curl B = 0 distributionally}$ . Then $M = {\overline{span {w_{j} : λ_{j} > 0}}}^{L^{2}} \oplus K_{M} .$ The declared first-order PEC modal energy class is $H \times M$ , with weak modal equations $a_{j}^{'} = v^{2} \sqrt{λ_{j}} b_{j} - γ a_{j}$ , $b_{j}^{'} = - \sqrt{λ_{j}} a_{j}$ for positive modes, $a_{j}^{'} = - γ a_{j}$ for electric zero modes, and $(B, K_{M})^{'} = 0$ for every kernel test. Define $D_{M}$ as the graph-norm completion of finite $w_{j}$ sums plus $K_{M}$ , with norm squared $∥ w ∥_{2}^{2} + ∥ curl w ∥_{2}^{2}$ . The equivalent weak-in-time variational equations are $⟨ E_{t}, ϕ ⟩ = v^{2} ⟨ B, curl ϕ ⟩ - γ ⟨ E, ϕ ⟩$ for $ϕ \in V$ , and $⟨ B_{t}, ψ ⟩ = - ⟨ E, curl ψ ⟩$ for $ψ \in D_{M}$ . They use the actual electric tangential-zero domain $V$ and magnetic normal-zero space $M$ , with no claim of maximal-trace equality. For every initial pair in $H \times M$ there is a unique $C (R; H \times M)$ solution of that declared realization, which also satisfies the displayed evolution equations distributionally in the interior. Finite electric and paired-magnetic modal sums, with the exact static magnetic kernel component retained, converge uniformly on compact time intervals in energy norm. No equality with every maximal rough tangential-trace realization is asserted. (c) [Exact energy balance, decay and zero sectors] In both declared classes, for every $t_{0} < t_{1}$ , $U (t_{1}) + σ \int_{t_{0}}^{t_{1}} ∥ E (t) ∥_{2}^{2} d t = U (t_{0}), U^{'} (t) = - σ ∥ E (t) ∥_{2}^{2} \leq 0.$ For positive smooth-cavity modes $a_{j}^{''} + γ a_{j}^{'} + v^{2} λ_{j} a_{j} = 0$ . If $Δ_{j} = γ^{2} / 4 - v^{2} λ_{j}$ is positive, the two roots are $- γ / 2 \pm \sqrt{Δ_{j}}$ ; if negative they are $- γ / 2 \pm i \sqrt{- Δ_{j}}$ ; at zero include $(A + B t) e^{- γ t / 2}$ . This finite scalar characteristic classification applies to every parameter $λ \geq 0$ and uses no choice principle; its general zero-eigenvalue associated-wave solutions and the narrower first-order compatibility are stated separately in (d). The full Choice premise belongs to the smooth-cavity spectral realization, not to these finite root formulas. For $σ > 0$ every positive mode decays, electric zero modes obey $a_{j} (t) = a_{j} (0) e^{- γ t}$ , and magnetic kernel data stay constant. In the smooth class the solution therefore tends in energy norm as $t \to + \infty$ to $(0, P_{K_{M}} B (0))$ . For $σ = 0$ energy is constant, electric zero modes are static and the original lossless branch is recovered. The zero sectors are retained, not divided by their eigenvalues; no universal decay to zero in the presence of magnetic kernel data is claimed. (d) [Associated second-order branch] Independently of first-order compatibility, the same spectral basis gives the damped associated electric wave for every $q_{0} \in V$ , $q_{1} \in H$ , with $q \in C (R; V)$ , $q^{'} \in C (R; H)$ and modal convergence uniformly on compact times. Its coefficients satisfy $a_{j}^{''} + γ a_{j}^{'} + v^{2} λ_{j} a_{j} = 0$ with arbitrary prescribed initial coefficients. At $λ_{j} = 0$ they are $α_{j} + β_{j} (1 - e^{- γ t}) / γ$ when $γ > 0$ , and $α_{j} + t β_{j}$ when $γ = 0$ . Such arbitrary second-order data need not arise from the first-order Maxwell system: its zero modes additionally satisfy $a_{j}^{'} = - γ a_{j}$ . This broader wave is retained alongside the physical first-order branch. All square roots and real derivatives use Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , The derivatives of sine and cosine are cosine and minus sine, 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 $g (c) \neq 0$ and the real-component exponential formula $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ . (e) [Scope] The conductor occupies the declared cavity interior while its stationary walls obey the independent ideal PEC boundary model. No arbitrary loss tensor, memory law, changing interface, open-end, rough-wall or empirical conclusion is made. Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) suffices for the rectangular/Fourier branch; the smooth branch retains its actual full Choice premise. Weak energy balance is proved by modal limits and does not assert an unprovided nonlinear weak local Poynting theorem.

Facts & Assumptions

Given: The constant isotropic inertial-rest branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch with constants $ε, μ > 0$ , $σ \geq 0$ , relations $D = ε E$ , $H = B / μ$ , $J_{e x t} = 0$ (the isotropic broadband condition $ε μ \geq c^{- 2}$ is additional and is not used); the smooth equations of The macroscopic Maxwell equations in terms of D, H and the free sources with initially zero volume charge; the PEC guide/cavity geometry and weak reflection class of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM; $v^{2} : = (ε μ)^{- 1}$ and $γ : = σ / ε$ .

[F1]

The first-order system and energy: with initially zero volume charge the macroscopic equations give $E_{t} = v^{2} curl B - γ E$ , $B_{t} = - curl E$ and zero divergences, and the quadratic stored energy is $U (t) = \frac{1}{2} (ε ∥ E (t) ∥_{2}^{2} + μ^{- 1} ∥ B (t) ∥_{2}^{2})$ , consistent with Energy balance and Ohmic dissipation for the constant nondispersive linear model (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, The macroscopic Maxwell equations in terms of D, H and the free sources, Energy balance and Ohmic dissipation for the constant nondispersive linear model).

[F2]

Rectangular energy class: Rectangular PEC reflection class: Fourier completeness, weak solutions and energy conservation states that for a rectangular box $Q = \prod_{j} (0, L_{j})$ , in the reflection-parity and distributional-divergence class defined there, the finite separated Fourier modes are dense and the class is complete for the $L^{2}$ pairing, and every initial datum of the class has a unique $C (R; L^{2})$ weak solution of the lossless system $\partial_{t} E = (ε μ)^{- 1} curl B$ , $\partial_{t} B = - curl E$ in the class with constant energy. Its system is written for the wavevector $k_{j} = π n_{j} / L_{j}$ and the finite-dimensional Fourier coefficients; the class and completeness are the objects used below, and Countable Choice is assumed as there (The Axiom of Countable Choice ( ${A C}_{ω}$ ), $L^{2}$ with the integral pairing is a Hilbert space).

[F3]

Smooth PEC cavity classes (supplied by the completed in-run items): Spectral completeness of the curl form in a smooth PEC cavity and Spectral completeness of the declared PEC waveguide and cavity classes declare, under the full Axiom of Choice, a bounded $C^{\infty}$ domain $Ω \subset R^{3}$ , the graph-norm completion $X$ in $H (curl) \cap H (div)$ of the smooth fields on the closure with $n \times u = 0$ , the subspace $V = {u \in X : div u = 0}$ and $H = {\overline{V}}^{L^{2}}$ , and supply a complete orthonormal $L^{2}$ eigenbasis ${u_{j}}$ of $H$ with nonnegative finite-multiplicity eigenvalues $λ_{j}$ , the distributional equation $curl curl u_{j} = λ_{j} u_{j}$ with PEC trace, the full- $X$ variational identity $\int_{Ω} curl u_{j} \cdot curl v = λ_{j} \int_{Ω} u_{j} \cdot v$ for every $v \in X$ , and $C^{\infty}$ eigenfields on the closure. Their statements are the sole premises taken from them, and the proof steps below name the supplier consumed at each use.

[F4]

Elliptic solvability: Weak and classical solvability of the elliptic boundary problems with Dirichlet, Neumann and transmission data (a) supplies, under the Axiom of Choice and Countable Choice, for a bounded $C^{\infty}$ domain and smooth data the unique smooth solution of the Dirichlet problem $- div (ε \nabla u) = ρ$ , $u ∣_{\partial Ω} = g$ .

[F5]

Hilbert-space tools: orthogonal decomposition by a closed subspace and Fourier expansion (Orthogonal decomposition by a closed subspace, Fourier expansion in a Hilbert space); Cauchy–Schwarz and completeness of the complex and real $L^{2}$ pairings (Cauchy-Schwarz inequality for $L^{2}$ , Complex completeness, density, and inner product: the consumer interface).

[F6]

Distributional and one-variable tools: distributional derivative and its compatibility with $C^{1}$ regular distributions, with differentiation commuting with the embedding (Distributional derivative, Distributional differentiation is continuous and commutes); a distribution with zero derivative on a connected open set is constant (A distribution with zero derivatives on a connected open set is constant); the oriented primitive of a continuous function is $C^{1}$ with the given derivative and continuous functions on compact intervals are Riemann integrable (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, 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 function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion); dominated convergence for passing integrals of uniformly convergent families (Dominated convergence).

[F7]

ODE tools: globally Lipschitz autonomous systems on a finite-dimensional real space have unique global solutions (A globally state-Lipschitz vector field on $R \times R^{n}$ has global solutions); the scalar characteristic alternatives are the elementary algebra of $a^{''} + γ a^{'} + v^{2} λ a = 0$ with the real square roots of Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , the real exponential limits (The exponential tends to $+ \infty$ at $+ \infty$ and to $0$ at $- \infty$ , The exponential dominates every fixed nonnegative integer power at $+ \infty$ ) and the sine/cosine derivative rules with the real cartesian form of the complex exponential (The derivatives of sine and cosine are cosine and minus sine, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ); for a nonnegative real $λ$ the positive square root $\sqrt{λ}$ is the one of Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ .

[F8]

Vector-calculus identities: $div (a \times b) = b \cdot curl a - a \cdot curl b$ , the curl of a gradient vanishes, the divergence theorem on bounded $C^{1}$ domains, the cross product is bilinear, alternating and orthogonal, and the Euclidean inner product is the coordinate dot product (The divergence and curl of a cross product, The curl of the gradient of a $C^{2}$ function vanishes, Divergence on a bounded C1 Euclidean domain, 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 $R^{n}$ , Divergence and curl of a $C^{1}$ vector field).

[F9]

Collar and cutoff: a bounded smooth domain has a tubular neighbourhood of its boundary, and the standard smooth step function supplies the cutoff used in the pairing argument (The Euclidean tubular neighbourhood theorem, The standard smooth step function). Complex modulus and conjugation laws are Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive.

Proof

technique · direct: solve the finite-dimensional damped modal systems, sum the exact energy identities, pass finite sums to the energy-norm limit, classify the scalar characteristics, and identify the limiting zero sectors

1.1F1F2

[Rectangular damped modal system] Assume Countable Choice. Fix the box $Q$ and the reflection class of [F2], and let $(E, B)$ be a $C (R; L^{2})$ distributional solution of the damped first-order system of [F1] in that class. The finite separated Fourier modes are dense in the class by [F2], and for a wavevector $k \in (π / L_{1}) Z \times (π / L_{2}) Z \times (π / L_{3}) Z$ the coefficient system of the damped system reads ${\hat{E}}_{t} = i v^{2} k \times \hat{B} - γ \hat{E}$ , ${\hat{B}}_{t} = - i k \times \hat{E}$ : this is [F2]'s recorded coefficient system with the additional scalar term $- γ \hat{E}$ of [F1], and the algebraic constraint $k \cdot (k \times w) = 0$ preserves $k \cdot \hat{E} = k \cdot \hat{B} = 0$ , while the finite Fourier modes carry the parities of the class because the reflection rules are diagonal in the separated basis.

1.2F3F4F5F8F9

[Smooth PEC cavity spaces and the magnetic decomposition] Assume the full Axiom of Choice as in [F3], let $Ω$ be a bounded $C^{\infty}$ domain and let $X, V, H$ , ${u_{j}}$ , $λ_{j} \geq 0$ be the classes and eigenbasis of [F3]. Put $M : = {B \in L^{2} (Ω; R^{3}) : \int_{Ω} B \cdot \nabla ψ = 0 for every smooth scalar ψ on \overline{Ω}}$ , the weak normal-zero divergence-free space, and $K_{M} : = {B \in M : curl B = 0 distributionally}$ . For each positive eigenvalue put $w_{j} : = curl u_{j} / \sqrt{λ_{j}}$ . Then the $w_{j}$ are orthonormal, $w_{j} \in M$ , and $M = {\overline{span {w_{j} : λ_{j} > 0}}}^{L^{2}} \oplus K_{M}$ . Orthonormality: the full- $X$ identity of [F3] with the test $v = u_{l} \in V \subseteq X$ gives $(curl u_{j}, curl u_{l}) = λ_{j} (u_{j}, u_{l}) = λ_{j} δ_{j l}$ , so $(w_{j}, w_{l}) = δ_{j l}$ . Membership and the forward inclusion: for a smooth scalar $ψ$ , [F8] gives $div (u_{j} \times \nabla ψ) = \nabla ψ \cdot curl u_{j} - u_{j} \cdot curl \nabla ψ = \nabla ψ \cdot curl u_{j}$ , and its integral over $Ω$ is the boundary integral $\int_{\partial Ω} (n \times u_{j}) \cdot \nabla ψ = 0$ because $u_{j}$ is smooth on the closure with $n \times u_{j} = 0$ [F3]; hence $w_{j} \in M$ , and the closed span of the $w_{j}$ lies in $M$ . For an element $u \in V$ , the $V$ -norm identity of [F3] with $∥ u_{j} ∥_{V}^{2} = 1 + λ_{j}$ gives the expansion $u = \sum_{j} (u, u_{j}) u_{j}$ in $L^{2}$ , and applying the bounded operator $curl$ on $V$ gives $curl u = \sum_{j} (u, u_{j}) \sqrt{λ_{j}} w_{j}$ in $L^{2}$ ; hence $\overline{curl V} = \overline{span {w_{j}}}$ , and both lie in the closed subspace $M$ . Decomposition: the orthogonal decomposition theorem [F5] applied to the closed subspace $\overline{span {w_{j}}}$ of $M$ writes $B \in M$ as the sum of its projections onto that span and onto its orthogonal complement inside $M$ ; it remains to identify the complement with $K_{M}$ . If $B \in M$ is orthogonal to every $curl v$ with $v \in V$ and $φ$ is smooth and compactly supported in $Ω$ , then the smooth Dirichlet solver [F4] with coefficient identity, zero boundary trace and source $- div φ$ supplies a smooth $f$ on the closure with $Δ f = div φ$ ; then $φ - \nabla f$ is smooth on the closure with $div (φ - \nabla f) = 0$ and with vanishing tangential trace on $\partial Ω$ (the first term is compactly supported, and $f ∣_{\partial Ω} = 0$ forces the tangential derivatives of $f$ , hence $n \times \nabla f$ , to vanish there), so $φ - \nabla f$ lies in $V$ ; since $curl (φ - \nabla f) = curl φ$ by [F8], we obtain $(B, curl φ) = 0$ ; as $φ$ is an arbitrary compactly supported smooth test, this is exactly the distributional statement $curl B = 0$ , i.e. $B \in K_{M}$ . Conversely, if $B \in K_{M}$ then for each eigenfield $u_{j}$ the collar cutoff of [F9] yields compactly supported smooth approximants $η_{δ} u_{j}$ with $η_{δ} u_{j} \to u_{j}$ and $curl (η_{δ} u_{j}) \to curl u_{j}$ in $L^{2}$ (the boundary layer has width $2 δ$ and $n \times u_{j} = 0$ makes the boundary term vanish in the limit), so the distributional pairing $\int B \cdot curl (η_{δ} u_{j}) = 0$ passes to the limit and gives $(B, curl u_{j}) = 0$ , hence $(B, w_{j}) = 0$ ; by continuity of the pairing and closedness of the span, $B$ is orthogonal to $\overline{span {w_{j}}}$ , which is the claimed complement and finishes the decomposition.

1.3F7

[Scalar characteristic classification] For parameters $γ \geq 0$ , $v > 0$ , $λ \geq 0$ , the solutions of $a^{''} + γ a^{'} + v^{2} λ a = 0$ are classified by $Δ : = γ^{2} / 4 - v^{2} λ$ : if $Δ > 0$ the characteristic roots are $- γ / 2 \pm \sqrt{Δ}$ and the general solution is $A e^{(- γ / 2 + \sqrt{Δ}) t} + B e^{(- γ / 2 - \sqrt{Δ}) t}$ ; if $Δ < 0$ the roots are $- γ / 2 \pm i \sqrt{- Δ}$ and the general solution is $e^{- γ t / 2} (A \cos (\sqrt{- Δ} t) + B \sin (\sqrt{- Δ} t))$ ; if $Δ = 0$ it is $(A + B t) e^{- γ t / 2}$ ; and if $λ = 0$ the solutions are $A + B e^{- γ t}$ for $γ > 0$ and $A + B t$ for $γ = 0$ . These are finite real computations with the square roots of [F7] and the real exponential and trigonometric solutions certified by [F7]; no choice principle is used.

2.1step 1.1F2F5F6F7F8F9

[Rectangular class: global solutions, energy and convergence] For every initial pair in the rectangular class of [F2] the damped system of [F1] has a unique $C (R; L^{2})$ distributional solution in the class, finite Fourier mode sums converge in energy norm uniformly on compact time intervals and are dense at every time, and the coefficient equations of step 1.1 hold for every wavevector. Indeed, for each $k$ the right-hand side of the finite-dimensional system of step 1.1 is globally state-Lipschitz (it is affine), so [F7] gives a unique global solution through the initial coefficients; differentiate the per-mode energy $ε ∣ \hat{E} ∣^{2} + μ^{- 1} ∣ \hat{B} ∣^{2}$ . With $C : = \overline{\hat{E}} \cdot (k \times \hat{B})$ and $\overline{\hat{B}} \cdot (k \times \hat{E}) = - \overline{C}$ , the curl contribution is $(2 / μ) Re (i C + i \overline{C}) = 0$ by the scalar triple-product identity of [F8] and complex conjugation [F9]; this gives $\frac{d}{d t} (ε ∣ \hat{E} ∣^{2} + μ^{- 1} ∣ \hat{B} ∣^{2}) = - 2 σ ∣ \hat{E} ∣^{2} \leq 0$ , so each modal energy is nonincreasing forward and bounded by $e^{2 γ T}$ times its initial value on $[- T, T]$ (multiply by $e^{2 γ t}$ and differentiate); Parseval for the complete class [F2] therefore controls the energy-norm of every omitted tail uniformly on compact times. Finite solutions converge in $C_{t} L^{2}$ to a limit in the class, and the linear distributional equations pass to the limit against compactly supported smooth periodic tests by Cauchy–Schwarz [F5] on a compact time support, exactly as in the lossless argument of [F2]; the constraints $k \cdot \hat{E} = k \cdot \hat{B} = 0$ and the parities are preserved by step 1.1, so the limit lies in the same class. Uniqueness: a $C_{t} L^{2}$ weak solution has continuous coefficients whose distributional derivatives equal the displayed continuous right-hand sides; for each coefficient equation, subtract the oriented primitive of the continuous right-hand side, which is $C^{1}$ with that derivative by [F6], use the $C^{1}$ regular-distribution compatibility of [F6] to identify its distributional derivative, conclude that the difference has zero distributional derivative, hence is constant by [F6], and match the initial value; global uniqueness of the finite system identifies the coefficients, and completeness of the Fourier class identifies the fields.

3.1step 1.2F3F5F7

[Smooth PEC cavity: modal system, variational equations and uniqueness] Assume the full Axiom of Choice as in [F3]. Let $H \times M$ be the energy class of steps 1.2, with $M = \overline{span {w_{j}}} \oplus K_{M}$ . For coefficients $a_{j} = (\cdot, u_{j})$ , $b_{j} = (\cdot, w_{j})$ declare the first-order modal system $a_{j}^{'} = v^{2} \sqrt{λ_{j}} b_{j} - γ a_{j}$ , $b_{j}^{'} = - \sqrt{λ_{j}} a_{j}$ for $λ_{j} > 0$ , and $a_{j}^{'} = - γ a_{j}$ for the electric zero modes, and declare the magnetic kernel data constant, $(B, K_{M})^{'} = 0$ for every kernel test. Endow the paired-magnetic space with the completion $D_{M}$ of finite $w_{j}$ -sums plus $K_{M}$ for the norm $∥ w ∥_{D_{M}}^{2} : = ∥ w ∥_{2}^{2} + ∥ curl w ∥_{2}^{2}$ , and consider the weak variational equations $⟨ E_{t}, ϕ ⟩ = v^{2} ⟨ B, curl ϕ ⟩ - γ ⟨ E, ϕ ⟩$ for $ϕ \in V$ and $⟨ B_{t}, ψ ⟩ = - ⟨ E, curl ψ ⟩$ for $ψ \in D_{M}$ . Then every initial pair in $H \times M$ has a unique $C (R; H \times M)$ solution of this declared realization, it satisfies the modal equations distributionally, and finite electric and paired-magnetic modal sums together with the static kernel component converge uniformly on compact time intervals in energy norm. Existence and uniqueness are finite-dimensional: each paired system is a globally Lipschitz constant-coefficient system solved by [F7], and the kernel data are constant, so a candidate solution is the norm-limit of its finite modal sums; the sums converge because the energy $\sum_{j} (ε ∣ a_{j} ∣^{2} + μ^{- 1} ∣ b_{j} ∣^{2})$ is controlled as in step 2.1 and the kernel component is retained exactly. Each finite sum with the static kernel satisfies the first-order equations distributionally in the interior: $curl w_{j} = \sqrt{λ_{j}} u_{j}$ follows from [F3], while the electric zero eigenfields and magnetic kernel have zero curl. Uniform compact-time $L^{2}$ convergence passes these linear equations to the limit against compactly supported smooth spacetime tests by Cauchy–Schwarz [F5]. For the variational equations, the finite sums satisfy them against finite spectral tests by the full- $X$ identity of [F3], and the density of the finite spectral tests in $V$ and in $D_{M}$ passes the identities to the limit by Cauchy–Schwarz [F5], since uniform energy convergence on compact times controls both pairings. Conversely, testing the variational equations against $u_{j}$ (for $ϕ$ ) and $w_{j}$ or kernel vectors (for $ψ$ ) recovers the modal equations, so the weak realization and the modal system are equivalent in the stated class; uniqueness follows from the scalar ODE uniqueness and the complete decomposition of step 1.2. For strong first-order solutions with $E \in C_{t}^{1} H \cap C_{t} V$ and $B \in C_{t}^{1} L^{2} \cap C_{t} H (curl)$ , the same pairing gives $a_{j}^{'} = v^{2} (B, curl u_{j}) - γ a_{j}$ and, differentiating with $B^{'} = - curl E$ and the full- $X$ identity, $a_{j}^{''} + γ a_{j}^{'} + v^{2} λ_{j} a_{j} = 0$ with $a_{j}^{'} = - γ a_{j}$ at $λ_{j} = 0$ . No maximal rough tangential-trace realisation is claimed.

4.1step 2.1step 3.1F6

[Exact energy balance] In both declared classes, for every $t_{0} < t_{1}$ , $U (t_{1}) + σ \int_{t_{0}}^{t_{1}} ∥ E (t) ∥_{2}^{2} d t = U (t_{0})$ and $U^{'} (t) = - σ ∥ E (t) ∥_{2}^{2} \leq 0$ . In the rectangular class, sum the finite per-mode identities of step 2.1 over the wavevectors of a finite modal projection, integrate over $[t_{0}, t_{1}]$ with [F6], and pass to the limit using the uniform energy-norm convergence of step 2.1 to pass both the energy and the continuous $∥ E (t) ∥_{2}^{2}$ -integral terms; in the smooth class, do the same with the paired modal energies of step 3.1 and the retained kernel component, whose curl vanishes. The derivative form then follows from the integral form by [F6]; the boundary term of a smooth local Poynting identity is not used, so no unprovided nonlinear weak local identity is asserted.

4.2step 1.2step 1.3step 3.1F3F6F7

[Associated second-order branch] Independently of first-order compatibility, the same spectral basis gives the damped associated electric wave for every $q_{0} \in V$ , $q_{1} \in H$ , with $q \in C (R; V)$ , $q^{'} \in C (R; H)$ and modal convergence uniform on compact times: its coefficients satisfy $a_{j}^{''} + γ a_{j}^{'} + v^{2} λ_{j} a_{j} = 0$ with prescribed initial coefficients $a_{j} (0) = (q_{0}, u_{j})$ , $a_{j}^{'} (0) = (q_{1}, u_{j})$ , and at $λ_{j} = 0$ they are $α_{j} + β_{j} (1 - e^{- γ t}) / γ$ for $γ > 0$ and $α_{j} + t β_{j}$ for $γ = 0$ . Indeed, for $λ_{j} > 0$ the function $Q_{j} = ∣ a_{j}^{'} ∣^{2} + v^{2} λ_{j} ∣ a_{j} ∣^{2}$ satisfies $Q_{j}^{'} = - 2 γ ∣ a_{j}^{'} ∣^{2} \leq 0$ by [F7], and the positive eigenvalues, being finite in multiplicity and tending to infinity, have a positive lower bound, so compact-time exponential bounds on $Q_{j}$ control the $V$ displacement and $H$ velocity tails by the weighted $V$ -norm identity of [F3]; the finitely many zero modes have the displayed solutions by step 1.3, and the finite-sum FTC estimate of [F6] passes the $H$ derivative to the limit and identifies the variational second-order equation against compact-time tests. Such arbitrary second-order data need not arise from the first-order Maxwell system: its zero modes additionally satisfy $a_{j}^{'} = - γ a_{j}$ , and the broader wave is retained alongside the physical first-order branch. All square roots and real derivatives are those of [F7].

5.1step 1.2step 1.3step 3.1step 4.1F7

[Decay, zero sectors and long-time limits] For $σ > 0$ (so $γ > 0$ ) every positive mode decays: if $Δ_{j} = γ^{2} / 4 - v^{2} λ_{j} > 0$ the two real roots $- γ / 2 \pm \sqrt{Δ_{j}}$ are negative; if $Δ_{j} < 0$ the complex roots have real part $- γ / 2 < 0$ , and the oscillatory factor is bounded by the cosine and sine identities of [F7]; if $Δ_{j} = 0$ the factor $t e^{- γ t / 2}$ tends to zero by [F7]. The positive-mode derivatives $a_{j}^{'}$ also tend to zero from the same explicit formulas, so $b_{j} = (a_{j}^{'} + γ a_{j}) / (v^{2} \sqrt{λ_{j}}) \to 0$ for each positive mode. Electric zero modes obey $a_{j} (t) = a_{j} (0) e^{- γ t}$ by step 1.3, and magnetic kernel data stay constant by step 1.2, so in the smooth class the solution tends in energy norm as $t \to + \infty$ to $(0, P_{K_{M}} B (0))$ , where $P_{K_{M}}$ is the orthogonal projection onto $K_{M}$ of step 1.2; the finite-head/finite-tail argument of step 3.1 passes the modewise convergence to the whole energy norm. For $σ = 0$ the energy is constant by step 4.1, the electric zero modes are static by step 1.3 and the original lossless branch is recovered. The zero sectors are retained, not divided by their eigenvalues, and no universal decay to zero in the presence of magnetic kernel data is claimed.

6.1step 1.2step 2.1step 3.1step 4.1step 4.2step 5.1F1∎

[Scope] Clauses (a) and (b) are steps 2.1 and 3.1: the rectangular class has a unique finite-energy distributional solution with dense finite modes at every time, and the smooth PEC class has the declared weak realization with a unique solution and energy-norm convergence of finite modal sums; the decomposition of the magnetic space, including arbitrary kernel data, is step 1.2. Clause (c) is steps 4.1 and 5.1. Clause (d) is step 4.2. Clause (e): the argument used only the declared constant Ohmic branch, the ideal PEC boundary model, the finite-energy classes and the recorded spectral and elliptic suppliers; Countable Choice suffices for the rectangular branch, the smooth branch retains its full Choice premise, no arbitrary loss tensor, memory law, changing interface, open-end, rough-wall or empirical claim is made, and weak energy balance was proved by modal limits without asserting an unprovided nonlinear weak local Poynting theorem.

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Complete longitudinal Fourier energy evolution in a rectangular PEC guide

Physical scope

The explicitly declared constant scalar Ohmic PEC energy classes of K10, including the lossless sigma=0 branch, with exact modal/Fourier completeness and energy balance. No arbitrary dispersive, nonlinear, hysteretic, nonsmooth-wall, open-end or junction-scattering claim; no empirical claim.

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and the constant isotropic material branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch with $ε, μ > 0$ , $σ \geq 0$ , $J_{e x t} = 0$ and initially zero volume charge. For an exact broadband physical interpretation retain the closure condition $ε μ \geq c^{- 2}$ ; without it this is the declared effective model with no universal front-speed claim. Put $v^{2} = (ε μ)^{- 1}$ , $γ = σ / ε$ . On the infinite straight rectangular guide $Q = (0, a) \times (0, b) \times R$ of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM, declare the finite-energy reflection class as follows. Extend in the two transverse variables to $T_{a b}^{2} \times R$ , $T_{a b}^{2} = R / (2 a Z) \times R / (2 b Z)$ . The electric components have transverse parities even/odd, odd/even, odd/odd respectively; magnetic components have odd/even, even/odd, even/even respectively. Require the extended real fields to be $L^{2}$ and have both divergences zero distributionally. When the reflected extensions are continuous across the walls, their odd components give the four-wall conditions $n \times E = 0$ , $n \cdot B = 0$ ; it is a precise weak PEC realization, not an assertion about every rough trace class. There is no end cap and no junction or open-end scattering datum. (a) [Complete mixed transform] Let $ϕ_{m n} (x_{t}) = (4 a b)^{- 1 / 2} e^{i π (m x_{1} / a + n x_{2} / b)}$ , $(m, n) \in Z^{2}$ , be the orthonormal transverse characters. For each component first form $F_{m n} (z) = \int_{T_{a b}^{2}} F (x_{t}, z) \overline{ϕ_{m n} (x_{t})} d x_{t}$ , then apply the exact negative-sign $2 π$ Fourier transform $F_{2}$ of Plancherel theorem in $z$ . Write ${\hat{F}}_{m n} (ξ) = F_{2} F_{m n} (ξ)$ and $β = 2 π ξ$ . The transform is a unitary representation of $L^{2} (T_{a b}^{2} \times R)$ by $ℓ^{2} (Z^{2}; L^{2} (R))$ , with inverse in $L^{2}$ from L2 Fourier inversion, and $∥ F ∥_{L^{2} (Q)}^{2} = \frac{1}{4} \sum_{m, n} \int_{R} ∣ {\hat{F}}_{m n} (ξ) ∣^{2} d ξ$ for these reflection fields. The parity and reality relations are retained, including ${\hat{F}}_{- m, - n} (- ξ) = \overline{{\hat{F}}_{m n} (ξ)}$ . With $K = (π m / a, π n / b, 2 π ξ)$ , the divergence constraints become $K \cdot \hat{E} = K \cdot \hat{B} = 0$ for almost every $ξ$ . This is a complete transverse sin/cos and longitudinal $L^{2}$ Fourier representation, not pointwise Fourier convergence or a finite-energy monochromatic plane wave. (b) [Unique energy evolution] Adopt in that weak reflection class the constant-Ohmic first-order material evolution $E_{t} = v^{2} curl B - γ E$ , $B_{t} = - curl E$ , distributionally, consistent with the smooth The macroscopic Maxwell equations in terms of D, H and the free sources and Energy balance and Ohmic dissipation for the constant nondispersive linear model branches. For every initial pair in the class there is a unique $C (R; L^{2} (Q))$ solution in the same class. Its mixed coefficients solve, for every transverse index and almost every $ξ$ , $\partial_{t} \hat{E} = i v^{2} K \times \hat{B} - γ \hat{E}, \partial_{t} \hat{B} = - i K \times \hat{E} .$ Finite transverse sums and bounded longitudinal frequency truncations reconstruct the solution in energy norm uniformly on compact time intervals; such reconstructed fields are dense in the class at every time. They retain the divergence constraints and the original parities, including zero coefficient cases without dividing by $K$ or $∣ K ∣$ . (c) [Energy and dispersion] The total stored energy of this whole infinite guide is finite and equals $U (t) = \frac{1}{8} \sum_{m, n} \int_{R} (ε ∣ \hat{E} ∣^{2} + μ^{- 1} ∣ \hat{B} ∣^{2}) d ξ .$ For every $t_{0} < t_{1}$ , $U (t_{1}) + σ \int_{t_{0}}^{t_{1}} ∥ E (t) ∥_{L^{2} (Q)}^{2} d t = U (t_{0})$ , and $U^{'} (t) = - σ ∥ E (t) ∥_{2}^{2}$ . At $σ = 0$ energy is conserved and each constrained electric coefficient satisfies ${\hat{E}}^{''} + v^{2} ∣ K ∣^{2} \hat{E} = 0$ . A nonzero harmonic coefficient $\hat{E} (t) = e e^{- i ω t}$ with $e \in C^{3} ∖ {0}$ and $ω > 0$ therefore requires $ω^{2} = v^{2} [(π m / a)^{2} + (π n / b)^{2} + β^{2}]$ ; the zero coefficient imposes no frequency condition. These are the exact finite transverse-mode matrices of the approved K10 branch; individual separated TE/TM candidate fields still have their own cutoff and zero-profile restrictions, and no TE $_{00}$ mode is created by a scalar constant coefficient. For $σ > 0$ the electric equation is ${\hat{E}}^{''} + γ {\hat{E}}^{'} + v^{2} ∣ K ∣^{2} \hat{E} = 0$ with the distinct/repeated-root alternatives of the choice-free finite-parameter clause of Constant-Ohmic PEC modal decay and exact energy-class completeness(c), here at $λ = ∣ K ∣^{2} \geq 0$ . Its full-Choice smooth-cavity realization is not used. At $K = 0$ keep the direct first-order equations $\hat{E}' = - γ \hat{E}$ , $\hat{B}' = 0$ , without dividing by $K$ ; the original parity and $L^{2}$ relations still apply. No signal-speed or empirical inference is made. (d) [Scope] This theorem supplies the previously missing complete longitudinal $L^{2}$ expansion of the declared straight rectangular guide, including its lossless and constant-Ohmic energy evolutions. It makes no completeness claim for arbitrary non-rectangular, dispersive, nonlocal, nonlinear, rough-wall or open-end/junction problems. The weak energy result is proved by linear mixed-transform evolution and modal limits, not by applying a nonlinear smooth Poynting identity to rough data.

Facts & Assumptions

Given: Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )); the constant isotropic material branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch with constants $ε, μ > 0$ and $σ \geq 0$ , $J_{e x t} = 0$ and initially zero volume charge (with the closure condition $ε μ \geq c^{- 2}$ retained for the exact broadband interpretation); $v^{2} = (ε μ)^{- 1}$ , $γ = σ / ε$ ; the infinite straight rectangular guide $Q = (0, a) \times (0, b) \times R$ and its geometry of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM; the finite-energy reflection class of the Statement, defined by transverse extension to $T_{a b}^{2} \times R$ with the stated electric and magnetic parities, $L^{2}$ membership and distributionally zero divergences.

[F1]

The material and evolution branch: $D = ε E$ , $B = μ H$ , and with initially zero volume charge the macroscopic equations of The macroscopic Maxwell equations in terms of D, H and the free sources give the first-order system $E_{t} = v^{2} curl B - γ E$ , $B_{t} = - curl E$ and zero divergences; the stored energy is $U = \frac{1}{2} \int_{Q} (ε ∣ E ∣^{2} + μ^{- 1} ∣ B ∣^{2})$ (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch (b), The macroscopic Maxwell equations in terms of D, H and the free sources (b), Energy balance and Ohmic dissipation for the constant nondispersive linear model).

[F2]

Transverse completeness: the characters $ϕ_{m n} (x_{t}) = (4 a b)^{- 1 / 2} e^{i π (m x_{1} / a + n x_{2} / b)}$ are orthonormal in $L^{2}$ of the fundamental rectangle of $T_{a b}^{2}$ by elementary exponential integration, and the product families of cosines and sines are complete orthogonal systems (The sine and cosine systems are complete orthogonal systems of L^2 on an interval and on a rectangle (a),(b), Rectangular PEC reflection class: Fourier completeness, weak solutions and energy conservation (i), Fourier expansion in a Hilbert space); Fubini and Tonelli apply to the product spaces (Fubini's theorem for L^1 functions on a sigma-finite product, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F3]

Fourier analysis in $z$ : the unitary Plancherel transform $F_{2}$ with inverse in $L^{2}$ , the derivative Fourier identity on Schwartz functions, and Schwartz cutoffs from the standard smooth step function (Plancherel theorem, L2 Fourier inversion, Fourier transform acts continuously on Schwartz space, Explicit compactly supported smooth cutoffs).

[F4]

Distributional and integration tools: distributional derivative and its regular-distribution compatibility, a distribution with zero derivative on a connected open set is constant, the oriented primitive of a continuous function is $C^{1}$ with that derivative, continuous functions on compact intervals are integrable, and dominated convergence (Distributional derivative, Distributional differentiation is continuous and commutes, A distribution with zero derivatives on a connected open set is constant, 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, 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 function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Dominated convergence).

[F5]

ODE tools: globally Lipschitz finite-dimensional systems have unique global solutions and their solutions depend continuously on initial data and parameters (A globally state-Lipschitz vector field on $R \times R^{n}$ has global solutions, Continuous dependence of ODE solutions on initial data and parameters); the characteristic alternatives for ${\hat{E}}^{''} + γ {\hat{E}}^{'} + v^{2} λ \hat{E} = 0$ at $λ \geq 0$ are the finite-parameter classification of Constant-Ohmic PEC modal decay and exact energy-class completeness (c), whose smooth-cavity, full-Choice clauses are not used.

[F6]

Algebra and estimates: the componentwise cross product and its identities, complex conjugation and modulus laws, the cartesian form of the complex exponential, Cauchy–Schwarz, and completeness/density in complex $L^{2}$ (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors, Divergence and curl of a $C^{1}$ vector field, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Cauchy-Schwarz inequality for $L^{2}$ , Complex completeness, density, and inner product: the consumer interface, 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: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Proof

technique · direct: build the mixed transverse-character and longitudinal-Fourier representation, transport the divergence constraints and the evolution into coefficient form, solve the finite dimensional modal systems with uniform energy bounds, and pass cutoffs to the limit

1.1F2F3algebra

[Complete mixed transform] With the characters of [F2], first form for each component the transverse coefficients $F_{m n} (z) = \int_{T_{a b}^{2}} F (x_{t}, z) \overline{ϕ_{m n} (x_{t})} d x_{t}$ and then ${\hat{F}}_{m n} (ξ) = F_{2} F_{m n} (ξ)$ for the exact $2 π$ -convention transform of [F3]; write $β = 2 π ξ$ . Then the map $F \mapsto ({\hat{F}}_{m n})_{(m, n) \in Z^{2}}$ is a unitary isomorphism of $L^{2} (T_{a b}^{2} \times R)$ onto $ℓ^{2} (Z^{2}; L^{2} (R))$ and, for the reflection fields of the Statement, $∥ F ∥_{L^{2} (Q)}^{2} = \frac{1}{4} \sum_{m, n} \int_{R} ∣ {\hat{F}}_{m n} (ξ) ∣^{2} d ξ$ . Indeed the characters are orthonormal on the fundamental rectangle and, being products of the complete one-dimensional character systems, their finite span is dense in the transverse $L^{2}$ of each torus fibre by [F2]; Fubini–Tonelli therefore gives, for almost every $z$ , the transverse expansion together with $\int_{T_{a b}^{2}} ∣ F (x_{t}, z) ∣^{2} d x_{t} = \sum_{m, n} ∣ F_{m n} (z) ∣^{2}$ and, after integration in $z$ , $∥ F ∥_{L^{2} (T_{a b}^{2} \times R)}^{2} = \sum_{m, n} ∥ F_{m n} ∥_{L^{2} (R)}^{2}$ . Applying the unitary Plancherel transform [F3] to each of the countably many scalar functions $F_{m n} \in L^{2} (R)$ gives the coefficient map and the isometry on the torus product, with inverse from the $L^{2}$ inversion theorem of [F3]; the reflection extension doubles each transverse period, so the torus fundamental rectangle has $4 a b$ -area and the physical norm is one quarter of the extended norm, which is the displayed factor $\frac{1}{4}$ . The parity relations follow from $ϕ_{- m, - n} = \overline{ϕ_{m n}}$ and the transverse parities by the coordinate changes $x_{j} \mapsto - x_{j}$ on the torus, and the reality relation ${\hat{F}}_{- m, - n} (- ξ) = \overline{{\hat{F}}_{m n} (ξ)}$ holds first for real integrable compactly supported longitudinal data by the defining integral and passes to $L^{2}$ by density and continuity of the transform [F3]. No pointwise reconstruction is asserted.

2.1step 1.1F2F3F6

[Divergence constraints in coefficient form] For $K : = (π m / a, π n / b, 2 π ξ)$ the distributionally zero divergence of the extended field is equivalent to $K \cdot \hat{E} = K \cdot \hat{B} = 0$ for almost every $ξ$ , and the finite-energy reflection class is exactly the class of coefficient families satisfying these almost-everywhere constraints together with the parities and $L^{2}$ summability. The derivative correspondence is proved on the dense test class of finite transverse-character sums times longitudinal Schwartz functions: for such a test $ψ$ , finite transverse integration by parts and the exact identity $F_{2} (\partial_{z} ψ) = 2 π i ξ \hat{ψ}$ of [F3] turn the pairing $\int F \cdot \nabla ψ$ into the sum of the coefficient pairings against $i K \cdot \hat{F}$ , and the compactly supported smooth tests used in the definition are obtained by multiplying with cutoffs $χ_{R}$ of [F3] and passing to the limit, since the tests and their first derivatives converge in $L^{2}$ and all pairings are bounded by Cauchy–Schwarz [F6]; conversely a test with $K \cdot \hat{F} = 0$ almost everywhere has zero weak divergence pairing, and the general compact smooth tests are reached through the transverse $L^{2}$ expansions of their derivatives and the Schwartz transform isomorphism [F3]. Hence the exact constrained class corresponds to the stated almost-everywhere coefficient constraints, not merely to a scalar basis without Gauss equations; for fields smooth in the interior the divergences vanish pointwise, and if their reflected extensions are continuous across the walls then the four-wall PEC conditions hold by Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM(c).

3.1step 1.1step 2.1F1F3F4F5F6

[Unique energy evolution in the class] Adopt the constant-Ohmic first-order system of [F1] distributionally in the reflection class of step 2.1. For every initial pair in the class there is a unique $C (R; L^{2} (Q))$ solution in the same class, its mixed coefficients satisfy, for every $(m, n)$ and almost every $ξ$ , $\partial_{t} \hat{E} = i v^{2} K \times \hat{B} - γ \hat{E}$ and $\partial_{t} \hat{B} = - i K \times \hat{E}$ , and finite transverse sums with bounded longitudinal frequency truncations reconstruct the solution in energy norm uniformly on compact time intervals and are dense in the class at every time, retaining the divergence constraints and parities without dividing by $K$ or $∣ K ∣$ . Existence: for each $(m, n, ξ)$ the real and imaginary parts of the displayed system form a real finite-dimensional constant-coefficient linear system on all of $R$ , hence a globally Lipschitz system with a unique global solution through the initial coefficients by [F5]; the solution matrix depends continuously on $ξ$ uniformly on compact parameter and time sets by the continuous-dependence theorem [F5] (applied on a common bounded state cylinder, which the uniform energy bound below supplies), so multiplying the measurable initial coefficient vector by the continuous fundamental matrix gives measurable coefficient trajectories [F6]; no uncountable choice of solutions is made, and the construction uses only the countable families of coefficients. Energy: with $q : = ε ∣ \hat{E} ∣^{2} + μ^{- 1} ∣ \hat{B} ∣^{2}$ , conjugation and the triple-product cancellation of the displayed system give $q^{'} = - 2 σ ∣ \hat{E} ∣^{2}$ , whence $q (t) \leq e^{2 γ T} q (0)$ for $∣ t ∣ \leq T$ and forward contractivity at $σ \geq 0$ ; the constraints $K \cdot \hat{E} = K \cdot \hat{B} = 0$ and the parity and reality relations of step 1.1 are preserved because the right-hand side is linear, $K \cdot (K \times \hat{B}) = 0$ and the parities are respected by the cross product with $K$ . Cutoffs $∣ m ∣, ∣ n ∣ \leq N$ , $∣ ξ ∣ \leq N$ preserve every relation; their initial energy tails tend to zero by Tonelli and dominated convergence [F4], and the uniform bound makes the time-dependent tails tend to zero uniformly on compact times; each truncated solution is strongly differentiable in $L^{2}$ with bounded multiplier $K$ on the cutoff, and its linear equations follow from the test derivative identity of step 2.1, so the compact-time $L^{2}$ limits are continuous, have the prescribed data and satisfy the equations against compact smooth tests. Uniqueness: for any weak $C_{t} L^{2}$ reflection solution and bounded spectral cutoff $N$ , project to the cutoff coefficients; the test identity of step 2.1 turns the evolution into $f^{'} = A_{N} f$ distributionally in time in the cutoff Hilbert space with bounded constant matrix $A_{N}$ , and the finite transverse tests and smooth frequency tests supported in the cutoff interval, whose inverse transforms are Schwartz, are dense in the cutoff space, so the identity extends to every cutoff vector by the bounded multiplier and Cauchy–Schwarz [F6]. Subtracting the continuous Hilbert-valued primitive of $A_{N} f$ , constructed as the norm limit of finite Riemann sums (which is independent of the tags by the uniform oscillation bound on a compact interval, exists by completeness of $L^{2}$ , has the correct derivative by the average-increment bound, and is the vector version of the primitive argument of [F4]), pairing with every fixed cutoff vector and applying the scalar zero-derivative distribution theorem [F4] identifies the difference with its constant initial value; the difference $d$ of two solutions with zero data satisfies $d (t) = \int_{0}^{t} A_{N} d (s) d s$ , so on an interval of length $h$ with $h ∥ A_{N} ∥ < 1$ the supremum norm forces $d = 0$ , and finitely iterating these intervals in either direction on any compact time interval gives $d = 0$ ; letting the cutoffs exhaust the representation gives uniqueness of the whole field. The zero coefficient $K = 0$ is treated directly by the same equations with $K = 0$ , without division by $K$ or $∣ K ∣$ .

4.1step 1.1step 3.1F4F5F6algebra

[Energy and dispersion] The total stored energy of the whole infinite guide is finite and equals $U (t) = \frac{1}{8} \sum_{m, n} \int_{R} (ε ∣ \hat{E} ∣^{2} + μ^{- 1} ∣ \hat{B} ∣^{2}) d ξ$ , and for every $t_{0} < t_{1}$ one has $U (t_{1}) + σ \int_{t_{0}}^{t_{1}} ∥ E (t) ∥_{L^{2} (Q)}^{2} d t = U (t_{0})$ with $U^{'} (t) = - σ ∥ E (t) ∥_{2}^{2}$ . Indeed, summing the finite/cutoff modal identities $q^{'} = - 2 σ ∣ \hat{E} ∣^{2}$ of step 3.1 over the bounded cutoffs, integrating in $ξ$ and summing the finite transverse indices gives the truncated balance, and dominated convergence with the compact-time tail bound of step 3.1 passes it to the limit; the physical energy carries the factor $\frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}$ from step 1.1, and the dissipation term is continuous in time, so ordinary FTC [F4] gives the derivative form. At $σ = 0$ the energy is constant, and on the constrained coefficient subspace differentiating the first-order equation once and using $K \times (K \times e) = K (K \cdot e) - ∣ K ∣^{2} e$ together with $K \cdot \hat{E} = 0$ gives ${\hat{E}}^{''} + v^{2} ∣ K ∣^{2} \hat{E} = 0$ . Substitution of a harmonic coefficient $e e^{- i ω t}$ gives $(v^{2} ∣ K ∣^{2} - ω^{2}) e = 0$ , so for $e \neq 0$ and $ω > 0$ the stated dispersion relation follows with $β = 2 π ξ$ ; for $e = 0$ it supplies no condition on $ω$ ; at $K = 0$ the direct equations ${\hat{E}}^{'} = - γ \hat{E}$ , ${\hat{B}}^{'} = 0$ are retained without dividing by $K$ . For $σ > 0$ the electric equation is ${\hat{E}}^{''} + γ {\hat{E}}^{'} + v^{2} ∣ K ∣^{2} \hat{E} = 0$ at $λ = ∣ K ∣^{2} \geq 0$ with the distinct/repeated-root alternatives of the finite-parameter classification [F5], including the zero- $K$ case; the zero-profile restrictions and individual cutoffs of the separated TE/TM candidates are unaffected, and no TE $_{00}$ mode is created by a scalar constant coefficient.

5.1step 1.1step 2.1step 3.1step 4.1F1∎

[Scope] Clause (a) is steps 1.1 and 2.1, clause (b) is step 3.1, and clause (c) is step 4.1. The theorem supplies the complete longitudinal $L^{2}$ expansion of the declared straight rectangular guide with its lossless and constant-Ohmic energy evolutions and makes no completeness claim for arbitrary non-rectangular, dispersive, nonlocal, nonlinear, rough-wall or open-end/junction problems; the weak energy result is proved by linear mixed-transform evolution and modal limits, not by applying a nonlinear smooth Poynting identity to rough data, and no signal-speed or empirical inference is made.

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Complete mode expansions of the declared guide and cavity classes: transverse profile completeness, the rectangular reflection class and the smooth PEC cavity eigenbasis

Physical scope

The completeness statements available for the declared classes of the lossless isotropic scalar constant branch: transverse L^2 profile completeness of the TE/TM families on the rectangular cross-section, the rectangular cavity reflection class with its unique finite-energy weak solutions and constant stored energy, and the smooth bounded PEC cavity curl eigenbasis with modal expansions retaining zero modes. The separate declared open-longitudinal rectangular guide and constant-Ohmic classes have their own completeness/energy theorems. No arbitrary lossy, non-smooth or exterior completeness claim, nor a general non-rectangular guide claim beyond the stated smooth-cavity spectral class. Displacement and velocity topologies and first-order zero-mode compatibility are explicit.

Statement

Assume the Axiom of Choice and Countable Choice as declared by the cited mathematical suppliers (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )), the lossless isotropic scalar constant branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch with $ε, μ > 0$ and $σ = 0$ , the source-free smooth macroscopic equations of The macroscopic Maxwell equations in terms of D, H and the free sources in the smooth branches, with $ρ_{f} = 0$ , $J_{f} = 0$ , and the explicitly declared weak first-order material equations in the weak cavity branches below, and the guide, cavity, PEC and reflection-class conventions of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM. (a) [Transverse profile completeness] For the cross-section $(0, a) \times (0, b)$ , the families ${h_{m n}}_{m, n \geq 0}$ and ${e_{m n}}_{m, n \geq 1}$ of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals are orthogonal complete systems in $L^{2}$ of the rectangle (The sine and cosine systems are complete orthogonal systems of L^2 on an interval and on a rectangle); The full scalar Neumann and Dirichlet families have these $L^{2}$ expansions. The physical TE candidate family excludes $h_{00} = 1$ by the full Maxwell equations: the remaining $h_{m n}$ are complete in the zero-mean subspace ${f \in L^{2} : \int_{R} f = 0}$ , not in all scalar $L^{2}$ . The Dirichlet TM profiles retain their full scalar completeness. The constant Neumann scalar profile is retained in the mathematical basis but is not asserted to be a nonzero positive-frequency TE mode. (b) [Rectangular cavity reflection class] For the box $Q = (0, a) \times (0, b) \times (0, d)$ with the reflection class and distributional constraints $div E = div B = 0$ of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM(c): the finite separated Fourier modes are dense in the class at every time; every initial datum in the class has a unique $C (R; L^{2})$ weak solution of the lossless first-order system $\partial_{t} E = (ε μ)^{- 1} curl B$ , $\partial_{t} B = - curl E$ in the same class, with derivatives as in Distributional derivative; this lossless first-order distributional system is the explicitly adopted weak material evolution of the reflection model, not inferred from the smooth-only hypotheses of The macroscopic Maxwell equations in terms of D, H and the free sources; along the solution the total stored energy $\int_{Q} u d x$ equals $\frac{1}{2} \int_{Q} (ε ∣ E ∣^{2} + ∣ B ∣^{2} / μ) d x$ and is constant, where the weak stored-energy density is defined almost everywhere by this nonnegative quadratic expression, consistent with the smooth local definition of Energy balance and Ohmic dissipation for the constant nondispersive linear model. Its integrability and constancy are furnished by the mathematical reflection-class energy result; no nonlinear weak Poynting identity is asserted. For $C^{1}$ -in-time solutions the four macroscopic equations with $D = ε E$ , $B = μ H$ hold distributionally. (c) [Smooth PEC cavity] For a bounded $C^{\infty}$ cavity domain $Ω$ and the PEC class $X, V, H$ of Spectral completeness of the declared PEC waveguide and cavity classes(a): the curl form on $V$ has a complete orthonormal $L^{2}$ eigenbasis ${u_{j}}$ with nonnegative eigenvalues $λ_{j}$ of finite multiplicity, $λ_{j} \to \infty$ whenever $H$ is infinite dimensional, and a finite-dimensional zero eigenspace that is retained; the eigen-equation $curl curl u_{j} = λ_{j} u_{j}$ holds distributionally with the PEC trace, and each $u_{j}$ has a $C^{\infty}$ representative on the closure. Put $v_{w a v e} = (ε μ)^{- 1 / 2} > 0$ . The associated second-order electric wave has the full supplied interface: for arbitrary initial displacement in $V$ and velocity in $H$ , its displacement is in $C (R; V)$ , its velocity in $C (R; H)$ , and finite modal sums converge in these respective norms uniformly on compact time intervals. Its zero eigenmodes may be affine in time. Separately, let $J$ be an open time interval and let a strong first-order solution obey $E \in C^{1} (J; H) \cap C (J; V)$ and $B \in C^{1} (J; L^{2} (Ω; R^{3})) \cap C (J; H (curl))$ , with the first-order equations as $L^{2}$ identities and both distributional divergence constraints. Electric tangential-zero boundary data are encoded by $E (t) \in V$ . The magnetic normal-zero condition is declared weakly by $\int_{Ω} B (t) \cdot \nabla ψ = 0$ for every smooth scalar $ψ$ on the closure; when $B \in C^{1} (\overline{Ω})$ this is $n \cdot B = 0$ , using $div B = 0$ . No magnetic membership in the electric tangential-zero class $X$ is assumed. Then $a_{j} (t) : = ⟨ E (t), u_{j} ⟩$ is $C^{2}$ and satisfies $a_{j}^{''} + v_{w a v e}^{2} λ_{j} a_{j} = 0$ . For $λ_{j} = 0$ , the admissible first-order coefficients satisfy $a_{j}^{'} = 0$ and are constant, with the zero eigenfields retained. This extra compatibility distinguishes first-order Maxwell data from arbitrary second-order $V \times H$ data; it does not remove the broader supplied second-order wave or its affine zero modes. (d) [Scope] The theorem asserts completeness only for the transverse $L^{2}$ profiles and for the two declared cavity classes with their exact hypotheses; this lossless theorem does not itself supply the longitudinal expansion or dissipative evolution. Those approved additional classes are supplied separately by Complete longitudinal Fourier energy evolution in a rectangular PEC guide and Constant-Ohmic PEC modal decay and exact energy-class completeness, respectively. None of these statements supplies arbitrary lossy, dispersive, open-end scattering, non-smooth or exterior problems, or arbitrary non-rectangular guide problems beyond the explicitly declared smooth-cavity spectral class, no trace regularity for arbitrary $H (curl)$ fields, and no identification of the weak classes with every maximal tangential-trace realization. The zero eigenvalues and zero modes are retained throughout.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice as declared by the cited suppliers (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )); the lossless isotropic scalar constant branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch with $ε, μ > 0$ , $σ = 0$ ; the source-free smooth macroscopic equations of The macroscopic Maxwell equations in terms of D, H and the free sources with $ρ_{f} = 0$ , $J_{f} = 0$ ; the explicitly declared weak first-order material equations in the weak cavity branches; the guide, cavity, PEC and reflection-class conventions of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM; the fields of The electric and magnetic fields as fixed-time spatial vector fields.

[F1]

Transverse completeness: for the cross-section $(0, a) \times (0, b)$ the families ${h_{m n}}_{m, n \geq 0}$ and ${e_{m n}}_{m, n \geq 1}$ of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals are orthogonal complete systems in $L^{2}$ of the rectangle, with the full-family expansions of The sine and cosine systems are complete orthogonal systems of L^2 on an interval and on a rectangle (b) and Fourier expansion in a Hilbert space; the profile identities include $\int_{R} h_{m n} h_{m^{'} n^{'}} = 0$ unless $(m, n) = (m^{'}, n^{'})$ and $\int_{R} h_{00}^{2} = a b > 0$ , so $\int_{R} h_{m n} = 0$ for $(m, n) \neq (0, 0)$ by pairing with $h_{00} = 1$ (The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals (c)); $L^{2}$ completeness supplies the Hilbert-space structure ( $L^{2}$ with the integral pairing is a Hilbert space).

[F2]

Rectangular reflection class: Rectangular PEC reflection class: Fourier completeness, weak solutions and energy conservation states that the finite separated Fourier modes are dense in the reflection class, that every initial datum of the class has a unique $C (R; L^{2})$ weak solution of $\partial_{t} E = (ε μ)^{- 1} curl B$ , $\partial_{t} B = - curl E$ in the class, and that the energy $\frac{1}{2} \int_{Q} (ε ∣ E ∣^{2} + ∣ B ∣^{2} / μ)$ is constant along the solution; distributions and their derivatives are those of Distributional derivative, and the class is the one of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM (c); the packaged rectangular branch of Spectral completeness of the declared PEC waveguide and cavity classes (b) repeats these assertions.

[F3]

Smooth PEC cavity (supplied by the completed in-run items): Spectral completeness of the declared PEC waveguide and cavity classes (a) and Spectral completeness of the curl form in a smooth PEC cavity state, under the full Axiom of Choice, that for a bounded $C^{\infty}$ domain $Ω$ the graph-norm class $X$ of smooth fields on the closure with $n \times u = 0$ , its divergence-free subspace $V$ and $H = {\overline{V}}^{L^{2}}$ carry a complete orthonormal $L^{2}$ eigenbasis ${u_{j}}$ of the curl form with nonnegative finite-multiplicity eigenvalues $λ_{j}$ , the distributional eigen-equation $curl curl u_{j} = λ_{j} u_{j}$ with the PEC trace, $C^{\infty}$ eigenfields on the closure, the full- $X$ variational identity $\int_{Ω} curl u_{j} \cdot curl v = λ_{j} \int_{Ω} u_{j} \cdot v$ for every $v \in X$ , and the associated second-order wave at $v_{w a v e} = (ε μ)^{- 1 / 2} > 0$ for arbitrary $V \times H$ data with $C_{t} V$ displacement and $C_{t} H$ velocity. Their statements are the premises used here, and the proof steps below name the supplier consumed at each use.

[F4]

Paring tools: for $F \in H (curl)$ , a compactly supported smooth test $φ$ satisfies the distributional identity $⟨ curl F, φ ⟩ = ⟨ F, curl φ ⟩$ (Distributional derivative); Cauchy–Schwarz bounds the pairings; a bounded smooth boundary has a collar with the inward normal coordinate, and the standard smooth step builds the cutoff $η_{δ}$ used below (The Euclidean tubular neighbourhood theorem, The standard smooth step function, Cauchy-Schwarz inequality for $L^{2}$ ).

[F5]

The smooth stored-energy density $u = \frac{1}{2} (E \cdot D + B \cdot H)$ of Energy balance and Ohmic dissipation for the constant nondispersive linear model reduces to $\frac{1}{2} (ε ∣ E ∣^{2} + μ^{- 1} ∣ B ∣^{2})$ in the lossless scalar branch.

Proof

technique · direct: instantiate the transverse completeness and its zero-mean restriction, read the rectangular weak branch from its supplier and verify the macroscopic rewriting, then derive the first-order modal equation for the smooth cavity by a collar pairing and differentiate it once

1.1F1algebra

[Clause (a)] The families ${h_{m n}}$ and ${e_{m n}}$ are orthogonal complete systems in $L^{2} (R_{a b})$ by [F1], which gives the expansion statement. Each member with $(m, n) \neq (0, 0)$ has zero integral over $R_{a b}$ by the orthogonality of [F1] against $h_{00} = 1$ , so the closed linear span $S$ of ${h_{m n} : (m, n) \neq (0, 0)}$ is contained in the closed zero-mean subspace $Z = {f \in L^{2} (R_{a b}) : \int_{R_{a b}} f = 0}$ ; conversely, if $f \in Z$ , the expansion theorem [F1] applied to the complete orthonormal family obtained by normalizing all the $h_{m n}$ writes $f$ as the norm limit of its finite partial sums, and the coefficient of $h_{00}$ is $⟨ f, h_{00} ⟩ / ∥ h_{00} ∥^{2} = \int_{R_{a b}} f / a b = 0$ , so those partial sums use only indices $(m, n) \neq (0, 0)$ and $f \in S$ . Hence $S = Z$ : the remaining Neumann profiles are complete in the zero-mean subspace and not in all of $L^{2} (R_{a b})$ because the constant function is orthogonal to them and nonzero. The constant $h_{00} = 1$ stays in the full mathematical basis, while the physical TE candidate family excludes it by the Maxwell index restriction of The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff (c), so no nonzero positive-frequency TE reconstruction is asserted from the constant scalar profile; the Dirichlet profiles keep their full scalar completeness.

1.2F2F5algebra

[Clause (b)] By [F2] the finite separated Fourier modes are dense in the rectangular reflection class at every time, every initial datum in the class has a unique $C (R; L^{2})$ weak solution of the lossless first-order system in the class, and the energy $\frac{1}{2} \int_{Q} (ε ∣ E ∣^{2} + ∣ B ∣^{2} / μ)$ is constant along the solution; this distributional system is the explicitly adopted weak material evolution of the reflection model. The energy equals $\int_{Q} u$ for the weak stored-energy density $u = \frac{1}{2} (ε ∣ E ∣^{2} + μ^{- 1} ∣ B ∣^{2})$ of [F5], which is defined almost everywhere and nonnegative; its integrability and constancy are furnished by the supplier, and no nonlinear weak Poynting identity is asserted. Finally, the four macroscopic equations with $D = ε E$ and $H = B / μ$ follow distributionally from the two evolution equations and the divergence constraints of the class: $div D = ε div E = 0$ and $div B = 0$ are the class constraints, $curl E = - \partial_{t} B$ is the second evolution equation, and $curl H = μ^{- 1} curl B = μ^{- 1} (ε μ) \partial_{t} E = \partial_{t} D$ uses the first evolution equation $\partial_{t} E = (ε μ)^{- 1} curl B$ .

1.3F3

[Clause (c), spectral content] Assume the full Axiom of Choice as in [F3], let $Ω$ be a bounded $C^{\infty}$ cavity and let $X, V, H$ , ${u_{j}}$ , $λ_{j} \geq 0$ be the classes and eigenbasis of [F3]. Then the curl form on $V$ has the complete orthonormal $L^{2}$ eigenbasis with nonnegative finite-multiplicity eigenvalues tending to infinity when $H$ is infinite dimensional and a finite-dimensional zero eigenspace retained; the eigen-equation holds distributionally with the PEC trace, each $u_{j}$ has a $C^{\infty}$ representative on the closure, and the full- $X$ variational identity holds for every $v \in X$ . With $v_{w a v e} = (ε μ)^{- 1 / 2} > 0$ , the associated second-order electric wave has displacement in $C (R; V)$ and velocity in $C (R; H)$ for arbitrary initial data in $V \times H$ , finite modal sums converge in these norms uniformly on compact times, and its zero eigenmodes may be affine in time. These are the spectral and wave assertions of [F3], taken with the exact hypotheses of the suppliers.

2.1step 1.3F3F4

[Clause (c), first-order modal equation] Let $J$ be an open time interval and let a strong first-order solution obey the hypotheses of the Statement: $E \in C^{1} (J; H) \cap C (J; V)$ , $B \in C^{1} (J; L^{2}) \cap C (J; H (curl))$ , with the first-order equations holding as $L^{2}$ identities and both distributional divergence constraints, the magnetic normal-zero condition $\int_{Ω} B (t) \cdot \nabla ψ = 0$ for every smooth scalar $ψ$ on the closure, and $E (t) \in V$ encoding the electric tangential-zero data; no magnetic membership in $X$ is assumed. Put $a_{j} (t) : = ⟨ E (t), u_{j} ⟩$ . To pair $curl B$ against the smooth eigenfield $u_{j}$ , use the collar cutoff of [F4]: write $s$ for the inward normal coordinate in a tubular neighbourhood of the compact boundary, so that $∣ n \times u_{j} ∣ \leq C s$ in the collar because $u_{j}$ is smooth on the closure with $n \times u_{j} = 0$ , and let $η_{δ}$ be the standard smooth step with $η_{δ} = 0$ for $s \leq δ$ and $1$ for $s \geq 2 δ$ . Then $η_{δ} u_{j} \to u_{j}$ and $curl (η_{δ} u_{j}) \to curl u_{j}$ in $L^{2}$ as $δ \to 0$ , since the derivative-cutoff term is supported on a layer of volume $O (δ)$ and is bounded by $C / δ$ times $O (s)$ ; the distributional identity of [F4] on the compactly supported smooth test $η_{δ} u_{j}$ gives $⟨ curl B, η_{δ} u_{j} ⟩ = ⟨ B, curl (η_{δ} u_{j}) ⟩$ , and Cauchy–Schwarz with the two limits gives $⟨ curl B, u_{j} ⟩ = ⟨ B, curl u_{j} ⟩$ for the actual graph-regular field $B$ . Hence $a_{j}^{'} = ⟨ E^{'}, u_{j} ⟩ = v_{w a v e}^{2} ⟨ curl B, u_{j} ⟩ = v_{w a v e}^{2} ⟨ B, curl u_{j} ⟩$ on $J$ . The assumed $B \in C^{1} (J; L^{2})$ permits one further derivative: $a_{j}^{''} = v_{w a v e}^{2} ⟨ B^{'}, curl u_{j} ⟩ = - v_{w a v e}^{2} ⟨ curl E, curl u_{j} ⟩ = - v_{w a v e}^{2} λ_{j} ⟨ E, u_{j} ⟩$ , where the last equality is the full- $X$ eigenvariational identity of [F3] tested at $v = E (t) \in V \subset X$ ; thus $a_{j}^{''} + v_{w a v e}^{2} λ_{j} a_{j} = 0$ , and $a_{j}$ is $C^{2}$ because $E$ and $B$ are $C^{1}$ in their norms. If $λ_{j} = 0$ , the same identity at $v = u_{j}$ gives $∥ curl u_{j} ∥_{2}^{2} = 0$ , so $curl u_{j} = 0$ and $a_{j}^{'} = v_{w a v e}^{2} ⟨ B, curl u_{j} ⟩ = 0$ : the electric zero modes are constant under first-order Maxwell evolution, while the unrestricted second-order wave of step 1.3 retains its affine zero modes. This distinguishes first-order Maxwell data from arbitrary second-order $V \times H$ data; the pairing imposed no magnetic electric-type trace, and the divergence constraints and the declared weak magnetic normal condition remain part of the model.

3.1step 1.1step 1.2step 1.3step 2.1F3∎

[Clause (d)] Clause (a) is step 1.1, clause (b) is step 1.2, and clause (c) is steps 1.3 and 2.1. Completeness is asserted only for the transverse $L^{2}$ profiles and for the two declared cavity classes with their exact hypotheses; this lossless theorem does not itself supply the longitudinal expansion or the dissipative evolution, which are supplied separately by Complete longitudinal Fourier energy evolution in a rectangular PEC guide and Constant-Ohmic PEC modal decay and exact energy-class completeness. None of the statements supplies arbitrary lossy, dispersive, open-end scattering, non-smooth or exterior problems, or arbitrary non-rectangular guide problems beyond the declared smooth-cavity spectral class; no trace regularity for arbitrary $H (curl)$ fields and no identification of the weak classes with every maximal tangential-trace realization is asserted, and the zero eigenvalues and zero modes are retained throughout. The inherited Axiom of Choice and Countable Choice cover the spectral, Fourier and collar suppliers, and no lossy or open-domain step was made.

Waveguides and cavities

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Calculus of separated phasor fields: real-part derivative identities and amplitude injectivity

Statement

Facts & Assumptions

Proof

Transverse vector-calculus identities for the splitting R3=R2⊕RezR3=R2⊕Rez​

Statement

Facts & Assumptions

Proof

The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals

Statement

Facts & Assumptions

Proof

The sine and cosine systems are complete orthogonal systems of L^2 on an interval and on a rectangle

Statement

Facts & Assumptions

Proof

Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM

Definition

The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff

Statement

Facts & Assumptions

Proof

The propagating candidate branch: cutoff, phase and group speeds, cross-section energy and flux, and energy transport at the group velocity

Statement

Facts & Assumptions

Proof

Constant-Ohmic PEC modal decay and exact energy-class completeness

Statement

Facts & Assumptions

Proof

Complete longitudinal Fourier energy evolution in a rectangular PEC guide

Statement

Facts & Assumptions

Proof

Complete mode expansions of the declared guide and cavity classes: transverse profile completeness, the rectangular reflection class and the smooth PEC cavity eigenbasis

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources

Transverse vector-calculus identities for the splitting $R^{3} = R^{2} \oplus R e_{z}$