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Induction and circuit approximations

1 · Prerequisites

2 · Summary

This page develops induction and the lumped circuit approximation from the framework's adopted equations. Two local mathematical lemmas provide the Newtonian linkage pairing and the curl-field energy identity for smooth compact conserved currents, and a moving-circuit definition fixes the geometry of a $C^{2}$ embedded material sheet with a single-valued $C^{1}$ velocity, its magnetic flux and its electromotive force line integral. The moving-loop theorem then derives transport by differentiating the parametrized flux and sums the elementary-piece Faraday identities of the integral Maxwell theorem to prove $e m f = - {\dot{Φ}}_{B}$ under explicit no-singularity-crossing hypotheses, and the lumped circuit model is adopted as a separate postulate with declared branch relations, constant element matrices and a qualitative quasistatic regime. Conditional on that model, the page proves the reciprocal linkage coefficients and frozen magnetic energy of smooth compact conserved profiles, the fixed-geometry inductive energy bookkeeping and its identification with the frozen field energy, and the scalar RL and LC dynamics with their bookkeeping-energy statements. The geometric capacitor identification invoked for the LC loop is supplied by the electrostatic-conductors page and remains an explicit conditional clause. Ohmic power is retained in the adopted RL model; no additional material-loss mechanism, radiation, distributed-element, thin-wire or empirical claim is made.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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

Symmetry and Fubini representation of the Newtonian linkage pairing

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $N$ be the Newtonian potential transform of Newtonian potential of compactly supported data in dimension $n = 3$ : for a measurable finite-valued function $f : R^{3} \to R$ whose support $supp f = \overline{{x \in R^{3} : f (x) \neq 0}}$ is compact (The support of a function on $R^{n}$ and its compactly supported Riemann integral), $N f (x) = \int_{R^{3}} Φ (x - y) f (y) d λ_{3} (y)$ at each $x \in R^{3}$ where this integral is absolutely finite, where $Φ$ is the normalized kernel of Fundamental solution for the positive operator minus Laplacian with its value at the pole assigned $0$ , so that $Φ (z) = 1 / (4 π ∣ z ∣)$ for $z \neq 0$ ; write also $g (0) : = 0$ and $g (z) : = ∣ z ∣^{- 1}$ for $z \neq 0$ , so that $Φ = g / (4 π)$ pointwise on $R^{3}$ . Use the coordinate indices $i \in {0, 1, 2}$ of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , with $u_{i} (x) : = u (x) (i)$ and $⟨ u (x), v (y) ⟩ = \sum_{i = 0}^{2} u_{i} (x) v_{i} (y)$ . Let $⟨ \cdot, \cdot ⟩$ be that Euclidean inner product, and let $C_{c}^{\infty} (R^{3}; R^{3})$ be the set of maps each of whose three components is smooth ( $C^{k}$ Euclidean maps and diffeomorphisms) with compact support in the above sense; for a vector map $u$ , set $supp u : = \overline{{x : u (x) \neq 0}}$ , equivalently the finite union of its component supports. For $u, v \in C_{c}^{\infty} (R^{3}; R^{3})$ write $N u : = (N u_{0}, N u_{1}, N u_{2})$ componentwise, so that the linkage pairing is the map $B (u, v) : = \int_{R^{3}} ⟨ u (x), N v (x) ⟩ d λ_{3} (x) .$ Then all three integrals $\int_{R^{3}} ⟨ u (x), N v (x) ⟩ d λ_{3} (x), \int_{R^{3}} ⟨ N u (x), v (x) ⟩ d λ_{3} (x), \frac{1}{4 π} \int_{R^{3}} \int_{R^{3}} \frac{⟨ u (x), v (y) ⟩}{∣ x - y ∣} d λ_{3} (y) d λ_{3} (x)$ converge absolutely (Integral over a measurable subset, The modulus of an integral is bounded by the integral of the modulus) and have the same value; therefore $B$ is well defined, real-valued, symmetric, $B (u, v) = B (v, u)$ , and bilinear over $R$ in each argument. In particular the common value of the three integrals is unchanged by exchanging the two density fields $u$ and $v$ .

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , and let $u, v : R^{3} \to R^{3}$ be smooth with compact supports, with $N$ , $B$ and $g$ as in the Statement.

[A1]

Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

For a measurable compactly supported $f$ , the Newtonian potential is $N f (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ at each $x$ where the integral is absolutely finite. (Newtonian potential of compactly supported data).

[F2]

The normalized Laplace kernel is $Φ_{n} (x) = ∣ x ∣^{2 - n} / ((n - 2) ω_{n - 1})$ for $n \geq 3$ and $x \neq 0$ ; its value at $0$ may be assigned arbitrarily, and $ω_{n - 1}$ is the chart surface measure, which agrees with the polar measure $σ$ . (Fundamental solution for the positive operator minus Laplacian).

[F3]

The chart surface measure on $S^{n - 1}$ equals the polar measure $σ$ , and $∣ S^{n - 1} ∣ = n ∣ B_{1} ∣$ . (Agreement with the existing polar sphere measure).

[F4]

For every $n \geq 1$ , the unit ball has $V_{n} (1) = π^{n / 2} / Γ (n / 2 + 1)$ . (The closed form for the volume of the unit $n$ -ball).

[F5]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ , and $Γ (1) = 1$ . (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

[F6]

$Γ (1 / 2) = \sqrt{π}$ . ( $Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral).

[F7]

The normalized Laplace kernel $Φ$ is locally integrable on $R^{n}$ for $n \geq 2$ . (Local integrability of the Laplace fundamental kernel).

[F8]

For $f : R^{n} \to R$ the support is $supp f : = \overline{{x \in R^{n} : f (x) \neq 0}}$ , and $f$ is compactly supported when $supp f$ is compact. (The support of a function on $R^{n}$ and its compactly supported Riemann integral).

[F9]

Let $k \in N$ , $m, q \geq 1$ and $U \subseteq R^{m}$ open; a map $f : U \to R^{q}$ is of class $C^{k}$ when each component is of class $C^{k}$ , and it is smooth, or $C^{\infty}$ , when it is $C^{k}$ for every $k \in N$ , in particular for every $k \geq 1$ . ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F10]

A scalar function of class $C^{k}$ on $U$ has every iterated derivative $\partial_{i_{r}} \dots \partial_{i_{1}} f$ for words of length $0 \leq r \leq k$ existing and continuous on $U$ ; the word of length $0$ denotes $f$ itself. ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[F11]

For $n \geq 1$ the Euclidean distance $d_{2}$ is a metric on $R^{n}$ . ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it).

[F12]

In a metric space, closed balls $\overset{ˉ}{B} (x, r)$ are closed. (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed).

[F13]

A subset $K$ of $R^{n}$ is compact if and only if it is closed and bounded for the Euclidean metric. (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F14]

A nonempty compact $A \subseteq R^{n}$ has the property that every continuous $f : A \to R$ attains a maximum and a minimum on $A$ . (For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).

[F15]

Every compact subset of a metric space is closed and bounded. (A compact subset of a metric space is closed and bounded).

[F16]

A subset $A$ of a metric space is bounded if $A = \emptyset$ or there are $x_{0}$ and $r > 0$ with $A \subseteq B (x_{0}, r)$ . (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

[F17]

For a measurable $f : X \to [0, + \infty]$ and $E \in A$ , the integral over $E$ is $\int_{E} f d μ : = \int f χ_{E} d μ$ . (Integral over a measurable subset).

[F18]

For measurable $f, g : X \to [0, + \infty]$ with $f \leq g$ , the integrals satisfy $\int f d μ \leq \int g d μ$ . (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F19]

If $f \in L^{1} (μ)$ , then $∣ \int f d μ ∣ \leq \int ∣ f ∣ d μ$ . (The modulus of an integral is bounded by the integral of the modulus).

[F20]

The Lebesgue integral is complex-linear on $L^{1} (μ)$ . (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F21]

For $n \geq 1$ , $x \in R^{n}$ and $r > 0$ , the Euclidean ball satisfies $0 < λ (B (x, r)) < \infty$ . (Euclidean balls have positive finite Lebesgue measure).

[F22]

Assume Countable Choice. Let $U, V \subseteq R^{n}$ be open and $T : U \to V$ a $C^{1}$ diffeomorphism. Then for every $f \in L^{1} (λ_{n})$ , $\int_{V} f (y) d λ_{n} (y) = \int_{U} f (T (x)) ∣ \det D T (x) ∣ d λ_{n} (x)$ . (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F23]

For nonnegative product-measurable $f$ on a product of sigma-finite measure spaces, the iterated integrals and the product integral agree. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

[F24]

For $f \in L^{1} (μ \times ν)$ on a product of sigma-finite measure spaces, the section integrals are integrable after the stated exceptional-set convention and the three integrals agree. (Fubini's theorem for L^1 functions on a sigma-finite product).

[F25]

Under the identification $R^{m + n} = R^{m} \times R^{n}$ , the Lebesgue measure $λ_{m + n}$ is the completion of the product measure $λ_{m} \times λ_{n}$ . (The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures).

[F26]

With coordinates indexed by $i < 3$ , $⟨ x, y ⟩ = \sum_{i = 0}^{2} x_{i} y_{i}$ ; it satisfies $⟨ x, y ⟩ = ⟨ y, x ⟩$ for all $x, y \in R^{n}$ . (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F27]

Under Countable Choice, every continuous map $R^{n} \to R^{m}$ is Borel measurable. (Continuous functions on Euclidean spaces are Borel measurable).

[F28]

If $\tilde{f}, \tilde{g} : R^{n} \to C$ are Borel measurable, then $(x, y) \mapsto \tilde{f} (x - y) \tilde{g} (y)$ is Borel measurable on $R^{2 n}$ . (Borel representatives make the convolution integrand Borel measurable).

[F29]

For positive Euclidean dimensions the Borel product sigma-algebra on $R^{m} \times R^{n}$ is the Euclidean Borel sigma-algebra of $R^{m + n}$ . (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}).

[F30]

Composition of Borel measurable maps preserves Borel measurability. (Composition with a Borel measurable outer map preserves measurability).

Proof

technique · direct

1.1F8F11F15F16algebra

The supports $K_{u} : = supp u$ and $K_{v} : = supp v$ are compact by [F8], hence closed and bounded in $(R^{3}, d_{2})$ by [F11] and [F15]; by [F16] there are $x_{u}, ρ_{u}$ and $x_{v}, ρ_{v}$ with $K_{u} \subseteq B (x_{u}, ρ_{u})$ and $K_{v} \subseteq B (x_{v}, ρ_{v})$ (for an empty support any pair works, since the empty set is bounded). Put $R : = \max {∣ x_{u} ∣ + ρ_{u}, ∣ x_{v} ∣ + ρ_{v}} > 0$ . For $x \in K_{u}$ the triangle inequality gives $∣ x ∣ \leq ∣ x_{u} ∣ + ∣ x - x_{u} ∣ < ∣ x_{u} ∣ + ρ_{u} \leq R$ , and likewise on $K_{v}$ , so $K_{u} \cup K_{v} \subseteq B_{R} (0) = B (0, R)$ ; moreover $u (x) = 0$ for $x \notin K_{u}$ and $v (y) = 0$ for $y \notin K_{v}$ by [F8].

1.2F1F2F3F4F5F6algebra

For $z \neq 0$ , [F2] gives $Φ (z) = ∣ z ∣^{2 - 3} / ((3 - 2) ω_{2}) = ∣ z ∣^{- 1} / ω_{2}$ , and [F3] gives $ω_{2} = ∣ S^{2} ∣ = 3 ∣ B_{1} ∣$ with $∣ B_{1} ∣ = V_{3} (1)$ . By [F4], $V_{3} (1) = π^{3 / 2} / Γ (5 / 2)$ ; by [F5] applied twice and [F6], $Γ (5 / 2) = \frac{3}{2} \cdot \frac{1}{2} \cdot Γ (1 / 2) = \frac{3}{4} \sqrt{π}$ , hence $V_{3} (1) = π^{3 / 2} / (\frac{3}{4} \sqrt{π}) = \frac{4}{3} π$ and $ω_{2} = 4 π$ . Therefore $4 π Φ (z) = ∣ z ∣^{- 1} = g (z)$ for $z \neq 0$ ; assigning the permitted pole value $Φ (0) = 0$ by [F2] gives $Φ = g / (4 π)$ on all of $R^{3}$ , so by [F1] the transform of the Statement is $N f (x) = \frac{1}{4 π} \int_{R^{3}} f (y) g (x - y) d λ_{3} (y)$ at each $x$ where this integral is absolutely finite.

1.3F9F10F26F27F28F29F30

For $i, j \in {0, 1, 2}$ , by [F9] the components $u_{i}, v_{j}$ are smooth, so by [F10] they are continuous, and by [F27] they are Borel measurable; likewise $g$ is Borel measurable because assigning the value $0$ at the pole of the continuous function $z \mapsto ∣ z ∣^{- 1}$ on $R^{3} ∖ {0}$ gives the function whose superlevel sets ${g > t}$ are $B_{1 / t} (0) ∖ {0}$ for $t > 0$ , $R^{3} ∖ {0}$ for $t = 0$ and $R^{3}$ for $t < 0$ , all open and hence Borel. Applying [F28] to $g$ and $v_{j}$ shows $(x, y) \mapsto g (x - y) v_{j} (y)$ is Borel on $R^{6}$ ; the maps $(x, y) \mapsto u_{i} (x)$ are Borel as compositions with the continuous projection, and for each pair $(i, j)$ the map $Φ_{i j} (x, y) : = (u_{i} (x), g (x - y) v_{j} (y))$ has Borel preimages of open rectangles by [F29]; multiplication $R^{2} \to R$ is continuous, so [F30] makes $(x, y) \mapsto u_{i} (x) v_{j} (y) g (x - y)$ Borel, and the finite sum $H (x, y) : = ⟨ u (x), v (y) ⟩ g (x - y)$ is Borel as well.

2.1F7F12F13F16F18step 1.2algebra

The closed ball ${\overset{ˉ}{B}}_{2 R} (0)$ is closed by [F12] and bounded by [F16], hence compact in $(R^{3}, d_{2})$ by [F13]. Since $Φ \geq 0$ in dimension three and $Φ = g / (4 π)$ by step 1.2, [F7] and [F18] give $K_{R} : = \int_{B_{2 R} (0)} g d λ_{3} = \int_{B_{2 R} (0)} 4 π Φ d λ_{3} \leq 4 π \int_{{\overset{ˉ}{B}}_{2 R} (0)} Φ d λ_{3} < \infty .$

3.1F18F22step 2.1step 1.2algebra

Fix $x \in B_{R} (0)$ . The translation-dilation map $T_{x} : R^{3} \to R^{3}$ , $T_{x} (y) : = x - y$ , is smooth with $T_{x}^{- 1} = T_{x}$ and $∣ \det D T_{x} (y) ∣ = 1$ for every $y$ , so it is a $C^{1}$ diffeomorphism; and $T_{x} (B_{R} (0)) = x - B_{R} (0) \subseteq B_{2 R} (0)$ , since $∣ x - y ∣ \leq ∣ x ∣ + ∣ y ∣ < 2 R$ for $∣ y ∣ < R$ . The function $g$ is nonnegative and $λ_{3}$ -measurable, and $\int_{B_{2 R} (0)} g < \infty$ by step 2.1, so $g 1_{B_{2 R} (0)} \in L^{1} (λ_{3})$ ; applying [F22] to it and using [F18] and $T_{x} (B_{R} (0)) \subseteq B_{2 R} (0)$ yields $\int_{B_{R} (0)} g (x - y) d λ_{3} (y) = \int_{T_{x} (B_{R} (0))} g (z) d λ_{3} (z) \leq \int_{B_{2 R} (0)} g d λ_{3} = K_{R} .$

4.1F17F18F21F23F24F25F26F29step 2.1step 3.1step 1.3step 1.1algebra

By step 1.1, $∣ u_{i} ∣ \leq M_{u}$ and $∣ v_{j} ∣ \leq M_{v}$ on all of $R^{3}$ for the finite constants $M_{u} : = \max_{{\overset{ˉ}{B}}_{R} (0)} ∣ u ∣$ and $M_{v} : = \max_{{\overset{ˉ}{B}}_{R} (0)} ∣ v ∣$ , which exist by [F14] applied to the continuous functions $∣ u ∣, ∣ v ∣$ on the nonempty compact set ${\overset{ˉ}{B}}_{R} (0)$ of steps 1.1 and 2.1 (a vanishing field has $M = 0$ ). The measure space is sigma-finite because $R^{3} = ⋃_{m = 1}^{\infty} B_{m} (0)$ and each ball has finite measure by [F21]. Hence, using [F17] and [F23] for the nonnegative Borel function $∣ u_{i} (x) ∣ ∣ v_{j} (y) ∣ g (x - y)$ on $R^{3} \times R^{3}$ , [F18] and step 3.1 give $\int_{R^{3}} \int_{R^{3}} ∣ u_{i} (x) ∣ ∣ v_{j} (y) ∣ g (x - y) d λ_{3} (y) d λ_{3} (x) = \int_{B_{R} (0)} \int_{B_{R} (0)} ∣ u_{i} (x) ∣ ∣ v_{j} (y) ∣ g (x - y) d λ_{3} (y) d λ_{3} (x) \leq M_{u} M_{v} λ_{3} (B_{R} (0)) K_{R} < \infty,$ with $λ_{3} (B_{R} (0)) < \infty$ by [F21] and $K_{R} < \infty$ by step 2.1. Summing over the three pairs $(i, j) = (0, 0), (1, 1), (2, 2)$ and using [F26] and $∣ ⟨ u, v ⟩ ∣ \leq \sum_{i} ∣ u_{i} ∣ ∣ v_{i} ∣$ , the double integral of $⟨ u (x), v (y) ⟩ g (x - y) / (4 π)$ converges absolutely; by [F25] and [F29] the Euclidean and product-measure structures on $R^{6}$ agree on Borel sets, so [F24] applies to this $L^{1}$ function and permits either order of integration.

5.1F7F18F19F20F22F23F24step 1.1step 1.3step 4.1algebra

For every $x \in R^{3}$ , $x - B_{R} (0) \subseteq B_{∣ x ∣ + R} (0)$ ; the change of variables $z = x - y$ of [F22] and local integrability [F7] give $\int ∣ v_{j} (y) ∣ Φ (x - y) d y \leq M_{v} (4 π)^{- 1} \int_{{\overset{ˉ}{B}}_{∣ x ∣ + R} (0)} g (z) d z < \infty$ . The same argument applies to $u$ , so $N u, N v$ are defined everywhere. Their components are measurable: apply [F23] to the nonnegative Borel integrands $Φ (x - y) v_{j}^{\pm} (y)$ and take the difference of the finite section integrals, and likewise for $u$ . By [F20], $⟨ u (x), N v (x) ⟩ = (4 π)^{- 1} \int ⟨ u (x), v (y) ⟩ g (x - y) d y$ . Pointwise, its absolute value is at most $(4 π)^{- 1} \sum_{j} \int ∣ u_{j} (x) ∣ ∣ v_{j} (y) ∣ g (x - y) d y$ ; integrating this bound and using step 4.1 proves absolute integrability. Integration of the preceding equality therefore gives the displayed double-integral representation in the Statement by [F24]. Every fraction with denominator $∣ x - y ∣$ is interpreted using $g (0) = 0$ as stipulated there.

6.1F24step 4.1step 5.1algebra

The same computation with the roles of $u$ and $v$ exchanged, using [F24] to interchange the two orders of integration of the common absolutely convergent double integral of step 4.1, gives $\int_{R^{3}} ⟨ N u (x), v (x) ⟩ d λ_{3} (x) = \frac{1}{4 π} \int_{R^{3}} \int_{R^{3}} \frac{⟨ u (x), v (y) ⟩}{∣ x - y ∣} d λ_{3} (y) d λ_{3} (x) .$

6.2F20step 4.1step 5.1algebra

For bilinearity let $u^{(1)}, u^{(2)}, v, α, β$ be given as in the Statement, with $w : = α u^{(1)} + β u^{(2)}$ . The componentwise linearity of the Newtonian integral [F20] and step 1.1's support bound for the union of the supports show $N w = α N u^{(1)} + β N u^{(2)}$ at every point, hence $B (w, v) = α B (u^{(1)}, v) + β B (u^{(2)}, v)$ by [F20] and the absolute finiteness of step 4.1 (the argument of steps 1.1-6.1 applies verbatim to $w$ , whose support is contained in the union of the two supports and hence bounded). The second-slot computation is the same with the roles exchanged.

7.1F24F26step 5.1step 6.1algebra

Steps 5.1, 6.1 and the definition of $B$ show that $B (u, v)$ is well defined and real-valued, that all three integrals of the Statement are absolutely convergent with that common value, and that the exchanged pairing satisfies $B (v, u) = \frac{1}{4 π} \int \int \frac{⟨ v (y), u (x) ⟩}{∣ y - x ∣} d λ_{3} (x) d λ_{3} (y)$ ; since $∣ x - y ∣ = ∣ y - x ∣$ and $⟨ v (y), u (x) ⟩ = ⟨ u (x), v (y) ⟩$ by [F26], [F24] identifies this with $B (u, v)$ , giving symmetry.

8.1A1F22F24F25step 1.1step 7.1step 6.2cases∎

If $u = 0$ or $v = 0$ then all integrands above vanish identically, all three integrals are $0$ , and the symmetry and bilinearity identities hold trivially; the empty support case was already admitted in step 1.1. The proof uses Countable Choice exactly through [A1] in the cited change-of-variables, product-measure, ball-measure and Borel-measurability interfaces, and no full Axiom of Choice.

Source notes

Hunter §2.6.1 and §2.7, printed pp. 33-37, treat the Newtonian potential of compactly supported data and its local kernel estimates; Teschl §5.3, equations (5.21)-(5.26), printed pp. 117-118, records the fundamental solution and the local integrability of the kernel. Neither text states the linkage pairing $B (u, v)$ or its symmetry; the statement and its proof assemble the two cited representations with the Tonelli/Fubini machinery of the measure-theory pages. The kernel normalization $ω_{2} = 4 π$ used for the $n = 3$ specialization is computed here from the chart-surface-measure convention, the unit-ball volume formula and the Gamma functional equation rather than quoted as a numerical convention.

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

Curl-field energy identity for Newtonian potentials of divergence-free compact currents

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $μ_{0} > 0$ be a real scalar parameter, let $J \in C_{c}^{\infty} (R^{3}; R^{3})$ ( $C^{k}$ Euclidean maps and diffeomorphisms) with $div J = 0$ (Divergence and curl of a $C^{1}$ vector field), and define componentwise $A : = μ_{0} N (J), A_{i} = μ_{0} N (J_{i}), N f (x) = \int_{R^{3}} \frac{f (y)}{4 π ∣ x - y ∣} d λ_{3} (y),$ the Newtonian vector potential attached to Newtonian potential of compactly supported data in dimension three. Then:

(a) $A \in C^{\infty} (R^{3}; R^{3})$ with $div A = 0$ and $- Δ A = μ_{0} J$ pointwise on $R^{3}$ , componentwise (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field);

(b) there is $C > 0$ with $∣ curl A (x) ∣ \leq C / (1 + ∣ x ∣^{2})$ for every $x \in R^{3}$ ; hence $curl A \in L^{2} (R^{3}; R^{3})$ and $⟨ A, J ⟩ \in L^{1} (R^{3})$ , both in the ordinary Lebesgue sense (Integral over a measurable subset);

(c) the curl-field energy identity $\int_{R^{3}} ∣ curl A ∣^{2} d λ_{3} = μ_{0} \int_{R^{3}} ⟨ A, J ⟩ d λ_{3}$ holds, and equivalently $\frac{1}{2} \int_{R^{3}} ⟨ A, J ⟩ d λ_{3} = \frac{1}{2 μ_{0}} \int_{R^{3}} ∣ curl A ∣^{2} d λ_{3} \geq 0$ .

The identity is stated for the explicit smooth compact currents and the Newtonian construction; no uniqueness, no decay of $A$ itself beyond the curl bound of (b), and no extension to singular currents is claimed.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , let $μ_{0} > 0$ and let $J \in C_{c}^{\infty} (R^{3}; R^{3})$ satisfy $div J = 0$ , with $A = μ_{0} N (J)$ componentwise as in the Statement.

[A1]

Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Assume ${A C}_{ω}$ , let $ε_{0} > 0$ and $f \in C_{c}^{\infty} (R^{3}; R)$ with $N f (x) = \int_{R^{3}} f (y) / (4 π ∣ x - y ∣) d y$ . Then $N f \in C^{\infty} (R^{3})$ , $- Δ N f = f$ pointwise, and for every multi-index $α$ one has $D^{α} N f = N (D^{α} f)$ . (Newtonian potential of smooth compact data is smooth and solves the Poisson equation pointwise).

[F2]

[F3]

Assume Countable Choice, let $n \geq 3$ , let $f$ be integrable and zero outside $B_{R} (0)$ , and put $M = \int f$ . For $r = ∣ x ∣ > 2 R$ , $∣ N f (x) - \frac{M r^{2 - n}}{(n - 2) ω_{n - 1}} ∣ \leq \frac{2^{n - 1} R}{ω_{n - 1}} ∥ f ∥_{1} r^{1 - n}$ . (Far-field asymptotics of compact-source Newtonian potentials).

[F4]

Assume ${A C}_{ω}$ , let $n \geq 3$ , let $f \in L^{1} (R^{n})$ vanish outside a closed ball of radius $R$ , and let $N f$ be the Newtonian potential. Then for every $x$ with $∣ x ∣ > 2 R$ the gradient is absolutely convergent and $∣ \nabla (N f) (x) ∣ \leq \frac{2^{n - 1} ∥ f ∥_{1}}{ω_{n - 1} ∣ x ∣^{n - 1}}$ . (The far-field gradient of a compactly supported Newtonian potential).

[F5]

For $C^{1}$ fields on an open $U \subseteq R^{3}$ , $div (F \times G) = ⟨ curl F, G ⟩ - ⟨ F, curl G ⟩$ . (The divergence and curl of a cross product).

[F6]

For a $C^{2}$ field $F$ on an open $U \subseteq R^{3}$ , $curl curl F = \nabla div F - Δ F$ . (The curl of a curl is the gradient of the divergence minus the Laplacian).

[F7]

The divergence and curl are defined pointwise from first partial derivatives; for $F = (F_{x}, F_{y}, F_{z})$ one has $div F = \partial_{x} F_{x} + \partial_{y} F_{y} + \partial_{z} F_{z}$ and $curl F = (\partial_{y} F_{z} - \partial_{z} F_{y}, \partial_{z} F_{x} - \partial_{x} F_{z}, \partial_{x} F_{y} - \partial_{y} F_{x})$ . (Divergence and curl of a $C^{1}$ vector field).

[F8]

For a $C^{2}$ function the Laplacian is $Δ f = div \nabla f$ , and for a $C^{2}$ map $Δ F$ is componentwise. (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F9]

The cross product in coordinates is $(u \times v)_{x} = u_{y} v_{z} - u_{z} v_{y}$ , with the cyclic permutations for the other two coordinates. (The cross product in $R^{3}$ ).

[F10]

The Euclidean inner product is $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ , and it is symmetric. (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F11]

$B (x, r) = {y : d (x, y) < r}$ is the open ball of the metric space. (Open ball, closed ball and sphere in a metric space).

[F12]

Let $0 \leq f_{1} \leq f_{2} \leq \dots$ be measurable with $f_{n} ↑ f$ pointwise. Then $\int f_{n} d μ ↑ \int f d μ$ . (Monotone convergence for the integral).

[F13]

Let $f_{n} \to f$ almost everywhere with $∣ f_{n} ∣ \leq g$ almost everywhere for a single integrable nonnegative $g$ . Then $f \in L^{1}$ and $\int f_{n} d μ \to \int f d μ$ . (Dominated convergence).

[F14]

For measurable $f : X \to [0, + \infty]$ and $E \in A$ , $\int_{E} f d μ : = \int f χ_{E} d μ$ . (Integral over a measurable subset).

[F15]

For measurable $f, g : X \to [0, + \infty]$ with $f \leq g$ one has $\int f d μ \leq \int g d μ$ . (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F16]

For a nonnegative simple measurable function $s = \sum_{j} c_{j} χ_{E_{j}}$ the simple integral is $\int s d μ = \sum_{j} c_{j} μ (E_{j})$ . (The integral of a nonnegative simple function).

[F17]

The nonnegative integral of a nonnegative simple function equals its simple integral. (The nonnegative integral agrees with the simple integral on simple functions).

[F18]

Let $k \in N$ and let $U \subseteq R^{3}$ be open; a map on $U$ is of class $C^{k}$ when each component is, and it is smooth when it is $C^{k}$ for every $k \in N$ , in particular for every $k \geq 1$ . ( $C^{k}$ Euclidean maps and diffeomorphisms).

[F19]

Assume ${A C}_{ω}$ and $0 < ε < R$ . Then $B_{R} = {x : ∣ x ∣ < R}$ is a bounded $C^{1}$ domain with boundary $S_{R}$ and outward normal $ν (x) = x / ∣ x ∣$ , the divergence theorem applies on $B_{R}$ , and each sphere $S_{r}$ is a compact embedded $C^{1}$ hypersurface carrying a finite Borel surface measure with the standard graph charts. (Euclidean balls, annuli and spheres as bounded smooth domains and hypersurfaces).

[F20]

On a bounded $C^{1}$ Euclidean domain the divergence theorem holds with the outward normal, $\int_{Ω} div F d λ = \int_{\partial Ω} ⟨ F, ν ⟩ d S$ . (Divergence on a bounded C1 Euclidean domain).

[F21]

On a compact embedded $C^{1}$ hypersurface the surface integral is defined by finitely many graph charts, and for a bounded continuous $g$ with finite area one has $\int_{S} ∣ g ∣ d S \leq \sup_{S} ∣ g ∣ \int_{S} 1 d S$ . (Surface integration on compact C1 hypersurfaces).

[F22]

The chart surface measure on $S^{n - 1}$ equals the polar measure $σ$ , scaling by $r$ multiplies surface measure by $r^{n - 1}$ , and $∣ S^{n - 1} ∣ = n ∣ B_{1} ∣$ . (Agreement with the existing polar sphere measure).

[F23]

For every $n \geq 1$ , $V_{n} (1) = π^{n / 2} / Γ (n / 2 + 1)$ . (The closed form for the volume of the unit $n$ -ball).

[F24]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ . (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

[F25]

$Γ (1 / 2) = \sqrt{π}$ . ( $Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral).

[F26]

A subset of $R^{n}$ is compact if and only if it is closed and bounded for the Euclidean metric. (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F27]

Every continuous $f : A \to R$ on a nonempty compact $A \subseteq R^{n}$ attains a maximum and a minimum on $A$ . (For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent).

[F28]

Closed balls $\overset{ˉ}{B} (x, r)$ in a metric space are closed. (Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed).

[F29]

[F30]

Every compact subset of a metric space is closed and bounded. (A compact subset of a metric space is closed and bounded).

[F31]

For $f : R^{n} \to R$ the support is $supp f = \overline{{x : f (x) \neq 0}}$ , and $f$ is compactly supported when that support is compact. (The support of a function on $R^{n}$ and its compactly supported Riemann integral).

[F32]

The Lebesgue integral is linear on $L^{1} (μ)$ . (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F33]

For $n \geq 1$ the Euclidean distance $d_{2}$ is a metric on $R^{n}$ . ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it).

[F34]

Under Countable Choice every continuous map $R^{n} \to R^{m}$ is Borel measurable. (Continuous functions on Euclidean spaces are Borel measurable).

[F35]

Assume ${A C}_{ω}$ , let $n \geq 1$ , $x \in R^{n}$ and $r > 0$ ; then the Euclidean ball $B (x, r)$ is Lebesgue measurable with $0 < λ_{n} (B (x, r)) < \infty$ . (Euclidean balls have positive finite Lebesgue measure).

Proof

technique · direct

1.1F11F29F30F31F33algebra

The supports of the components $J_{i}$ are compact by [F31] and bounded by [F30], [F33] and [F29]. Fix $R > 0$ with $supp J \subseteq B_{R} (0) = B (0, R)$ , an open ball in the sense of [F11] (for the empty case any $R$ works), and put $r_{0} : = \max {2 R, 1}$ .

1.2F1F2F6F7F8F18F32algebra

By [F1] applied to each component $J_{i}$ (with [F18] supplying the smoothness of the components of $J$ ), each $A_{i} = μ_{0} N (J_{i})$ lies in $C^{\infty} (R^{3})$ and satisfies $- Δ A_{i} = μ_{0} J_{i}$ and $D^{α} A_{i} = μ_{0} N (D^{α} J_{i})$ for every multi-index $α$ ; in particular $A$ is $C^{\infty}$ , hence $C^{2}$ for [F6], and [F8] gives $- Δ A = μ_{0} J$ componentwise. Moreover $div A = \sum_{i} \partial_{i} A_{i} = μ_{0} \sum_{i} N (\partial_{i} J_{i}) = μ_{0} N (div J) = 0$ pointwise, where the middle equality uses [F1] with $α = e_{i}$ and the linearity [F32] (with [F2] supplying the integral representation).

1.3F3F4F26F27F28F33algebra

For $r > r_{0} \geq 2 R$ and each $i$ , [F3] with $n = 3$ gives $∣ N J_{i} (x) - \frac{M_{i}}{4 π r} ∣ \leq \frac{R ∥ J_{i} ∥_{1}}{π r^{2}}$ , where $M_{i} = \int J_{i}$ and $r = ∣ x ∣$ ; hence $∣ A_{i} (x) ∣ \leq μ_{0} (\frac{∣ M_{i} ∣}{4 π} + \frac{R ∥ J_{i} ∥_{1}}{π}) \frac{1}{r}$ , and [F4] with $n = 3$ gives $∣ \nabla A_{i} (x) ∣ \leq \frac{μ_{0} ∥ J_{i} ∥_{1}}{π r^{2}}$ . By [F28], [F26], [F33] and [F27], the closed ball ${\overset{ˉ}{B}}_{r_{0}} (0)$ is compact and each of the continuous functions $A_{i}$ and $\partial_{k} A_{i}$ attains a finite maximum on it, so there are finite constants $c_{1}, c_{2}$ with $∣ A (x) ∣ \leq c_{1}$ and $∣ \partial_{k} A_{i} (x) ∣ \leq c_{2}$ for all $x$ and $i, k$ ; consequently $A$ and all its first partials are bounded on $R^{3}$ .

2.1F7F9F10F14F15F16F17F35step 1.2step 1.3algebra

By [F9] each coordinate of $curl A$ is a difference of two entries of $\nabla A$ , so [F7] yields $∣ curl A (x) ∣ \leq 2 \sum_{i, k} ∣ \partial_{k} A_{i} (x) ∣ \leq 18 c_{2}$ for $∣ x ∣ \leq r_{0}$ , while for $∣ x ∣ > r_{0}$ the gradient bound of step 1.3 gives $∣ curl A (x) ∣ \leq 2 \sum_{i, k} \frac{μ_{0} ∥ J_{i} ∥_{1}}{π ∣ x ∣^{2}} \leq \frac{36 μ_{0} \max_{i} ∥ J_{i} ∥_{1}}{π ∣ x ∣^{2}}$ . Hence $∣ curl A (x) ∣ \leq C / (1 + ∣ x ∣^{2})$ on all of $R^{3}$ with $C : = \max {1, 18 c_{2} (1 + r_{0}^{2}), 72 μ_{0} \max_{i} ∥ J_{i} ∥_{1} / π}$ . Moreover $⟨ A, J ⟩$ is continuous because $A$ is $C^{\infty}$ by step 1.2 and $J$ is smooth, it is supported in the compact set $supp J \subseteq B_{R} (0)$ with $R > 0$ , and it is bounded by $3 c_{1} \max_{i} ∥ J_{i} ∥_{\infty}$ : by [F10], $∣ ⟨ A, J ⟩ ∣ \leq \sum_{i} ∣ A_{i} ∣ ∣ J_{i} ∣ \leq c_{1} \sum_{i} ∣ J_{i} ∣ \leq 3 c_{1} \max_{i} ∥ J_{i} ∥_{\infty}$ , since each $∣ A_{i} ∣ \leq ∣ A ∣ \leq c_{1}$ . Since $∣ ⟨ A, J ⟩ ∣ 1_{B_{R} (0)} \leq 3 c_{1} \max_{i} ∥ J_{i} ∥_{\infty} 1_{B_{R} (0)}$ pointwise, [F14] and [F15] give $\int_{R^{3}} ∣ ⟨ A, J ⟩ ∣ d λ_{3} = \int_{B_{R} (0)} ∣ ⟨ A, J ⟩ ∣ d λ_{3} \leq 3 c_{1} \max_{i} ∥ J_{i} ∥_{\infty} \int_{B_{R} (0)} 1 d λ_{3} = 3 c_{1} \max_{i} ∥ J_{i} ∥_{\infty} λ_{3} (B_{R} (0)) < \infty,$ the middle equality being the simple integral of the indicator by [F16] and [F17], and the finiteness of $λ_{3} (B_{R} (0))$ being [F35].

2.2F5F6F10step 1.2algebra

On $R^{3}$ one has, by [F5], [F6], [F10] and step 1.2, $div (A \times curl A) = ⟨ curl A, curl A ⟩ - ⟨ A, curl curl A ⟩ = ∣ curl A ∣^{2} - μ_{0} ⟨ A, J ⟩,$ because $curl curl A = \nabla div A - Δ A = - Δ A = μ_{0} J$ . The fields $A$ and $curl A$ are $C^{1}$ by step 1.2, so [F5] applies.

3.1F19F20step 2.2

Fix $r > r_{0}$ . By [F19] the ball $B_{r} (0)$ is a bounded $C^{1}$ domain with boundary $S_{r}$ and outward normal $ν (x) = x / ∣ x ∣$ , and [F20] applied to $F = A \times curl A$ together with step 2.2 gives $\oint_{S_{r}} ⟨ A \times curl A, ν ⟩ d S = \int_{B_{r} (0)} ∣ curl A ∣^{2} d λ_{3} - μ_{0} \int_{B_{r} (0)} ⟨ A, J ⟩ d λ_{3} .$

3.2F9F21F22F23F24F25step 1.3step 2.1algebra

For $r > r_{0}$ the boundary term is bounded by the decay estimates: [F21] and the componentwise bound $∣ (u \times v)_{x} ∣ \leq 2 \max_{k} ∣ u_{k} ∣ \max_{k} ∣ v_{k} ∣$ from [F9] give $∣ \oint_{S_{r}} ⟨ A \times curl A, ν ⟩ d S ∣ \leq \int_{S_{r}} ∣ A \times curl A ∣ d S \leq 4 \sup_{S_{r}} ∣ A ∣ \sup_{S_{r}} ∣ curl A ∣ \int_{S_{r}} 1 d S .$ By [F22], [F23], [F24] and [F25] the sphere area is $\int_{S_{r}} 1 d S = ∣ S^{2} ∣ r^{2} = 3 V_{3} (1) r^{2} = 4 π r^{2}$ , and by steps 1.3 and 2.1, $\sup_{S_{r}} ∣ A ∣ \leq c_{1}^{'} r^{- 1}$ and $\sup_{S_{r}} ∣ curl A ∣ \leq c_{2}^{'} r^{- 2}$ for constants $c_{1}^{'}, c_{2}^{'}$ independent of $r$ . Hence the boundary term is at most $16 π c_{1}^{'} c_{2}^{'} / r$ , which tends to $0$ as $r \to \infty$ .

4.1F12F13F34step 2.1step 3.1step 3.2algebra

Let $r_{n} ↑ \infty$ with $r_{n} > r_{0}$ . The functions $∣ curl A ∣^{2} 1_{B_{r_{n}} (0)}$ are nonnegative and measurable by [F34], and increase to $∣ curl A ∣^{2}$ ; [F12] therefore gives $\int_{B_{r_{n}} (0)} ∣ curl A ∣^{2} d λ_{3} ↑ L : = \int_{R^{3}} ∣ curl A ∣^{2} d λ_{3} \in [0, + \infty] .$ The functions $⟨ A, J ⟩ 1_{B_{r_{n}} (0)}$ converge pointwise to $⟨ A, J ⟩$ and are dominated by the integrable function $∣ ⟨ A, J ⟩ ∣$ of step 2.1, so [F13] gives $\int_{B_{r_{n}} (0)} ⟨ A, J ⟩ d λ_{3} \to \int_{R^{3}} ⟨ A, J ⟩ d λ_{3}$ , a finite real number. Passing to the limit in the identity of step 3.1 using the vanishing boundary term of step 3.2 yields $L = μ_{0} \int_{R^{3}} ⟨ A, J ⟩$ , so $curl A \in L^{2} (R^{3}; R^{3})$ and the identity of (c) holds; the equivalent form follows by dividing by $2 μ_{0} > 0$ , and $L \geq 0$ gives the final inequality.

5.1A1F1F3F4F12F13F20step 1.1step 1.2step 1.3step 2.1step 4.1∎

Clauses (a), (b), (c) are now established: (a) in step 1.2, the decay bound and the $L^{1}$ statement (b) in steps 2.1 and 4.1, and the identity with its equivalent form in step 4.1. The hypothesis $div J = 0$ was used only in step 1.2; smoothness and compact support only in steps 1.1 through 1.3; no uniqueness of $A$ , no decay of $A$ beyond the curl bound, and no singular-current extension is asserted. Countable Choice enters only through the cited Newtonian, measure and surface interfaces, not through any selection.

Source notes

Hunter §2.6.1 and §2.7, printed pp. 33-37, and Teschl §5.3, equations (5.21)-(5.26), printed pp. 117-118, supply the Newtonian potential, the local kernel estimates and the far-field behaviour behind the decay steps. Neither text states the curl-field energy identity in this exact form; the present proof assembles it from the supplied smoothness and decay of the Newtonian potential, the vector-calculus identity for $div (F \times G)$ , the divergence theorem on a ball and the monotone-convergence argument. The local identities $div A = 0$ and $- Δ A = μ_{0} J$ are recorded pointwise for the explicit smooth compact currents; the fine behaviour of the fields at a singular current is not treated.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force

Physical scope

Kinematic and definitional content only: the C2 moving material circuit (a compact embedded swept patch with a single-valued C1 material velocity), its magnetic flux and its electromotive force line integral, in the laboratory frame and units of the classical-electromagnetism framework. It excludes circuits crossing singular sources, requires subluminal material speeds when the relativistic force interpretation is used, and asserts no relation between emf and flux derivative and no dynamical or empirical claim.

Definition

Let $Ω \subseteq R^{3}$ be nonempty and open and $I \subseteq R$ a nonempty open interval, with the laboratory frame, fields $E, B : Ω \times I \to R^{3}$ , units and $c > 0$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) A moving material circuit is data $(D, r, v)$ . The nonempty compact parameter set $D = D_{1} \cup \dots \cup D_{N} \subseteq R^{2}$ has a supplied finite elementary Green decomposition, including the opposite orientations on shared arcs, as in Type I, Type II, and elementary regions for Green's theorem. Its positive boundary chain $\partial D = (σ_{1}, \dots, σ_{m})$ is the finite list of surviving oriented piecewise $C^{1}$ arcs after internal pairs cancel (Positive orientation of elementary-region boundaries). The map $r : D \times I \to Ω$ extends to a $C^{2}$ map on an open neighbourhood of $D \times I$ , has $r_{u} \times r_{w} \neq 0$ on $D^{\circ} \times I$ , and is injective on $D$ at each time. These conditions are the meaning of an embedded material sheet here. Its swept spacetime graph is $G : = {(r (p, t), t) : p \in D, t \in I} \subseteq Ω \times I$ . A compatible material velocity is a $C^{1}$ map $v : W \to R^{3}$ on an open $W \subseteq Ω \times I$ containing $G$ , satisfying $v (r (p, t), t) = r_{t} (p, t)$ . Thus coincident images must have the same velocity. Material labels are the points $p = (u, w) \in D$ .

(b) Put $S (t) : = r (D, t)$ and $γ_{t} : = (r (\cdot, t) \circ σ_{k})_{k = 1}^{m}$ , a boundary chain, whose trace is $C (t) : = r (\partial D, t)$ . We use $\partial S (t)$ for this parametrized boundary chain, rather than the ambient topological boundary of the subset $S (t) \subseteq R^{3}$ . A finite Green region can have several components or holes, so $γ_{t}$ need not be one closed path. For $B$ jointly $C^{1}$ on a spacetime neighbourhood of $G$ , the magnetic flux is $Φ_{B} (t) : = \int_{D} ⟨ B (r (u, w, t), t), r_{u} (u, w, t) \times r_{w} (u, w, t) ⟩ d u d w .$ This is the parametrized moving-patch flux convention of Moving parametrized surface patches and their flux, specified here with its spacetime arguments explicit. Its unit is $W b = T m^{2}$ .

(c) For $E, B$ continuous on a spacetime neighbourhood of $G$ , the electromotive force is the finite sum of vector line integrals $e m f (t) : = \oint_{γ_{t}} (E (\cdot, t) + v (\cdot, t) \times B (\cdot, t)) \cdot d r : = \sum_{k = 1}^{m} \int_{r (\cdot, t) \circ σ_{k}} (E (\cdot, t) + v (\cdot, t) \times B (\cdot, t)) \cdot d r,$ with Scalar line integrals with respect to arc length and vector-field line integrals and The cross product in $R^{3}$ . Its unit is $V = k g m^{2} s^{- 3} A^{- 1}$ . For the force interpretation, additionally adopt The Lorentz coupling: external force on a test charge and the continuum force and power densities(i): a constant $q \neq 0$ , a mass $m > 0$ , a $C^{2}$ future timelike test-charge worldline and a prescribed external $C^{1}$ electromagnetic tensor. In the worldline conventions of Worldlines, four-velocity, chart readings and the frame/observer distinction and Norms of four-velocity and four-acceleration, and the observer rest-space decomposition(e), $U = γ (c, v)$ , $d / d τ = γ d / d t$ and $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ . The tensor convention The electromagnetic two-form, its observer decomposition and the four-current gives $F_{i 0} = E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ and $Φ (U)^{i} = F_{i β} U^{β} = γ (E + v \times B)^{i}$ . For the spatial momentum $p : = m γ v$ , the spatial components of $m d U / d τ = q Φ (U)$ therefore yield $γ d p / d t = q γ (E + v \times B)$ , hence $d p / d t = q (E + v \times B)$ . When the charge velocity agrees with the material velocity, the integrand is this lab momentum force divided by $q$ . This interpretation adds those external-coupling hypotheses; the emf definition itself only names the integral.

(d) The stated field regularity must hold near every point of $G$ ; singularity crossings are excluded. The force interpretation additionally requires $∣ v ∣ < c$ along the material motion. No conservation law, flux-derivative relation, empirical assertion or parametrization-invariance assertion is made.

Well-definedness and conventions

Each $D_{i}$ is compact and Jordan measurable; its boundary has content zero. The finite union $D$ is Jordan measurable, and its continuous flux integrand is Riemann integrable by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set. At a fixed time each elementary piece is an admissible regular $C^{2}$ patch: its interior is connected and dense in that piece, the parameter cross product is nonzero there, and injectivity prevents interior image identifications (Regular parametrized surface patches on compact Jordan parameter regions). All integrals over the full sheet mean the sum over the pieces, with internal boundary arcs cancelling. Neither connectedness nor a single boundary loop is assumed for their union.

Composition of each $σ_{k}$ with the $C^{2}$ slice gives a piecewise $C^{1}$ path. Its trace lies in the time- $t$ slice of $W$ , where $E + v \times B$ is continuous, so each line integral exists. The chain sum is independent of its ordering. Spatial derivatives of two $C^{2}$ extensions agreeing on $D \times I$ agree first on $D^{\circ} \times I$ by difference quotients and then on its closure in each time slice by continuity; since $I$ is open the same argument gives the time derivative. The flux and chain therefore do not depend on the extension. Boundary rank failures remain allowed.

Source notes

Tong §4.1–4.1.1, printed pp. 67–71, supplies the moving-loop flux and force-per-charge emf convention. The finite Green chain, spacetime domains and precise regularity above are the local formulation; the source is not claimed to impose them.

PostulateStatement: AI-adaptedProof: Not applicableOpen item page →

Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime

Physical scope

An adopted reduced lumped-circuit model: fixed finite branch data with constant symmetric L, positive-definite C and symmetric positive-semidefinite R, the Faraday/capacitive/Ohmic branch relations and Kirchhoff laws, confined to a declared quasistatic smallness condition with negligible radiation and leakage and ideal terminals. It is not derived from the vacuum Maxwell equations, asserts no error bound, and excludes radiation, distributed, nonlinear, hysteretic, leakage, proximity and material-dissipative effects beyond the cited material branch.

Postulate

Adopt the following reduced circuit model in the laboratory frame and SI units of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values. The continuum material reference is Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch; the terminal laws below are additional assumptions, rather than consequences of its pointwise constitutive relations or of Maxwell's equations.

(a) Branch data. Fix a nonempty interval $J \subseteq R$ , a finite set of nodes $N$ , and a finite set of oriented branches $E$ , each with specified tail and head nodes. Designate $n$ inductive/resistive branches, $m$ capacitive branches and any source branches. On the first family specify currents $I : J \to R^{n}$ and separate voltage-drop vectors $V_{L}, V_{R} : J \to R^{n}$ ; on the second specify charges $Q : J \to R^{m}$ , currents $I_{C} : J \to R^{m}$ and voltage drops $V_{C} : J \to R^{m}$ . Currents $I$ and charges $Q$ admit $C^{1}$ extensions to an open interval containing $J$ ; included endpoint derivatives are their one-sided derivatives, and at a singleton the chosen extensions are part of the data supplying the jets. Voltages and source currents are continuous. The matrices $L, R \in R^{n \times n}$ and $C \in R^{m \times m}$ are constant, with $L^{T} = L$ , $C^{T} = C$ , $z^{T} C z > 0$ for every $z \neq 0$ , $R^{T} = R$ , and $y^{T} R y \geq 0$ for every $y$ . These inequalities define positive definiteness and positive semidefiniteness here. Symmetry alone is imposed on a declared $L$ ; a smooth compact-current realization can supply its additional positivity. Geometry, orientations and source data are fixed.

(b) Element and network relations. Adopt $V_{L} = L \dot{I}, V_{R} = R I, I_{C} = \dot{Q}, V_{C} = C^{- 1} Q .$ Positive definiteness makes $C^{- 1}$ well defined: $C z = 0$ implies $z^{T} C z = 0$ , hence $z = 0$ ; an injective square finite matrix is invertible by Invertible matrix theorem: invertibility, full pivot rank, RREF $I$ , trivial nullspace and unique solvability are equivalent. The voltage drop on an inductive/resistive branch is $(V_{L} + V_{R})_{α}$ , with either element omitted by setting its corresponding matrix to zero; a capacitive branch has drop $(V_{C})_{β}$ . Thus the three voltage symbols refer to distinct element drops. In a purely inductive use, abbreviate $V_{L}$ by $V$ . The linkage is $L I$ and the induced emf convention is $e_{L} : = - V_{L}$ . The capacitor scalar law is $V_{C} = Q / C$ when $m = 1$ .

For a branch $b$ let $i_{b}$ be its oriented current and $v_{b}$ its voltage drop from tail to head (a source may have a negative drop). Define the incidence matrix by $A_{a b} : = 1_{{a = tail (b)}} - 1_{{a = head (b)}}$ . Kirchhoff's current law is $A i (t) = 0$ , expressing no node-charge accumulation. A closed walk is a finite sequence of traversals of branches with matching successive nodes and final node equal to initial node. If its signed traversal count is $z_{b}$ (forward minus backward), Kirchhoff's voltage law is $\sum_{b} z_{b} v_{b} (t) = 0$ . Adopt it for every closed walk. Equivalently it suffices to impose it on a chosen linearly independent spanning set (basis) of the real span of their traversal-count vectors; this is the meaning of independent loops here. These are imposed ideal network constraints, not statements about arbitrary field circulations.

The bookkeeping energies are $\frac{1}{2} I^{T} L I$ and $\frac{1}{2} Q^{T} C^{- 1} Q$ . If a smooth compact conserved-profile realization of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b) is specified, identification of its linkage matrix with the declared $L$ is an additional model premise, not automatic.

(c) Quasistatic regime. Specify a geometric size $ℓ > 0$ , a time scale $T > 0$ and a chosen dimensionless tolerance $0 < ϵ_{q s} < 1$ . The declaration $ℓ / (c T) ≪ 1$ means $ℓ / (c T) \leq ϵ_{q s}$ with a tolerance regarded as small in the chosen application. Negligible radiation and leakage, ideal terminals and the fixed conductor geometry are separately adopted idealizations. Neither this inequality nor these idealizations give a quantitative error bound for an arbitrary full Maxwell solution; no universal accuracy is asserted.

(d) Units and scope. $V_{L}, V_{R}, V_{C}, v_{b}$ carry volts, $I, I_{C}, i_{b}$ amperes, $Q$ coulombs, $t$ seconds, $L$ henries ( $H = V s A^{- 1}$ ), $C$ farads ( $F = C V^{- 1}$ ), and $R$ ohms ( $Ω = V A^{- 1}$ ). The model excludes radiation, distributed-element, nonlinear, hysteretic, leakage and proximity effects; resistor loss is retained through $R$ . Subluminal material speeds are required if a relativistic moving-matter interpretation is added. The model is adopted and contains no measurement or derivation from the vacuum Maxwell equations.

Status and conventions

The scalar series RL loop consists of one inductive/resistive branch and a source: voltage balance gives $L \dot{I} + R I = V_{s o u r c e}$ . The source-free scalar LC loop uses $R = 0$ , equal oriented currents $I = I_{C} = \dot{Q}$ from the node law, and voltage balance $L \dot{I} + Q / C = 0$ . These identify the exact network assumptions used by the later conditional deductions. General network solvability is not asserted.

Source notes

Tong §4.1.2–4.1.3, printed pp. 71–77, supplies inductive energy, the induced-emf sign and the adoption of Ohm's law; its worked dynamical example there is a sliding bar. Fitzpatrick §7.4–7.6, printed pp. 222–234, supplies the distinct inductive/resistive voltage drops, RL equation (7.37), and series LCR law (7.76), including capacitor current as the charge derivative. The finite network and matrix formulation here is explicitly adopted model content; no source is represented as proving a Maxwell-to-network error estimate.

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Motional emf for a moving material circuit: emf = -flux derivative

Physical scope

The moving-loop flux rule for the strict branch of the adopted framework: a C2 embedded material sheet swept over a compact finite elementary Green region with a single-valued C1 material velocity, in a jointly C2 vacuum-Maxwell field branch on a compact subinterval with no singularity crossing. It asserts no formula for loops crossing singular sources, no fixed-loop formula for arbitrary moving loops, no field uniqueness and no empirical claim; the Lorentz reading of the integrand is the adopted coupling and is not used in the algebraic identity.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ), inherited from the Maxwell and transport suppliers below. Let $Ω \times I$ be the laboratory region with $c > 0$ , $ε_{0}$ , $μ_{0}$ the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values; let $E, B$ be fields and $ρ, J$ source fields of The electric and magnetic fields as fixed-time spatial vector fields and Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, jointly of class $C^{2}$ and $C^{1}$ respectively on $Ω \times I$ , satisfying the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches pointwise on $Ω \times I$ . Let $(D, r, v)$ be a moving material circuit of C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force with time interval $I$ , and fix a compact subinterval $K \subseteq I$ such that the swept set $r (D \times K)$ is compactly contained in $Ω$ (no singularity crossing). Put $S (t) = r (D, t)$ and let $γ_{t}$ be the oriented boundary chain of (b) of the definition, with trace $C (t) = \partial S (t)$ .

Then the flux $Φ_{B} : I \to R$ of C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force is differentiable at every $t \in I$ with $Φ_{B}^{'} (t) = \int_{S (t)} \partial_{t} B (\cdot, t) \cdot n d S - \oint_{C (t)} (v \times B) (\cdot, t) \cdot d r,$ for every $t \in K$ each elementary-piece frozen patch $(D_{i}, r (\cdot, \cdot, t) ∣_{D_{i}})$ is a test geometry of Fixed volume and patch test geometries, outward normals and induced orientations, the fixed-patch Faraday identity $\int_{S (t)} \partial_{t} B (\cdot, t) \cdot n d S = - \oint_{C (t)} E (\cdot, t) \cdot d r$ holds, and consequently the electromotive force of the circuit satisfies $e m f (t) = \oint_{C (t)} (E + v \times B) (\cdot, t) \cdot d r = - Φ_{B}^{'} (t) for every t \in K .$ For subluminal material speeds and the external test-charge premises of The Lorentz coupling: external force on a test charge and the continuum force and power densities(i) with the tensor and worldline conventions in C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force(c), $E + v \times B$ is the instantaneous laboratory momentum force per unit nonzero charge. Its oriented circulation equals minus the flux derivative; this adds a force interpretation without asserting a charge's actual work along a spacetime traversal of the circuit. The result is conditional on the joint regularity, the smooth Maxwell branch and the no-singularity-crossing hypothesis; it asserts no fixed-loop formula for arbitrary moving loops, no field uniqueness, no statement about singular supports and no empirical claim.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , the laboratory region $U ≅ I \times Ω$ , constants $c > 0, ε_{0} > 0, μ_{0} > 0$ , fields $E, B$ jointly $C^{2}$ and sources $ρ, J$ of class $C^{1}$ satisfying the smooth branch of the vacuum Maxwell equations pointwise on $Ω \times I$ , a moving material circuit $(D, r, v)$ with time interval $I$ , and a compact subinterval $K \subseteq I$ with the swept set $r (D \times K)$ compactly contained in $Ω$ .

[A1]

Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

A moving material circuit is the data $(D, r, v)$ : $D$ a compact finite elementary Green region with its positive boundary chain, $r : D \times I \to Ω$ extending to a $C^{2}$ map on a neighbourhood of $D \times I$ with $r_{u} \times r_{v} \neq 0$ on the interior of $D$ and $r (\cdot, t)$ injective on $D$ for every $t \in I$ , and $v$ a $C^{1}$ single-valued velocity field with $v (r (u, w, t), t) = \partial_{t} r (u, w, t)$ on a neighbourhood of the swept set. (C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force).

[F2]

The electromotive force of the circuit at $t$ is the vector line integral $e m f (t) = \oint_{γ_{t}} ⟨ E (\cdot, t) + v (\cdot, t) \times B (\cdot, t), d r ⟩$ over the oriented boundary chain $γ_{t}$ with trace $C (t) = r (\partial D, t)$ , in the piecewise $C^{1}$ convention, with unit V; the definition asserts no relation between emf and $Φ_{B}^{'}$ . (C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force).

[F3]

The transport formula needed here is established in step 1.1 from the explicitly typed circuit data and the calculus and Stokes suppliers below. Transport of flux through a moving patch supplies the general parametrized formula; the present argument independently derives the identity for the explicitly typed material circuit.

[F4]

A patch test geometry in $Ω$ is a regular $C^{2}$ surface patch $(D, r)$ over a finite elementary Green region: $D \subseteq R^{2}$ finite elementary Green region, $r$ of class $C^{2}$ on an open neighbourhood of $D$ , $r_{u} \times r_{v} \neq 0$ on the interior of $D$ , and no interior parameter point shares its image with a distinct point of $D$ ; its induced boundary chain is $r (\partial D)$ with the circulation of the piecewise $C^{1}$ convention, and its oriented flux is $\int_{S} F \cdot n_{S} d S = \int_{D} (F \circ r) \cdot (r_{u} \times r_{v}) d u d v$ . (Fixed volume and patch test geometries, outward normals and induced orientations).

[F5]

Assume the smooth branch of the vacuum Maxwell postulate holds pointwise with $E, B$ jointly $C^{2}$ and $ρ, J$ of class $C^{1}$ . Then for every patch test geometry $(D, r)$ the flux functions $t \mapsto \int_{S} B (\cdot, t) \cdot n_{S} d S$ and $t \mapsto \int_{S} E (\cdot, t) \cdot n_{S} d S$ are differentiable on $I$ and, for every $t \in I$ , $\oint_{\partial S} E (\cdot, t) \cdot d r = - \frac{d}{d t} \int_{S} B (\cdot, t) \cdot n_{S} d S$ . (The integral form of the vacuum Maxwell equations and its equivalence with the smooth differential form).

[F6]

A field of class $C^{k}$ is a map $Ω \times I \to R^{3}$ of class $C^{k}$ ; $E$ carries V m $^{- 1}$ and $B$ carries T. (The electric and magnetic fields as fixed-time spatial vector fields).

[F7]

The framework adopts the SI constants $c > 0$ , $μ_{0} > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ , with time in seconds and the emf unit V; no numerical value of $μ_{0}$ is asserted. (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F8]

The source fields $ρ$ (C m $^{- 3}$ ) and $J$ (A m $^{- 2}$ ) are maps on $Ω \times I$ . (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current).

[F9]

In the smooth branch, jointly $C^{1}$ fields with the additional derivatives required by each equation existing and continuous and continuous sources satisfy $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ , $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ pointwise. (The microscopic vacuum Maxwell equations: smooth and weak branches).

[F10]

The adopted coupling gives the continuum force density $ρ E + J \times B$ for matter interacting with the field through an explicitly stated matter model, and the covariant per-charge equation of motion for a test charge; the nonrelativistic coordinate statement is not adopted as exact. (The Lorentz coupling: external force on a test charge and the continuum force and power densities).

[F11]

The Euclidean inner product is additive in each argument: $⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩$ . (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F12]

For $a < b$ and integrable $f, g$ on $[a, b]$ , $f + g$ is integrable and $\int_{a}^{b} (f + g) = \int_{a}^{b} f + \int_{a}^{b} g$ . (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ).

[F13]

For a piecewise $C^{1}$ path $γ : [a, b] \to R^{n}$ with $a < b$ and a continuous vector field $F$ on a set containing the trace, $\int_{γ} F \cdot d r : = \sum_{i} \int_{t_{i}}^{t_{i + 1}} ⟨ F (γ (t)), v_{i} (t) ⟩ d t$ ; the value is independent of the admissible partition. (Scalar line integrals with respect to arc length and vector-field line integrals).

[F14]

For a compact measurable parameter set and an integrand with continuous time derivative on its neighbourhood, differentiation passes through its Lebesgue integral on a compact nondegenerate time interval (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface). Under ${A C}_{ω}$ , bounded Borel Riemann-integrable functions on a rectangle have equal Riemann and Lebesgue integrals; this includes zero extensions of continuous integrands from compact Jordan regions (Riemann–Lebesgue comparison for distribution test integrands).

[F15]

Stokes on each admissible regular $C^{2}$ elementary patch gives the curl flux as the circulation of its induced boundary chain (The classical Stokes theorem for a $C^{2}$ patch over a finite elementary Green region). Internal pairs of arcs cancel in the positive finite Green boundary chain (Positive orientation of elementary-region boundaries).

[F16]

The chain, product and mixed-partial rules apply to the stated $C^{2}$ map; for $C^{1}$ spatial fields, $curl (v \times B) = v div B - B div v + (D_{x} v) B - (D_{x} B) v$ (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 $g (c) \neq 0$ , Clairaut--Schwarz theorem for continuous second partial derivatives, The divergence and curl of a cross product).

Proof

technique · direct differentiation on the parameter region and finite-piece Stokes

1.1F1F2F6F9F14F15F16algebra

Write $a = r_{u} \times r_{w}$ and $h (p, t) = ⟨ B (r (p, t), t), a (p, t) ⟩$ . These functions have continuous time derivatives on a neighbourhood of $D \times I$ . Fix any $t \in I$ and a compact nondegenerate time interval around it contained in $I$ ; [F14], with the Riemann/Lebesgue comparison on a bounding rectangle, gives $Φ_{B}^{'} (t) = \int_{D} \partial_{t} h d p$ . Put $M = D_{x} v$ . Compatibility and [F16] give $\partial_{t} r_{u} = M r_{u}$ , $\partial_{t} r_{w} = M r_{w}$ . The elementary determinant identity $\det [M q_{0}, q_{1}, q_{2}] + \det [q_{0}, M q_{1}, q_{2}] + \det [q_{0}, q_{1}, M q_{2}] = tr (M) \det [q_{0}, q_{1}, q_{2}]$ follows by expanding in the three columns and checking the standard-basis triples (repeated indices cancel; distinct indices give the trace). With $q_{0} = B$ , $q_{1} = r_{u}$ , $q_{2} = r_{w}$ it yields $⟨ B, \partial_{t} a ⟩ = ⟨ (div v) B - M B, a ⟩$ . Thus [F16] gives $\partial_{t} h = ⟨ \partial_{t} B + (D_{x} B) v + (div v) B - (D_{x} v) B, a ⟩ = ⟨ \partial_{t} B + v div B - curl (v \times B), a ⟩$ . Here all spatial derivatives are at fixed laboratory time on the open time slice of $W \cap (Ω \times I)$ . Since $div B = 0$ by [F9], integrating and applying [F15] on each elementary piece, then cancelling oppositely oriented internal arcs, gives $Φ_{B}^{'} (t) = \int_{S (t)} \partial_{t} B \cdot n d S - \oint_{γ_{t}} (v \times B) \cdot d r$ . Parameter integrals add over the pieces because they have disjoint interiors and null boundaries; [F14] transfers this finite Lebesgue additivity to the Riemann integrals. This argument applies at every $t \in I$ .

1.2F1F4F5F6F8F9F14F15algebra

Fix $t \in K$ . Every elementary $D_{i}$ is the closure of its nonempty connected interior, hence is a compact Jordan parameter region. The frozen restriction of $r$ is $C^{2}$ , has nonzero parameter cross product on that interior, and is injective, so it is a patch test geometry of [F4]. Apply [F5] to each such fixed spatial patch, differentiating its flux with respect to a separate field time $s$ and evaluating at $s = t$ . Sum the resulting identities and cancel internal arcs as in [F15]. This gives $\int_{S (t)} \partial_{t} B (\cdot, t) \cdot n d S = - \oint_{γ_{t}} E (\cdot, t) \cdot d r$ . The separate variable $s$ keeps this fixed-patch derivative distinct from the moving flux derivative.

2.1F2F11F12F13step 1.1step 1.2algebra

On each arc of $γ_{t}$ , additivity of the Euclidean pairing [F11] and Riemann integral linearity [F12] split the line integral of $E + v \times B$ into those of $E$ and $v \times B$ ; summing the arcs gives $e m f = \oint_{γ_{t}} E \cdot d r + \oint_{γ_{t}} (v \times B) \cdot d r$ by [F2] and [F13]. Substitution of step 1.2 into step 1.1 proves $e m f (t) = - Φ_{B}^{'} (t)$ for every $t \in K$ .

3.1A1F3F6F7F10F14step 1.1step 1.2step 2.1∎

The claimed differentiability, transport formula, elementary-piece test geometries, fixed-patch Faraday formula and emf identity follow from steps 1.1–2.1. The transport identity agrees with the general parametrized formula of [F3]. The additional subluminal external-test-charge reading is the local coordinate-force calculation of C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force(c) from The Lorentz coupling: external force on a test charge and the continuum force and power densities(i); it does not replace relativistic momentum by $m v$ and is unused in the integral algebra. Countable Choice is carried through the comparison and Maxwell interfaces. The regular fields and compact swept graph on $D \times K$ exclude singularity crossings. No empirical or full matter-dynamical assertion is made.

Source notes

Tong §4.1.1, printed pp. 69–71, gives the moving-loop transport signs and emf convention. The strict regularity and finite-piece proof are supplied locally.

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Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents

Physical scope

Fixed compact smooth conserved current profiles of the adopted framework and their Coulomb-gauge Newtonian vector potential: finite reciprocal linkage coefficients, the linear linkage identity, the exact positive-semidefinite quadratic magnetic energy and the vanishing/positive-definiteness criterion. It asserts no thin-wire, sheet or singular self-inductance, no time-dependent, moving, radiating or material configuration and no uniqueness of the vector potential; the identification of the coefficients with the lumped circuit inductance matrix is the separately adopted circuit model.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $μ_{0} > 0$ , $c > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ be the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let $j_{1}, \dots, j_{n} \in C_{c}^{\infty} (Ω; R^{3})$ with $div j_{α} = 0$ be fixed compact smooth conserved current profiles of the class of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b), extended by zero to $R^{3}$ . Fix amplitudes $I \in R^{n}$ , label them by $I_{α} : = I (α - 1)$ for $1 \leq α \leq n$ in the zero-based convention of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , and put $J_{I} : = \sum_{α = 1}^{n} I_{α} j_{α}$ , let $A_{J_{I}}$ be the Coulomb-gauge Newtonian vector potential of Coulomb-gauge Newtonian vector potential for a compact smooth conserved current applied to the conserved current $J_{I}$ , and let $B_{J_{I}} = curl A_{J_{I}}$ . Define the linkage coefficients $L_{α β} : = \frac{μ_{0}}{4 π} \int_{R^{3}} \int_{R^{3}} \frac{⟨ j_{α} (x), j_{β} (y) ⟩}{∣ x - y ∣} d λ_{3} (y) d λ_{3} (x), α, β = 1, \dots, n .$ The amplitudes $I_{α}$ carry amperes and the profiles $j_{α}$ are shapes per unit current, with units $m^{- 2}$ ; their numerical SI maps belong to the stated smooth class. Consequently $J_{I}$ has unit $A m^{- 2}$ , $A_{J_{I}}$ has $T m$ , $L_{α β}$ has H, the linkage has Wb and $U$ has joules. $A_{j_{α}}$ denotes the linear potential response per ampere. Then:

(a) [Finiteness and self-energy] every $L_{α β}$ is a finite real number: the defining double integral converges absolutely, and in particular every self-coefficient $L_{α α}$ is finite, so the compact smooth profiles carry a finite self-energy. This is a statement about the smooth profiles only; no finite value is assigned to a zero-radius thin-wire self-inductance.

(b) [Reciprocity and flux linkage] $L_{α β} = L_{β α}$ for all $α, β$ ; $A_{J_{I}} = \sum_{α} I_{α} A_{j_{α}}$ componentwise; and for every $β$ the linkage integral is $\int_{R^{3}} ⟨ j_{β}, A_{J_{I}} ⟩ d λ_{3} = \sum_{α = 1}^{n} L_{β α} I_{α} .$ Thus the numbers $L_{α β}$ are the linkage coefficients of the profiles: the linkage of profile $β$ with the field of the superposition is the linear combination $\sum_{α} L_{β α} I_{α}$ .

(c) [Magnetic energy and positivity] with $U (I) : = \frac{1}{2} \int_{R^{3}} ⟨ J_{I}, A_{J_{I}} ⟩ d λ_{3}$ , $U (I) = \frac{1}{2} \sum_{α, β = 1}^{n} L_{α β} I_{α} I_{β} = \frac{1}{2 μ_{0}} \int_{R^{3}} ∣ B_{J_{I}} ∣^{2} d λ_{3} \geq 0,$ all quantities finite; moreover $U (I) = 0$ holds precisely when $J_{I} = 0$ on $R^{3}$ . Identify these profile-labelled coefficients with the standard matrix by $L (i, j) : = L_{i + 1, j + 1}$ for $0 \leq i, j < n$ . Consequently the symmetric matrix $L$ is positive semidefinite, and it is positive definite exactly when the profiles are pointwise linearly independent, i.e. when $\sum_{α} I_{α} j_{α} \equiv 0$ forces $I = 0$ .

(d) [Scope] the theorem concerns fixed compact smooth conserved current profiles and their Coulomb-gauge fields. It asserts no thin-wire, sheet or singular self-inductance; no time-dependent, moving, radiating or material configuration; and no uniqueness of the vector potential. The identification of these coefficients with the inductance matrix of the adopted lumped circuit model, and of the linkage derivatives with terminal voltages, is adopted in Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime; the strict material-loop theorem applies separately only to an actual moving sheet and compatible smooth Maxwell fields satisfying its hypotheses. No terminal-voltage law for arbitrary varying frozen profiles is inferred from the present magnetostatic computation.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , the constants $μ_{0} > 0$ , $c > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ , compact smooth divergence-free profiles $j_{α}$ , amplitudes $I \in R^{n}$ with $I_{α} = I (α - 1)$ and matrix entries $L (i, j) = L_{i + 1, j + 1}$ , the current $J_{I} = \sum_{α} I_{α} j_{α}$ , and the Coulomb-gauge field $A_{J_{I}}$ of the magnetostatic potential theorem.

[A1]

Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

For $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , the Coulomb-gauge Newtonian potential $A = μ_{0} N (J)$ is $C^{\infty}$ with $- Δ A = μ_{0} J$ and $div A = 0$ pointwise, and $B = curl A$ is $C^{\infty}$ with $div B = 0$ , $curl B = μ_{0} J$ pointwise. (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current).

[F2]

For $u, v \in C_{c}^{\infty} (R^{3}; R^{3})$ the three integrals $\int ⟨ u, N v ⟩$ , $\int ⟨ N u, v ⟩$ and $\frac{1}{4 π} \iint \frac{⟨ u (x), v (y) ⟩}{∣ x - y ∣}$ converge absolutely and have a common value; the pairing is symmetric and bilinear. (Symmetry and Fubini representation of the Newtonian linkage pairing).

[F3]

For $J \in C_{c}^{\infty} (R^{3}; R^{3})$ with $div J = 0$ and $A = μ_{0} N (J)$ : $\int ∣ curl A ∣^{2} = μ_{0} \int ⟨ A, J ⟩$ and $\frac{1}{2} \int ⟨ A, J ⟩ = \frac{1}{2 μ_{0}} \int ∣ curl A ∣^{2} \geq 0$ , the integrals finite, and $curl A \in L^{2}$ . (Curl-field energy identity for Newtonian potentials of divergence-free compact currents).

[F4]

The compact smooth current class consists of the currents $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero to $R^{3}$ . (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models).

[F5]

The framework constants are $μ_{0} > 0$ , $c > 0$ and $ε_{0} = (μ_{0} c^{2})^{- 1}$ . (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F6]

The lumped circuit model of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime is an adopted reduced model: its matrix $L$ is declared model data, the flux linkage is $L I$ , and its terminal law is a branch relation of the model, not a derivation from arbitrary time-dependent Maxwell fields. (Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime).

[F7]

The Newtonian potential is $N f (x) = \int_{R^{n}} Φ (x - y) f (y) d y$ at each $x$ where the integral is absolutely finite. (Newtonian potential of compactly supported data).

[F8]

The Lebesgue integral is linear on $L^{1}$ . (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F9]

For a $C^{2}$ field $F$ on open $U \subseteq R^{3}$ , $curl curl F = \nabla div F - Δ F$ . (The curl of a curl is the gradient of the divergence minus the Laplacian).

[F10]

For a $C^{2}$ map $Δ F$ is componentwise and the Laplacian is $div \nabla$ . (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

[F11]

The divergence and curl are defined pointwise from the first partial derivatives. (Divergence and curl of a $C^{1}$ vector field).

[F12]

The Euclidean inner product is symmetric and real bilinear. (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F13]

A nonnegative measurable function has zero integral exactly when it vanishes almost everywhere. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere).

[F14]

Under Countable Choice every continuous map $R^{n} \to R^{m}$ is Borel measurable. (Continuous functions on Euclidean spaces are Borel measurable).

[F15]

For measurable $f, g : X \to [0, + \infty]$ with $f \leq g$ , $\int f \leq \int g$ . (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F16]

For $n \geq 1$ , $x \in R^{n}$ , $r > 0$ the ball satisfies $0 < λ (B (x, r)) < \infty$ . (Euclidean balls have positive finite Lebesgue measure).

[F17]

On a measure space $(X, A, μ)$ , for measurable $f : X \to [0, + \infty]$ and $E \in A$ , $\int_{E} f d μ : = \int f χ_{E} d μ$ , with the nonnegative Lebesgue integral on the right. (Integral over a measurable subset).

Proof

technique · direct

1.1F1F4F7F8F10F11algebra

For each $α$ the profile $j_{α}$ lies in the class of [F4], and so does $J_{I}$ ; by [F1] applied to $J_{I}$ the field $A_{J_{I}}$ is $C^{\infty}$ with $- Δ A_{J_{I}} = μ_{0} J_{I}$ and $div A_{J_{I}} = 0$ pointwise, and $B_{J_{I}} = curl A_{J_{I}}$ is $C^{\infty}$ with $curl B_{J_{I}} = μ_{0} J_{I}$ , the Laplacian here being the componentwise operator of [F10]. Since each $A_{j_{α}} = μ_{0} N (j_{α})$ is the Newtonian potential of [F7] and $N$ is componentwise linear by [F8], $A_{J_{I}} = \sum_{α} I_{α} A_{j_{α}}$ componentwise, and $B_{J_{I}} = \sum_{α} I_{α} B_{j_{α}}$ by linearity of the pointwise curl [F11].

1.2F2F3algebra

By [F2] applied to $(J_{I}, J_{I})$ and the definitions of $U$ and the coefficients, $U (I) = \frac{1}{2} \int ⟨ J_{I}, A_{J_{I}} ⟩ = \frac{μ_{0}}{2} \frac{1}{4 π} \iint \frac{⟨ J_{I} (x), J_{I} (y) ⟩}{∣ x - y ∣} = \frac{1}{2} \sum_{α, β} L_{α β} I_{α} I_{β}$ , all finite. On the other hand [F3] applied to the divergence-free current $J_{I}$ with parameter $μ_{0}$ gives $\int ⟨ J_{I}, A_{J_{I}} ⟩ = \frac{1}{μ_{0}} \int ∣ B_{J_{I}} ∣^{2}$ , hence $U (I) = \frac{1}{2 μ_{0}} \int ∣ B_{J_{I}} ∣^{2} \geq 0$ ; this is the energy identity of (c).

2.1F2F8F12step 1.1algebra

Apply [F2] to the pair $(j_{α}, j_{β})$ : the double integral defining $L_{α β}$ (with the prefactor $μ_{0}$ ) converges absolutely, so $L_{α β}$ is a finite real number, and exchanging the two fields and the two integration variables leaves it unchanged, so $L_{α β} = L_{β α}$ ; this is (a) and the reciprocity of (b). Applying [F2] to $(j_{β}, J_{I})$ and using the symmetry of the Euclidean pairing [F12] and the linearity of the integrals [F8] gives $\int ⟨ j_{β}, A_{J_{I}} ⟩ = μ_{0} \frac{1}{4 π} \iint \frac{⟨ j_{β} (x), J_{I} (y) ⟩}{∣ x - y ∣} = \sum_{α} I_{α} L_{β α}$ ; this is the linkage identity of (b).

2.2F9F13F14F15F16F17step 1.1step 1.2algebra

If $U (I) = 0$ then $\int ∣ B_{J_{I}} ∣^{2} = 0$ by step 1.2. The function $∣ B_{J_{I}} ∣^{2}$ is nonnegative, continuous by step 1.1 and Borel by [F14], so [F13] gives $∣ B_{J_{I}} ∣ = 0$ almost everywhere; were $B_{J_{I}} (x_{0}) \neq 0$ for some $x_{0}$ , continuity would give $∣ B_{J_{I}} ∣^{2} \geq ε > 0$ on a ball $B (x_{0}, r)$ of finite positive measure [F16], and [F15] with [F17] would give $\int ∣ B_{J_{I}} ∣^{2} \geq ε λ_{3} (B (x_{0}, r)) > 0$ , a contradiction; hence $B_{J_{I}} = 0$ everywhere. Then $curl curl A_{J_{I}} = \nabla div A_{J_{I}} - Δ A_{J_{I}} = μ_{0} J_{I}$ by [F9] and step 1.1, while $curl B_{J_{I}} = 0$ ; so $μ_{0} J_{I} = 0$ and $J_{I} = 0$ . Conversely $J_{I} = 0$ gives $A_{J_{I}} = 0$ componentwise and $U (I) = 0$ .

3.1step 2.1step 1.2step 2.2algebra

By step 1.2, $U$ is the quadratic form $I \mapsto \frac{1}{2} I^{T} L I$ of the symmetric matrix $L$ of step 2.1, and $U \geq 0$ ; hence $L$ is positive semidefinite. By step 2.2, $U (I) = 0$ exactly when $\sum_{α} I_{α} j_{α} \equiv 0$ ; therefore $L$ is positive definite exactly when that vanishing forces $I = 0$ , that is, exactly when the profiles are pointwise linearly independent.

4.1A1F5F6step 1.1step 2.1step 1.2step 2.2step 3.1∎

The theorem establishes (a)-(c) for the fixed compact smooth conserved profiles and their Coulomb-gauge fields, with no thin-wire, singular, moving, radiating or material configuration treated; the identification of $L$ with the lumped-model inductance matrix and of the linkage derivatives with terminal voltages is the adopted model content of [F6], not a consequence of this magnetostatic computation, and no terminal-voltage law for arbitrary varying frozen profiles is inferred. The constants used are those of [F5], and Countable Choice is inherited only through the named Newtonian, measure and vector-calculus interfaces.

Source notes

Griffiths §7.2.3, printed pp. 310-317, derives $M_{21} = M_{12}$ and the Neumann formula; Fitzpatrick §7.2, printed pp. 216-218, treats stationary-loop linkage and reciprocity in (7.3)-(7.10); §7.5, printed pp. 225-231, treats coil energy and the mutual term $M I_{1} I_{2}$ in (7.37)-(7.71); Tong §4.1.2, printed pp. 71-73, defines the inductance as flux per unit current and writes the magnetostatic energy $\frac{1}{2} L I^{2}$ . The present theorem replaces the thin-loop formulas by the explicitly defined smooth compact conserved profiles, makes the reciprocity an instance of the symmetry of the Newtonian linkage pairing, and identifies the energy with the frozen curl-field energy of the Newtonian construction; the thin-wire self-inductance and every singular limit remain outside the statement.

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Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy

Physical scope

The restricted inductive-energy bookkeeping of the adopted lumped model: for fixed geometry and a constant symmetric inductance matrix the accumulated terminal energy is the change of $\frac{1}{2} I^{T} L I$ , and for a smooth compact conserved-current realization this equals the change of the frozen magnetic field energy $\frac{1}{2 μ_{0}} \int ∣ B ∣^{2}$ . It excludes changing geometry, radiation, distributed and material-loss effects, thin-wire self-inductance, and any empirical claim.

Statement

Assume the lumped circuit model of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and assume Countable Choice ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )) for clause (b), where it is inherited from the linkage suppliers.

For $y \in R^{n}$ write $y_{α} : = y (α - 1)$ , and for $L \in R^{n \times n}$ write $L_{α β} : = L (α - 1, β - 1)$ , where $1 \leq α, β \leq n$ , consistently with the zero-based coordinates of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and the profile labels of Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents. All sums over $α, β$ below range from $1$ to $n$ .

(a) [Terminal-energy bookkeeping] Let $J \subseteq R$ be an interval (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), let $t_{0}, t_{1} \in J$ , let $L \in R^{n \times n}$ be constant and symmetric, let $I : J \to R^{n}$ be of class $C^{1}$ in the following sense: $I$ extends to a $C^{1}$ map on an open interval containing $J$ , so that the derivatives used below are those of the extension (one-sided at included endpoints of $J$ , and not intrinsic to a value restriction when $J$ is a singleton), and let $V (t) : = L \dot{I} (t)$ be the terminal-voltage vector of the inductive branches as adopted in Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime (the Faraday branch: the terminal voltage of a branch is the rate of change of its flux linkage $Φ = L I$ ). Then $\int_{t_{0}}^{t_{1}} ⟨ V (t), I (t) ⟩ d t = \frac{1}{2} ⟨ I (t_{1}), L I (t_{1}) ⟩ - \frac{1}{2} ⟨ I (t_{0}), L I (t_{0}) ⟩,$ and for $n = 1$ with $L > 0$ this reads $\int_{t_{0}}^{t_{1}} V I d t = \frac{1}{2} L (I (t_{1})^{2} - I (t_{0})^{2})$ : the accumulated terminal energy equals the change of the inductor bookkeeping energy $\frac{1}{2} L I^{2}$ . The identity is exact for every such $C^{1}$ current and fixed geometry; it uses only symmetry of $L$ , the product rule and the fundamental theorem of calculus. For a changing geometry the mechanical work done on the circuit is an additional term and is not covered here.

(b) [Identification with the frozen field energy] Suppose in addition that the currents are realized by fixed compact smooth conserved profiles $j_{1}, \dots, j_{n}$ of the class of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b), with instantaneous source $J_{I} (t) : = \sum_{α} I_{α} (t) j_{α}$ , and that the inductance matrix is the linkage matrix of those profiles in the sense of Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents: $L_{α β} = \frac{μ_{0}}{4 π} \iint \frac{⟨ j_{α} (x), j_{β} (y) ⟩}{∣ x - y ∣} d λ_{3} (y) d λ_{3} (x)$ . Then for every $t \in J$ $\frac{1}{2} ⟨ I (t), L I (t) ⟩ = \frac{1}{2} \int_{R^{3}} ⟨ J_{I} (t), A_{J_{I} (t)} ⟩ d λ_{3} = \frac{1}{2 μ_{0}} \int_{R^{3}} ∣ B_{J_{I} (t)} ∣^{2} d λ_{3} \geq 0,$ where $A_{J_{I} (t)}$ is the Coulomb-gauge Newtonian vector potential and $B_{J_{I} (t)} = curl A_{J_{I} (t)}$ of that theorem: the right-hand side is the magnetic field energy of the frozen instantaneous configuration, and the accumulated terminal energy of (a) between $t_{0}$ and $t_{1}$ is the change of this field energy. The relation $V = L \dot{I}$ is the explicitly adopted terminal law of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime; the identification of the model's inductance matrix with the smooth-profile linkage matrix is part of the adopted model of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime.

(c) [Scope] no radiation, distributed-element, material-loss, moving-geometry or thin-wire self-inductance claim is made; the frozen-configuration energy is not asserted to be the energy of a time-dependent full Maxwell solution.

Facts & Assumptions

Given: Assume the lumped model data of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime, a constant symmetric $L \in R^{n \times n}$ , an interval $J$ , points $t_{0}, t_{1} \in J$ , a current $I : J \to R^{n}$ extending to a $C^{1}$ map on an open interval containing $J$ , the terminal voltage $V = L \dot{I}$ , and ${A C}_{ω}$ for clause (b).

[A1]

Countable Choice, written ${A C}_{ω}$ , says every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The adopted lumped circuit model carries branch currents $I_{α} : J \to R$ and constant matrices $L, C, R$ with $L$ symmetric, the Faraday branch relation $V (t) = L \dot{I} (t)$ (each terminal potential is the rate of change of the branch flux linkage $L I$ ), and the circuit-level bookkeeping quantities $L I$ , $\frac{1}{2} I^{T} L I$ ; the matrices, terminal geometry and loop choice are declared model data, not derived from the vacuum Maxwell equations. (Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime).

[F2]

A subset of $R$ is an interval when it has one of the nine order-convex forms listed in the definition; in particular every interval is order-convex. (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[F3]

If $f, g$ are differentiable at a point $c$ of their common domain and $α \in R$ , then $f + g$ , $α f$ and $f g$ are differentiable at $c$ , with $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ and $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ . (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 $g (c) \neq 0$ ).

[F4]

If $a < b$ and $f : [a, b] \to R$ is continuous on $[a, b]$ , then $f$ is bounded and Riemann (Darboux) integrable on $[a, b]$ . (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).

[F5]

If $a < b$ , $G : [a, b] \to R$ is differentiable at every point of $[a, b]$ as a function on $[a, b]$ (one-sided at the endpoints), and $G^{'}$ is integrable on $[a, b]$ , 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 $\int_{a}^{b} f = G (b) - G (a)$ ).

[F6]

The oriented integral is defined for all $u, v \in R$ by: the Darboux integral over $[u, v]$ when $u < v$ ; $0$ when $u = v$ ; and $- \int_{v}^{u} f$ when $u > v$ ; on $u < v$ it is the published integral verbatim. (The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ).

[F7]

For fixed compact smooth conserved profiles $j_{1}, \dots, j_{n}$ the linkage coefficients $L_{α β} = \frac{μ_{0}}{4 π} \iint ⟨ j_{α} (x), j_{β} (y) ⟩ / ∣ x - y ∣$ are finite real numbers with $L_{α β} = L_{β α}$ , the linkage of the superposition is $\int ⟨ j_{β}, A_{J_{I}} ⟩ = \sum_{α} L_{β α} I_{α}$ , and with $U (I) : = \frac{1}{2} \int ⟨ J_{I}, A_{J_{I}} ⟩$ one has $U (I) = \frac{1}{2} \sum_{α, β} L_{α β} I_{α} I_{β} = \frac{1}{2 μ_{0}} \int ∣ B_{J_{I}} ∣^{2} \geq 0$ , with $U (I) = 0$ exactly when $J_{I} = 0$ . (Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents).

[F8]

[F9]

The framework uses SI units: $V$ in volts, $I$ in amperes, $t$ in seconds and $L$ in henries ( $H = V s A^{- 1}$ ). (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F10]

The Euclidean inner product is $⟨ x, y ⟩ = \sum_{k = 0}^{n - 1} x (k) y (k)$ ; it is symmetric and bilinear over $R$ , so with the Statement's profile labels $⟨ L y, x ⟩ = \sum_{α, β = 1}^{n} L_{α β} y_{β} x_{α}$ for every $L \in R^{n \times n}$ and $x, y \in R^{n}$ . (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

technique · direct

1.1F1F3F10algebra

Fix the $C^{1}$ extension of $I$ to an open interval $U \supseteq J$ , and define $W (y) : = \frac{1}{2} ⟨ y, L y ⟩ = \frac{1}{2} \sum_{α, β} L_{α β} y_{α} y_{β}$ for $y \in R^{n}$ , so that $t \mapsto G (t) : = W (I (t)) = \frac{1}{2} ⟨ I (t), L I (t) ⟩$ is the bookkeeping-energy function of the Statement. For each pair $(α, β)$ the product $t \mapsto I_{α} (t) I_{β} (t)$ is differentiable on $U$ with derivative ${\dot{I}}_{α} I_{β} + I_{α} {\dot{I}}_{β}$ by [F3], and $G$ is a finite linear combination of these products; applying the sum and scalar rules of [F3] repeatedly therefore gives, for every $t \in J$ , $G^{'} (t) = \frac{1}{2} \sum_{α, β} L_{α β} ({\dot{I}}_{α} (t) I_{β} (t) + I_{α} (t) {\dot{I}}_{β} (t)) = \sum_{α, β} L_{α β} {\dot{I}}_{α} (t) I_{β} (t) = ⟨ L \dot{I} (t), I (t) ⟩ = ⟨ V (t), I (t) ⟩,$ where the second equality relabels $α \leftrightarrow β$ in one half of the sum and uses the symmetry $L_{α β} = L_{β α}$ , and the last two use [F10] and $V = L \dot{I}$ ; the derivative $G^{'}$ is continuous on $J$ , being a finite sum of products of continuous functions.

1.2F7F8F10algebra

Assume now the realization hypotheses of clause (b). For each fixed $t \in J$ put $J_{I} (t) : = \sum_{α} I_{α} (t) j_{α}$ , a member of the compact smooth conserved class; applying [F7] to the amplitudes $I (t) = (I_{1} (t), \dots, I_{n} (t))$ (its coefficients are the linkage numbers of the profiles $j_{1}, \dots, j_{n}$ , which by hypothesis coincide with the entries of $L$ ) gives $\frac{1}{2} ⟨ I (t), L I (t) ⟩ = \frac{1}{2} \sum_{α, β} L_{α β} I_{α} (t) I_{β} (t) = U (I (t)) = \frac{1}{2} \int_{R^{3}} ⟨ J_{I} (t), A_{J_{I} (t)} ⟩ d λ_{3} = \frac{1}{2 μ_{0}} \int_{R^{3}} ∣ B_{J_{I} (t)} ∣^{2} d λ_{3} \geq 0,$ the outer equalities by [F10] and the definition of $U$ in [F7]; this is clause (b)'s energy identity at each fixed $t$ .

2.1F2F4F5F6F10step 1.1algebracases

Let $u : = \min {t_{0}, t_{1}}$ and $v : = \max {t_{0}, t_{1}}$ ; then $[u, v] \subseteq J$ , since $J$ is order-convex by [F2] and contains both $u$ and $v$ . If $u < v$ , then $G$ is differentiable at every point of $[u, v]$ as a function on $[u, v]$ (interior points lie in the open set $U$ , and at $u, v$ the one-sided derivatives exist by restriction of the extension), while $G^{'}$ is continuous on $[u, v]$ by step 1.1 and hence integrable there by [F4]; [F5] gives $\int_{u}^{v} G^{'} = G (v) - G (u)$ , that is $\int_{u}^{v} ⟨ V, I ⟩ = G (v) - G (u)$ by step 1.1. If $u = v$ the same display is $0 = G (v) - G (u)$ , which holds identically and is the $u = v$ reading of [F6]. If $t_{0} > t_{1}$ , applying the already-proved case to the pair $(t_{1}, t_{0})$ and the sign clause of [F6] gives $\int_{t_{0}}^{t_{1}} ⟨ V, I ⟩ = - \int_{t_{1}}^{t_{0}} ⟨ V, I ⟩ = - (G (t_{0}) - G (t_{1})) = G (t_{1}) - G (t_{0})$ . In every case $\int_{t_{0}}^{t_{1}} ⟨ V (t), I (t) ⟩ d t = \frac{1}{2} ⟨ I (t_{1}), L I (t_{1}) ⟩ - \frac{1}{2} ⟨ I (t_{0}), L I (t_{0}) ⟩$ , which is the identity of (a); for $n = 1$ the quadratic form is $\frac{1}{2} L I^{2}$ by [F10].

3.1step 2.1step 1.2F1algebra

Combining steps 2.1 and 1.2 gives, for all $t_{0}, t_{1} \in J$ , $\int_{t_{0}}^{t_{1}} ⟨ V (t), I (t) ⟩ d t = \frac{1}{2} \int_{R^{3}} ⟨ J_{I} (t_{1}), A_{J_{I} (t_{1})} ⟩ d λ_{3} - \frac{1}{2} \int_{R^{3}} ⟨ J_{I} (t_{0}), A_{J_{I} (t_{0})} ⟩ d λ_{3} = \frac{1}{2 μ_{0}} \int_{R^{3}} ∣ B_{J_{I} (t_{1})} ∣^{2} d λ_{3} - \frac{1}{2 μ_{0}} \int_{R^{3}} ∣ B_{J_{I} (t_{0})} ∣^{2} d λ_{3},$ so that the accumulated terminal energy is the change of the frozen field energy, as asserted in (b). The terminal law $V = L \dot{I}$ is the adopted branch relation [F1], and the identification of the model's inductance matrix with the linkage matrix is the additional realization hypothesis of (b), not a consequence of the magnetostatic theorem alone.

4.1A1F1F9step 2.1step 1.2step 3.1∎

Clause (c) holds as stated: the computation is fixed-geometry bookkeeping of the adopted model; changing geometry would add the mechanical work term, and the frozen realization is not asserted to solve any time-dependent Maxwell system. Units are those of [F9]; the singleton case of (a) was handled in step 2.1, where the extension supplies the derivative and no intrinsic derivative is inferred from the value restriction. Countable Choice enters only through clause (b), inherited from the linkage theorem [F7] and its Newtonian and measure interfaces; clause (a) is choice-free.

Source notes

Fitzpatrick §7.5, printed pp. 225-231, writes the battery work for coupled coils as Ohmic terms plus the inductive energy and states that the energy of an inductor is stored in the magnetic field; Tong §4.1.2, printed pp. 71-73, integrates the work against the induced emf to $W = \frac{1}{2} L I^{2}$ and rewrites it as the magnetostatic field energy under a decay assumption; Griffiths §7.2.4, printed pp. 317-320, records $\frac{1}{2} L I^{2}$ and $U = \frac{1}{2 μ_{0}} \int B^{2} d τ$ . The present theorem replaces the decay assumption by the compact smooth conservative profiles of Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents and keeps the terminal law and the matrix identification as explicitly adopted model content; no error bound and no singular thin-wire self-inductance are claimed.

Physics theoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Restricted RL decay and LC oscillation under the adopted circuit model

Physical scope

Conditional scalar RL/LC dynamics of the adopted lumped model, with arbitrary positive model parameters, endpoint intervals and explicit singleton candidate jets retained. Algebraic bookkeeping is choice-free; the optional identification with a specified geometric capacitor $C = C_{11}$ carries full AC and the bounded smooth conductor hypotheses of its supplier, and the frozen magnetic realization separately carries the compact-profile and ${A C}_{ω}$ premises of the linkage theorem. Neither frozen realization is a time-dependent full Maxwell solution; no radiation or distributed effect is included.

Statement

Assume the lumped circuit model and its units as in Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime and Framework-local SI units, the vacuum constants and the exact/measured status of their values. Let $J \subseteq R$ be an interval (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length) and let $t_{0} \in J$ . For a nondegenerate $J$ , derivatives are domain-relative, with one-sided values at included endpoints, in the sense of First-order systems, initial value problems, and solutions on intervals; the functions constructed below also admit the stated $C^{k}$ open-neighbourhood extensions. If $J = {t_{0}}$ , the formulas specify the initial values and the jets of the explicitly constructed extensions below, rather than an intrinsic derivative of a singleton restriction. The RL solution, the LC candidate and the bookkeeping-energy algebra use no choice principle. Only the optional geometric capacitor identification in (b) assumes full Axiom of Choice (The Axiom of Choice) and its ${A C}_{ω}$ consequence (The Axiom of Countable Choice ( ${A C}_{ω}$ )), exactly as Capacitance and energy for specified linear conductor geometry requires. The inductive energy is that of Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy.

(a) [RL circuit] Let $L > 0$ and $R \geq 0$ be constants and let $V : J \to R$ be continuous. Then the initial value problem $L \dot{I} (t) + R I (t) = V (t), I (t_{0}) = I_{0} \in R,$ has exactly one $C^{1}$ solution on every nondegenerate $J$ , namely $I (t) = \exp ⁣ (- \frac{R (t - t_{0})}{L}) [I_{0} + \frac{1}{L} \int_{t_{0}}^{t} \exp ⁣ (\frac{R (s - t_{0})}{L}) V (s) d s],$ and along it the inductive bookkeeping energy satisfies the power identity $\frac{d}{d t} (\frac{1}{2} L I (t)^{2}) = L \dot{I} (t) I (t) = V (t) I (t) - R I (t)^{2},$ i.e. the terminal power splits into the rate of change of the inductive energy and the Ohmic term $R I^{2}$ . (For $R = 0$ the same formula reduces to $I = I_{0} + L^{- 1} \int_{t_{0}}^{t} V$ .) For $J = {t_{0}}$ its unique restriction is $I (t_{0}) = I_{0}$ ; the canonical extension obtained using the constant source $V (t_{0})$ has jet $\dot{I} (t_{0}) = (V (t_{0}) - R I_{0}) / L$ and satisfies the same power algebra. No uniqueness of arbitrary extension germs is asserted.

(b) [LC oscillation] Let $L > 0$ , $C > 0$ and set $Ω : = (L C)^{- 1 / 2} > 0$ , $z_{0} : = Q_{0} - i I_{0} / Ω$ , $w (t) : = z_{0} \exp (i Ω (t - t_{0}))$ with the complex exponential of The complex exponential by its power series, and $Q (t) : = Re w (t)$ . Then $Q$ is of class $C^{2}$ on $J$ , $Q (t_{0}) = Q_{0}$ , $Q^{'} (t_{0}) = I_{0}$ , and $L Q^{''} (t) + \frac{Q (t)}{C} = 0 on J;$ equivalently, with the real trigonometric functions of Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , $Q (t) = Q_{0} \cos (Ω (t - t_{0})) + \frac{I_{0}}{Ω} \sin (Ω (t - t_{0})) .$ Moreover the total bookkeeping energy $E (t) : = \frac{1}{2} L Q^{'} (t)^{2} + \frac{Q (t)^{2}}{2 C}$ satisfies $E^{'} (t) = Q^{'} (t) (L Q^{''} (t) + Q (t) / C) = 0$ on $J$ , hence $E$ is constant with value $\frac{1}{2} L I_{0}^{2} + \frac{Q_{0}^{2}}{2 C}$ : the two terms are the adopted capacitive and inductive bookkeeping energies, the latter as in Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy(a), so in the lossless loop their sum is conserved. For the additional geometric identification, explicitly choose the fixed compact separated conductor arrangement of Capacitance and energy for specified linear conductor geometry with bounded smooth ambient domain, connected vacuum, grounded outer boundary, zero volume source and its unit-Dirichlet solutions; drive only conductor 1 and ground the others, set the circuit parameter $C$ equal to that arrangement's $C_{11} > 0$ , and at each time set $V_{1} = Q / C_{11}$ . Under full AC that supplier gives the frozen electrostatic field energy $Q^{2} / (2 C_{11})$ . An arbitrary declared $C > 0$ is model data and is not automatically the capacitance of a particular arrangement. Any magnetic field-energy identification likewise requires the extra smooth-profile linkage realization of Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy(b), under its ${A C}_{ω}$ hypothesis. Neither frozen realization is an asserted time-dependent full Maxwell solution. At a singleton time the entire LC candidate supplies the jets $\dot{Q} = I_{0}$ , $\ddot{Q} = - Q_{0} / (L C)$ and the same constant energy.

(c) [Scope] both statements are consequences of the expressly adopted circuit reduction, with the sign conventions and relations of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime; they assert no general network completeness, no derivation from the full time-dependent Maxwell system, no radiation or distributed-element effect and no empirical claim.

Facts & Assumptions

Given: Assume the lumped circuit data of Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime and the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, an interval $J$ with $t_{0} \in J$ , constants $L > 0$ , $R \geq 0$ in (a) and $C > 0$ in (b), a continuous source $V : J \to R$ , and the initial data $I_{0}, Q_{0} \in R$ . For clause (b)'s geometric identification assume full $A C$ with its ${A C}_{ω}$ consequence.

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. (The Axiom of Choice).

[A2]

Countable Choice, written ${A C}_{ω}$ , states that every sequence of nonempty sets has a choice function. (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The adopted lumped model carries the Faraday branch relation $V = L \dot{I}$ for an inductive branch, the capacitive branch relation $V = C^{- 1} Q$ (in the scalar case $V = Q / C$ ), the resistive branch relation $V = R I$ as the circuit form of the Ohmic branch of the material model, and Kirchhoff's loop voltage balance around each independent loop; the matrices and terminal data are declared model data, not derived from the vacuum Maxwell equations. (Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime).

[F2]

A subset of $R$ is an interval when it has one of the nine order-convex forms of the definition; in particular intervals are order-convex, nondegenerate intervals have at least two elements, and a nondegenerate interval contains the two-point set ${u, v}$ for suitable $u < v$ . (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[F3]

Let $I \subseteq R$ be order-convex with at least two elements, let $x_{0} \in I$ and let $p, q : I \to R$ be continuous. The initial value problem $y^{'} + p y = q$ , $y (x_{0}) = y_{0}$ , has exactly one solution on $I$ , namely $y (x) = \exp (- P (x)) (y_{0} + \int_{x_{0}}^{x} \exp (P (t)) q (t) d t)$ with $P (x) = \int_{x_{0}}^{x} p (t) d t$ . (A scalar first-order linear ODE has a unique solution given by the integrating-factor formula).

[F4]

A solution of the first-order system $x^{'} = F (t, x (t))$ with $x (t_{0}) = x_{0}$ on an interval $I$ is a function $x : I \to R^{n}$ with $t_{0} \in I$ , differentiable on $I$ in the domain-relative sense, endpoints of $I$ using one-sided, domain-relative derivatives, and satisfying the equation and initial condition. (First-order systems, initial value problems, and solutions on intervals).

[F5]

If $f, g$ are differentiable at a point of their common domain and $α \in R$ , then $f + g$ , $α f$ and $f g$ are differentiable there with the sum, scalar and product rules; in particular $\frac{d}{d t} \frac{1}{2} L f (t)^{2} = L f (t) f^{'} (t)$ . (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 $g (c) \neq 0$ ).

[F6]

Sums, scalar multiples, products, absolute values and (where defined) quotients of continuous functions are continuous, and every constant function and the identity are continuous. (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).

[F7]

The complex exponential is $\exp z = \sum_{n \geq 0} z^{n} / n!$ , convergent for every $z \in C$ . (The complex exponential by its power series).

[F8]

The complex exponential is entire and $\exp^{'} (z) = \exp z$ for every $z \in C$ . (The complex exponential is entire and its complex derivative is itself).

[F9]

For $U \subseteq C$ open and $f : U \to C$ , the following are equivalent: $f$ is complex differentiable at $a$ ; under $C ≅ R^{2}$ the map $f$ is real totally differentiable at $a$ and $D f (a)$ is multiplication by a complex number; and the Cauchy--Riemann equations hold at $a$ with real total differentiability. When these hold, $f^{'} (a) = \partial_{z} f (a)$ . (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[F10]

Complex derivatives obey the sum, scalar, product and quotient rules, the derivative of a constant is $0$ , and the derivative of the identity is $1$ . (Linearity, product, reciprocal, and quotient rules for complex derivatives).

[F11]

If $f$ is complex differentiable at $a$ and $g$ at $f (a)$ , then $g \circ f$ is complex differentiable at $a$ with $(g \circ f)^{'} (a) = g^{'} (f (a)) f^{'} (a)$ . (The chain rule for complex derivatives).

[F12]

If $f$ is totally differentiable at $a$ and $g$ at $f (a)$ , then $g \circ f$ is totally differentiable at $a$ with $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ . (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F13]

For every real $θ$ , $\exp (i θ) = \cos θ + i \sin θ$ with the real cosine and sine. (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ ).

[F14]

The complex trigonometric functions restrict to their real power-series counterparts on the real axis. (The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over $C$ ).

[F15]

If $I \subseteq R$ is order-convex, $f : I \to R$ is continuous on $I$ and differentiable at every interior point of $I$ with $f^{'} (x) = 0$ there, then $f$ is constant on $I$ . (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

[F16]

For clause (b) of the Statement, the inductive bookkeeping energy and its smooth-profile realization are those of Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy: the terminal energy $\int ⟨ V, I ⟩$ is the change of $\frac{1}{2} L I^{2}$ for constant symmetric $L$ , and for a smooth compact conserved-profile realization it is the change of the frozen field energy $\frac{1}{2 μ_{0}} \int ∣ B ∣^{2}$ . (Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy).

[F17]

The SI units are those of the framework: $V$ in volts, $I$ in amperes, $Q$ in coulombs, $t$ in seconds, $L$ in henries and $C$ in farads. (Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[L1]

The optional geometric identification clause uses Capacitance and energy for specified linear conductor geometry: for the fixed compact separated conductor arrangement with bounded smooth ambient domain of its clause (f), its clause (c) gives the field-energy identity $U (V) = \frac{ε_{0}}{2} \int_{Ω_{v}} ∣ \nabla φ^{V} ∣^{2} d λ_{3} = \frac{1}{2} V^{T} C V = \frac{1}{2} \sum_{j} V_{j} Q_{j} (V)$ , valid under full AC and its stated Dirichlet solvability hypotheses, and its clause (d) gives, with only conductor $1$ driven and the others grounded, $C_{11} > 0$ , $Q_{1} = C_{11} V_{1}$ and the frozen electrostatic field energy $U = \frac{1}{2} C_{11} V_{1}^{2} = Q_{1}^{2} / (2 C_{11})$ . This consumer relies on exactly that clause with $C = C_{11}$ and no independent derivation of the electrostatic energy is substituted.

Proof

technique · direct

1.1F2F3F4F6algebracases

(a) Assume first that $J$ has at least two elements. Put $p : = R / L$ , a constant of class $C^{1}$ , and $q : = V / L$ , continuous on $J$ by [F6]; $J$ is order-convex with at least two elements by [F2]. With $P (x) : = \int_{t_{0}}^{x} p = \frac{R}{L} (x - t_{0})$ one has $\exp (- P (x)) = \exp (- \frac{R (x - t_{0})}{L})$ and $\exp (P (s)) = \exp (\frac{R (s - t_{0})}{L})$ , so the unique solution supplied by [F3] is exactly the displayed $I$ . The ODE gives $\dot{I} = (V - R I) / L$ , continuous on $J$ . At each included finite endpoint extend $I$ to the outer side by the affine function with that endpoint value and slope; its derivative matches the one-sided ODE value, giving a $C^{1}$ extension to an open interval containing $J$ . Its derivative on $J$ is the domain-relative derivative of [F4] (one-sided at included endpoints), and the uniqueness is the uniqueness assertion of [F3], which is the uniqueness of the $C^{1}$ solution in the sense of [F4]. If $J = {t_{0}}$ , the restriction of any solution is $I_{0}$ by [F4]; choosing the constant extension $V \equiv V (t_{0})$ on an open interval and applying the already-proved nondegenerate case there produces a $C^{1}$ extension of the candidate whose jet at $t_{0}$ is obtained by substituting $t = t_{0}$ into the differential equation, $\dot{I} (t_{0}) = (V (t_{0}) - R I_{0}) / L$ , and no other extension is distinguished.

1.2F7F8F9F10F11F12F13F14algebracases

(b) Define the entire function $G (z) : = z_{0} \exp (i Ω (z - t_{0}))$ on $C$ . By [F10] and [F11], applied to the affine map $z \mapsto i Ω (z - t_{0})$ (derivative $i Ω$ ) and to [F8], $G$ is complex differentiable with $G^{'} (z) = i Ω G (z)$ for every $z$ ; differentiating once more gives $G^{''} = (i Ω)^{2} G = - Ω^{2} G$ . The function $w$ of the Statement is the restriction of $G$ to the real interval $J$ , so by [F9] and the real chain rule [F12] it is real differentiable with $w^{'} (t) = i Ω w (t)$ and $w^{''} (t) = - Ω^{2} w (t)$ for every $t \in J$ ; taking real parts, $Q = Re w$ is of class $C^{2}$ with $Q^{'} (t) = Re w^{'} (t)$ and $Q^{''} (t) = Re w^{''} (t)$ , because the real part is real-linear. Hence $Q^{''} = - Ω^{2} Q$ , which is $L Q^{''} + Q / C = 0$ since $Ω^{2} = 1 / (L C)$ by definition. At $t_{0}$ : $w (t_{0}) = z_{0}$ gives $Q (t_{0}) = Re z_{0} = Q_{0}$ , and $w^{'} (t_{0}) = i Ω z_{0} = i Ω Q_{0} + I_{0}$ gives $Q^{'} (t_{0}) = I_{0}$ . Euler's formula [F13] expands $\exp (i Ω (t - t_{0})) = \cos Ω (t - t_{0}) + i \sin Ω (t - t_{0})$ , and multiplying out $w = z_{0} \exp (i Ω (t - t_{0}))$ gives $Re w = Q_{0} \cos Ω (t - t_{0}) + (I_{0} / Ω) \sin Ω (t - t_{0})$ , with the real trigonometric functions of [F13] and the real restriction recorded in [F14]. For $J = {t_{0}}$ the same formulas are evaluated at $t_{0}$ , the entire function supplying the jets $\dot{Q} (t_{0}) = I_{0}$ and $\ddot{Q} (t_{0}) = - Ω^{2} Q_{0} = - Q_{0} / (L C)$ by the same computation, so the LC candidate needs no separate singleton convention.

2.1F1F5step 1.1algebra

(a) Along the solution of step 1.1 the function $t \mapsto \frac{1}{2} L I (t)^{2}$ is differentiable at every interior point of $J$ (and one-sidedly at included endpoints) by [F5], with derivative $L I \dot{I}$ ; substituting the circuit equation $L \dot{I} = V - R I$ gives $L I \dot{I} = V I - R I^{2}$ , which is the power identity.

2.2A1A2F16L1F5F6F15step 1.2algebracases

(b) Define $E (t) : = \frac{1}{2} L Q^{'} (t)^{2} + \frac{Q (t)^{2}}{2 C}$ ; by [F6] and step 1.2 the functions $Q$ and $Q^{'}$ are continuous, so $E$ is continuous on $J$ , and at every interior point of $J$ the product and scalar rules of [F5] give $E^{'} = L Q^{'} Q^{''} + Q Q^{'} / C = Q^{'} (L Q^{''} + Q / C) = 0$ by the equation of step 1.2. By [F15], $E$ is constant on $J$ , and its value is $E (t_{0}) = \frac{1}{2} L I_{0}^{2} + \frac{Q_{0}^{2}}{2 C}$ by the initial values of step 1.2. The first summand is the inductive bookkeeping energy of [F16], and for the optional geometric identification of the Statement the second summand is the frozen electrostatic field energy supplied by the declared clause of [L1] under full AC and the stated conductor arrangement, with $C = C_{11}$ and $V_{1} = Q / C_{11}$ ; the magnetic realization of the inductive term is the separate smooth-profile clause of [F16], under its ${A C}_{ω}$ premise. For $J = {t_{0}}$ the constant-energy display is the value at $t_{0}$ supplied by step 1.2, so no derivative of a singleton restriction is used.

3.1A1A2F16step 1.1step 1.2step 2.1step 2.2algebra

The full choice hypothesis of the geometric identification is [A1], and the inductive realization of [F16] carries the ${A C}_{ω}$ of [A2]; full AC yields it, because for a sequence $(X_{n})_{n \in N}$ of nonempty sets the family ${X_{n} : n \in N}$ of its values consists of nonempty sets, so [A1] provides a choice function $g$ on that family and $n \mapsto g (X_{n})$ is a choice function for the sequence, as [A2] requires. Hence clause (b) is available under its stated hypotheses, and the RL and LC algebra of steps 1.1, 1.2, 2.1 and 2.2 is choice-free and uses neither [A1] nor [A2].

4.1F1F17step 1.1step 1.2step 2.1step 2.2step 3.1∎

Clauses (a)-(c) are established: step 1.1 and step 2.1 give the existence, uniqueness, formula and power identity of (a); step 1.2 and step 2.2 give the solution, initial values, cosine-sine form, energy identity and conservation of (b), with the geometric identification delegated to [L1]; step 3.1 accounts for the choice use. The units are [F17]; no general network completeness, no full time-dependent Maxwell solution, no radiation or distributed-element effect and no empirical claim is asserted, and both statements are consequences of the expressly adopted relations of [F1].

Source notes

Fitzpatrick §7.5-7.6, printed pp. 225-234, writes the RL equation and the LCR circuit equation with the ideal element relations; Griffiths §7.2.3, printed pp. 310-317, records the battery-plus-RL exponential approach with time constant $L / R$ ; Tong §4.1.3, printed pp. 74-77, presents resistance as an additional material input in the circuit treatment. The RL solution is proved from the published integrating-factor theorem; the lossless LC candidate is verified by explicit complex-exponential differentiation with the real cosine-sine form displayed through Euler's formula, rather than through a real-trigonometric derivative supplier outside this page's declared prerequisite closure; and the optional capacitive identification is referred to its own supplier with its exact conductor and Choice hypotheses recorded. No numerical value, measurement or error bound is asserted.

5 · Examples, experiments and counterexamples

None yet.

Induction and circuit approximations

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Symmetry and Fubini representation of the Newtonian linkage pairing

Statement

Facts & Assumptions

Proof

Source notes

Curl-field energy identity for Newtonian potentials of divergence-free compact currents

Statement

Facts & Assumptions

Proof

Source notes

C2 moving material circuit: swept patch, single-valued material velocity, flux and electromotive force

Definition

Well-definedness and conventions

Source notes

Lumped circuit approximation: branch data, terminal laws and the declared quasistatic regime

Postulate

Status and conventions

Source notes

Motional emf for a moving material circuit: emf = -flux derivative

Statement

Facts & Assumptions

Proof

Source notes

Reciprocal linkage coefficients and magnetic energy of smooth compact conserved currents

Statement

Facts & Assumptions

Proof

Source notes

Restricted inductor energy LI^2/2 and its identification with the frozen magnetic field energy

Statement

Facts & Assumptions

Proof

Source notes

Restricted RL decay and LC oscillation under the adopted circuit model

Statement

Facts & Assumptions

Proof

Source notes

5 · Examples, experiments and counterexamples

Sources