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Magnetostatics in vacuum

1 · Prerequisites

2 · Summary

This page develops magnetostatics in vacuum from the framework's adopted equations: the steady-current regime, the compact smooth conserved-current class, and three explicitly declared ideal source models (the closed thin loop, the infinite straight wire and the infinite cylindrical sheet). Two local mathematical lemmas compute the distributional curl of the azimuthal line field and of the piecewise constant cylinder field, and a third differentiates the Newtonian kernel; these are the exact distributional ingredients of the source equations for the ideal models. The Coulomb-gauge Newtonian vector potential is constructed for compact smooth conserved currents, the Biot-Savart representation and the controlled far-field dipole estimate with its explicit uniform remainder follow, and the force and torque on a declared thin-wire loop in a prescribed external field are computed under an explicitly adopted neutral-wire coupling. The ideal noncompact models are declared idealisations with stated symmetry and decay data, adopted physically by a separate postulate; no measurement, apparatus or empirical claim is involved on this page, and no finite-energy, self-energy, uniqueness or apparatus-accuracy claim is made for them.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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

The piecewise constant cylinder field $e_{3} 1_{{ρ < a}}$ : distributional curl on the cylinder

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $a > 0$ , write $x = (x_{1}, x_{2}, x_{3})$ and $ρ = \sqrt{x_{1}^{2} + x_{2}^{2}}$ , let $S : = {x \in R^{3} : ρ = a}$ be the circular cylinder of radius $a$ about the third coordinate axis, with unit normal $e_{ρ} = ρ^{- 1} (x_{1}, x_{2}, 0)$ pointing from the interior region ${ρ < a}$ to the exterior region ${ρ > a}$ , and let $e_{φ} = ρ^{- 1} (- x_{2}, x_{1}, 0)$ and $e_{3} = (0, 0, 1)$ . Let $g : R^{3} \to R^{3}$ be the piecewise constant field $g (x) : = {\begin{cases} e_{3}, & ρ < a, \\ 0, & ρ > a, \end{cases}$ extended arbitrarily on the three-dimensional Lebesgue-null set $S$ , and let $T_{g}$ be its regular distribution (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions). For an $R^{3}$ -valued distribution $T = (T_{1}, T_{2}, T_{3})$ whose components are scalar distributions of Distribution, a scalar test $φ \in C_{c}^{\infty} (R^{3})$ and a vector test $ψ \in C_{c}^{\infty} (R^{3}; R^{3})$ , the pairings used below are the componentwise readings of Distributional derivative: $⟨ div T, φ ⟩ : = - \sum_{i = 1}^{3} ⟨ T_{i}, \partial_{i} φ ⟩, ⟨ curl T, ψ ⟩ : = \sum_{i = 1}^{3} ⟨ T_{i}, (curl ψ)_{i} ⟩ .$ Then for every vector test $ψ$ and every scalar test $φ$ , $⟨ curl T_{g}, ψ ⟩ = \int_{S} e_{φ} \cdot ψ d S, ⟨ div T_{g}, φ ⟩ = 0,$ that is, $curl T_{g} = e_{φ} δ_{S}$ and $div T_{g} = 0$ in the distributional sense of Distributional derivative. The surface integral is that of Surface integration on compact C1 hypersurfaces; since $ψ$ has compact support it equals the integral over any compact cylinder segment containing $supp ψ \cap S$ . Equivalently, with $g_{\pm}$ the one-sided values and the normal $e_{ρ}$ from the minus to the plus side, $g_{+} - g_{-} = - e_{3}$ , $e_{ρ} \times (g_{+} - g_{-}) = e_{φ}$ and $e_{ρ} \cdot (g_{+} - g_{-}) = 0$ . No product of distributions is formed.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $a > 0$ ; let $ψ \in C_{c}^{\infty} (R^{3}; R^{3})$ and $φ \in C_{c}^{\infty} (R^{3})$ ; let $S$ , $e_{ρ}$ , $e_{φ}$ , $e_{3}$ , $g$ and $T_{g}$ be as in the Statement.

[F1]

$C_{c}^{\infty} (R^{3})$ and its finite componentwise variant $C_{c}^{\infty} (R^{3}; R^{3})$ are the test spaces of Test function space d of an open set: their members are smooth and compactly supported, and every pairing below is against such a test.

[F2]

The regular distribution $T_{h}$ of a locally integrable function acts by $⟨ T_{h}, χ ⟩ = \int h χ$ and depends only on the almost-everywhere class of $h$ ; locally integrable functions embed into distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions).

[F3]

Distributional derivatives act by signed differentiation of the test, $⟨ \partial^{α} u, χ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} χ ⟩$ (Distributional derivative), and a vector distribution is a triple of scalar distributions in $D^{'} (R^{3})$ (Distribution).

[F4]

Under ${A C}_{ω}$ , Tonelli applies to the $σ$ -finite Euclidean Lebesgue products; $λ_{m} \times λ_{n}$ agrees with $λ_{m + n}$ on Borel sets for $m, n \geq 1$ , and every finite subset of $R$ is $λ_{1}$ -null (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ).

[F5]

Assuming ${A C}_{ω}$ , Lebesgue measure is finite on every bounded Lebesgue measurable set, so a bounded Borel function on $R^{3}$ is locally integrable (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure). Lebesgue measure is complete (Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume); hence arbitrary finite changes of a measurable function on a null set remain measurable, because each changed preimage differs by a subset of that null set.

[F6]

The cylinder $V = {ρ < a, ∣ x_{3} ∣ < R}$ , $R > a$ , has the finite piecewise $C^{1}$ presentation of Specified finite piecewise C1 boundary presentations given by the two planar disc caps and four lateral faces $X ([j π / 2, (j + 1) π / 2] \times [- R, R])$ , $j = 0, 1, 2, 3$ , where $X (θ, z) = (a \cos θ, a \sin θ, z)$ . Each angular interval lies inside an open interval of length less than $2 π$ , on which $X$ is an injective regular chart of Gram density $a$ . The edge set consists of the four vertical seams and the two rim circles. In a lateral chart these have parameter sets contained in finitely many horizontal or vertical lines, which are planar-null by [F4], so the edges are surface-null; they are null in each planar cap as well, by applying [F4] to the at-most-two-point sections of its rim circle. The actual outward normals are $e_{ρ}$ on the lateral faces and $\pm e_{3}$ on the caps.

[F7]

Under that presentation the divergence theorem of Divergence for finite piecewise C1 presentations holds: for every $C^{1}$ field $F$ on a neighborhood of the compact closure $\overline{V}$ , $\int_{V} div F d λ_{3} = \sum_{j} \int_{S_{j}} F \cdot ν_{j} d S$ , each face counted once off the edge set.

[F8]

For $u = (u_{x}, u_{y}, u_{z})$ and $v = (v_{x}, v_{y}, v_{z})$ one has $u \times v = (u_{y} v_{z} - u_{z} v_{y}, u_{z} v_{x} - u_{x} v_{z}, u_{x} v_{y} - u_{y} v_{x})$ ; in particular $A \times e_{3} = (A_{2}, - A_{1}, 0)$ , $e_{3} \times e_{ρ} = e_{φ}$ , $(ψ \times e_{3}) \cdot e_{ρ} = ψ \cdot e_{φ}$ , $e_{ρ} \times (- e_{3}) = e_{φ}$ and $e_{ρ} \cdot (- e_{3}) = 0$ (The cross product in $R^{3}$ ).

[F9]

On a $C^{1}$ field $F$ the operators are $div F = \partial_{1} F_{1} + \partial_{2} F_{2} + \partial_{3} F_{3}$ and $curl F = (\partial_{2} F_{3} - \partial_{3} F_{2}, \partial_{3} F_{1} - \partial_{1} F_{3}, \partial_{1} F_{2} - \partial_{2} F_{1})$ (Divergence and curl of a $C^{1}$ vector field).

[F10]

The surface integral over a compact Borel face contained in a regular patch of a compactly supported integrand is the chart-defined integral of Surface integration on compact C1 hypersurfaces, and it does not depend on the choice of a larger compact band containing the support of the integrand.

[F11]

${A C}_{ω}$ is the countable-choice principle invoked by the measure, surface and distribution interfaces above (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct; the nullity of the cylinder surface, the divergence theorem on the cylinder, and two coordinate computations

1.1givenF4

The cylinder $S = {x : x_{1}^{2} + x_{2}^{2} = a^{2}}$ is closed, hence Borel. For each $(x_{2}, x_{3}) \in R^{2}$ , its section $S^{(x_{2}, x_{3})} : = {x_{1} \in R : (x_{1}, x_{2}, x_{3}) \in S}$ consists of at most the two values $\pm \sqrt{a^{2} - x_{2}^{2}}$ when $∣ x_{2} ∣ \leq a$ , and is empty otherwise. Each section is $λ_{1}$ -null by [F4]. Tonelli and product-measure agreement [F4] give $λ_{3} (S) = \int_{R^{2}} λ_{1} (S^{(x_{2}, x_{3})}) d λ_{2} = 0$ .

2.1givenF1F2F3F5step 1.1

The bounded Borel function $g_{0} : = e_{3} 1_{{ρ < a}}$ agrees with $g$ off the null set $S$ of step 1.1, so completeness [F5] makes $g$ Lebesgue measurable with the same almost-everywhere class as $g_{0}$ ; by [F5] this class is locally integrable. Hence $T_{g} : = T_{g_{0}}$ is a well-defined distribution by [F2], and for the tests of [F1] the componentwise pairings of [F3] read $⟨ curl T_{g}, ψ ⟩ = \int_{R^{3}} g_{0} \cdot curl ψ d λ_{3}$ and $⟨ div T_{g}, φ ⟩ = - \int_{R^{3}} g_{0} \cdot \nabla φ d λ_{3}$ , where the componentwise sums against the vector tests are the ordinary finite sums of the stated integrals.

3.1givenstep 1.1step 2.1

Choose $R > a$ with $supp ψ \cup supp φ \subset W : = {ρ < R, ∣ x_{3} ∣ < R}$ and put $V : = {ρ < a, ∣ x_{3} ∣ < R}$ . Since $curl ψ$ and $\nabla φ$ are supported in $W$ , and $g_{0}$ equals $e_{3}$ on $V$ , equals $0$ on the open set $W ∖ \overline{V}$ , and the remaining part $W \cap \partial V$ lies in the null set $S$ of step 1.1, the two integrals of step 2.1 reduce to integrals over $V$ : $\int_{R^{3}} g_{0} \cdot curl ψ d λ_{3} = \int_{V} e_{3} \cdot curl ψ d λ_{3}$ and $\int_{R^{3}} g_{0} \cdot \nabla φ d λ_{3} = \int_{V} e_{3} \cdot \nabla φ d λ_{3}$ .

4.1step 2.1step 3.1F6F7F8F9

The field $F_{0} : = φ e_{3}$ is $C^{1}$ on $R^{3}$ , hence on a neighborhood of $\overline{V}$ , and by [F9] its divergence is $div F_{0} = \partial_{3} φ = e_{3} \cdot \nabla φ$ . The divergence theorem [F7] on the presentation [F6] gives $\int_{V} e_{3} \cdot \nabla φ d λ_{3} = \int_{\partial V} φ e_{3} \cdot ν d S$ . On the lateral face $ν = e_{ρ}$ and $e_{3} \cdot e_{ρ} = 0$ by [F8]; on the two disc caps $x_{3} = \pm R$ the test $φ$ vanishes because $supp φ \subset {∣ x_{3} ∣ < R}$ ; hence $\int_{V} e_{3} \cdot \nabla φ d λ_{3} = 0$ . With steps 2.1 and 3.1 this proves $⟨ div T_{g}, φ ⟩ = 0$ for every scalar test.

4.2step 2.1step 3.1F6F7F8F9F10

The field $G_{0} : = ψ \times e_{3}$ is $C^{1}$ on $R^{3}$ , and by [F8] its coordinates are $G_{0} = (ψ_{2}, - ψ_{1}, 0)$ , so by [F9] $div G_{0} = \partial_{1} ψ_{2} - \partial_{2} ψ_{1} = (curl ψ)_{3} = e_{3} \cdot curl ψ$ . The divergence theorem [F7] on the presentation [F6] gives $\int_{V} e_{3} \cdot curl ψ d λ_{3} = \int_{\partial V} G_{0} \cdot ν d S$ . The disc caps carry $ψ = 0$ by the support condition, while on the lateral face $ν = e_{ρ}$ and $G_{0} \cdot e_{ρ} = (ψ \times e_{3}) \cdot e_{ρ} = ψ \cdot e_{φ}$ by [F8]; therefore $\int_{V} e_{3} \cdot curl ψ d λ_{3} = \int_{ρ = a, ∣ x_{3} ∣ < R} e_{φ} \cdot ψ d S$ , and by [F10] the last integral equals $\int_{S} e_{φ} \cdot ψ d S$ because $e_{φ} \cdot ψ$ is compactly supported and vanishes off the compact band ${ρ = a, ∣ x_{3} ∣ \leq R}$ . With steps 2.1 and 3.1 this is $⟨ curl T_{g}, ψ ⟩ = \int_{S} e_{φ} \cdot ψ d S$ for every vector test, which is the claimed identification $curl T_{g} = e_{φ} δ_{S}$ .

5.1step 4.1step 4.2F2F4F8F10F11∎

The one-sided values of $g$ across $S$ are $g_{+} = 0$ and $g_{-} = e_{3}$ , so $g_{+} - g_{-} = - e_{3}$ and, by [F8], $e_{ρ} \times (g_{+} - g_{-}) = e_{ρ} \times (- e_{3}) = e_{φ}$ and $e_{ρ} \cdot (g_{+} - g_{-}) = e_{ρ} \cdot (- e_{3}) = 0$ , which is the stated equivalence. The two distributional identities of steps 4.1 and 4.2 involve only integrals of locally integrable functions against smooth tests, so no product of two distributions is formed anywhere; the only multiplication on $S$ is that of the compactly supported function $e_{φ} \cdot ψ$ with the surface measure of [F10]. Countable Choice is used in step 1.1 through the product-measure and finite-null-set suppliers [F4] and is inherited only through the cited measure, surface and distribution interfaces [F11]; no stronger choice principle occurs.

Source notes

Tong, Section 3.1.2, printed pp. 43-46, and Chew, Lecture 4 Section 4.3.3, printed pp. 46-48, present the current-sheet jump with exactly the pillbox bookkeeping carried out above at the distributional level; Griffiths, Section 5.3.3, printed pp. 226-228, computes the uniform-surface-current and infinite-solenoid fields whose interface content this lemma records as the identity $curl T_{g} = e_{φ} δ_{S}$ . The lemma itself is a pure distributional computation on the declared piecewise constant field and uses no physical assumption.

LemmaStatement: AI-adaptedProof: AI-adaptedOpen item page →

The azimuthal line field $ρ^{- 1} e_{φ}$ : local integrability and distributional curl on the axis

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $Z : = {(0, 0, x_{3}) : x_{3} \in R} \subset R^{3}$ be the third coordinate axis and let $f : R^{3} ∖ Z \to R^{3}, f (x) = \frac{(- x_{2}, x_{1}, 0)}{x_{1}^{2} + x_{2}^{2}}, x = (x_{1}, x_{2}, x_{3}),$ so that, in cylindrical notation $ρ = \sqrt{x_{1}^{2} + x_{2}^{2}}$ and $e_{φ} = ρ^{- 1} (- x_{2}, x_{1}, 0)$ , one has $f = e_{φ} / ρ$ on the punctured space. Let $T_{f}$ be the regular distribution of $f$ on $R^{3}$ (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions) and let $A$ be the $R^{3}$ -valued distribution $A (ψ) : = \int_{R} ψ_{3} (0, 0, x_{3}) d x_{3}, ψ \in C_{c}^{\infty} (R^{3}; R^{3}),$ the third-direction line measure on $Z$ . For an $R^{3}$ -valued distribution $T = (T_{1}, T_{2}, T_{3})$ whose components are scalar distributions of Distribution, the pairings used below are the componentwise readings of Distributional derivative: $⟨ div T, φ ⟩ : = - \sum_{i = 1}^{3} ⟨ T_{i}, \partial_{i} φ ⟩, ⟨ curl T, ψ ⟩ : = \sum_{i = 1}^{3} ⟨ T_{i}, (curl ψ)_{i} ⟩ .$ Then: (a) $f \in L_{l o c}^{1} (R^{3})$ ; (b) $div f = 0$ and $curl f = 0$ pointwise on $R^{3} ∖ Z$ ; (c) for every vector test $ψ \in C_{c}^{\infty} (R^{3}; R^{3})$ and every scalar test $φ \in C_{c}^{\infty} (R^{3})$ , $⟨ curl T_{f}, ψ ⟩ = 2 π A (ψ), ⟨ div T_{f}, φ ⟩ = 0,$ that is, $curl T_{f} = 2 π e_{3} δ_{Z}$ and $div T_{f} = 0$ in the distributional sense of Distributional derivative; and (d) for every $r > 0$ and $h \in R$ , the circulation of $f$ along the positively oriented circle $γ_{r, h} : [0, 2 π] \to R^{3} ∖ Z$ , $γ_{r, h} (t) = (r \cos t, r \sin t, h)$ , traversed once with increasing $t$ , is $2 π$ (Scalar line integrals with respect to arc length and vector-field line integrals, read componentwise). No product of distributions is formed.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $Z$ , $f$ , $e_{φ}$ , $ρ$ be as in the Statement; let $ψ \in C_{c}^{\infty} (R^{3}; R^{3})$ and $φ \in C_{c}^{\infty} (R^{3})$ .

[F1]

$C_{c}^{\infty} (R^{3})$ and $C_{c}^{\infty} (R^{3}; R^{3})$ are the test spaces of Test function space d of an open set, and $T_{f}$ is the regular distribution of the almost-everywhere class of $f$ , acting by integration and depending only on that class (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions).

[F2]

Distributional derivatives act by signed test differentiation, $⟨ \partial^{α} u, χ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} χ ⟩$ (Distributional derivative), and a vector distribution is a triple of scalar distributions in $D^{'} (R^{3})$ (Distribution); integrals over measurable subsets are those of Integral over a measurable subset.

[F3]

Under ${A C}_{ω}$ , Tonelli's theorem gives $\int_{X \times Y} h d (μ \times ν) = \int_{X} \int_{Y} h d ν d μ$ for nonnegative measurable $h$ on a $σ$ -finite product, and $λ_{m + n}$ agrees with $λ_{m} \times λ_{n}$ on Borel subsets of $R^{m + n} = R^{m} \times R^{n}$ for $m, n \geq 1$ (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}).

[F4]

Under ${A C}_{ω}$ , the polar-coordinate formula holds for every nonnegative Borel function on $R^{n}$ , and the polar surface measure $σ$ on the unit sphere is a finite Borel measure (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F5]

Under ${A C}_{ω}$ , every continuous map $R^{n} \to R^{m}$ , $n, m \geq 1$ , is Borel measurable (Continuous functions on Euclidean spaces are Borel measurable), and every at most countable subset of $R^{m}$ , in particular every singleton, is Lebesgue null (Every at most countable subset of $R^{n}$ is Lebesgue null; in particular $λ_{1} (Q) = 0$ ). The Borel sigma-algebra is generated by the open sets (The Borel sigma-algebra of a topological space); a map $g : R^{n} \to R^{m}$ is Borel measurable when $g^{- 1} (B)$ is Borel for every Borel $B \subseteq R^{m}$ (Borel measurable and Lebesgue measurable functions on $R^{n}$ ). Measurability of the zero extension from the punctured domain is proved locally in step 2.1.

[F6]

On each coordinate line, a partial derivative is an ordinary one-variable derivative, so the sum, scalar-multiple, product and quotient rules apply pointwise to $C^{1}$ components; in particular $\partial_{j} (g / h) = (\partial_{j} g h - g \partial_{j} h) / h^{2}$ wherever $h \neq 0$ (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 $g (c) \neq 0$ ).

[F7]

Divergence and curl of $C^{1}$ fields are $div F = \partial_{1} F_{1} + \partial_{2} F_{2} + \partial_{3} F_{3}$ and $curl F = (\partial_{2} F_{3} - \partial_{3} F_{2}, \partial_{3} F_{1} - \partial_{1} F_{3}, \partial_{1} F_{2} - \partial_{2} F_{1})$ (Divergence and curl of a $C^{1}$ vector field).

[F8]

Write $e_{ρ} (x) = ρ^{- 1} (x_{1}, x_{2}, 0)$ on ${ρ > 0}$ and $e_{3} = (0, 0, 1)$ . For $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ one has $u \cdot (v \times w) = \det (u, v, w) = (u \times v) \cdot w$ , $e_{φ} \times e_{ρ} = - e_{3}$ , and on a circle of radius $r$ about $Z$ with parameter $t \mapsto (r \cos t, r \sin t, h)$ the velocity is $r e_{φ}$ (The cross product in $R^{3}$ ).

[F9]

The cylindrical annulus $V_{ε} = {ε < ρ < R, ∣ x_{3} ∣ < R}$ , $0 < ε < R$ , has the finite piecewise $C^{1}$ presentation of Specified finite piecewise C1 boundary presentations given by two planar cap annuli and four quarter-cylinder faces on each of $ρ = ε$ and $ρ = R$ . At radius $b \in {ε, R}$ the faces are $X_{b} ([j π / 2, (j + 1) π / 2] \times [- R, R])$ , $j = 0, 1, 2, 3$ , with $X_{b} (θ, z) = (b \cos θ, b \sin θ, z)$ , each restricted from an injective regular chart on a larger angular interval of length less than $2 π$ . The seams and rims are surface-null: in lateral coordinates they lie on finitely many horizontal or vertical parameter lines, null by [F3] and [F5], and a planar rim circle has at-most-two-point sections, giving nullity by the same facts. The outward normal on the inner cylinder is $- e_{ρ}$ . The divergence theorem of Divergence for finite piecewise C1 presentations applies to every $C^{1}$ field on a neighborhood of the compact closure.

[F10]

The surface integral over a compact Borel face inside a regular chart is the chart formula of Surface integration on compact C1 hypersurfaces: for the cylinder chart $X (θ, z) = (ε \cos θ, ε \sin θ, z)$ the Gram density is $ε$ .

[F11]

If $h_{ε} \to h$ pointwise on a set and $∣ h_{ε} ∣ \leq M$ for all $ε$ with $M$ integrable on a finite-measure region, then $\int h_{ε} \to \int h$ (Dominated convergence); a continuous function is uniformly continuous on a compact metric set (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous). The Lebesgue integral is linear on integrable functions (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[F12]

The circulation of a continuous field $F$ around a piecewise $C^{1}$ path $γ$ is the line integral $\int_{γ} F \cdot d r$ of Scalar line integrals with respect to arc length and vector-field line integrals.

[F13]

${A C}_{ω}$ is the countable-choice principle invoked by the measure and distribution interfaces above (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct; local integrability by polar coordinates, one excision limit, and the divergence theorem on a cylindrical annulus

1.1givenF3F5

The axis $Z = {(0, 0, x_{3}) : x_{3} \in R}$ is closed, hence Borel. By [F3] in the product $R^{2} \times R$ , $λ_{3} (Z) = \int_{R} λ_{2} ({(0, 0)}) d x_{3} = 0$ , because the singleton in $R^{2}$ is null by [F5].

1.2givenF6F7

On $R^{3} ∖ Z$ the components of $f$ are quotients of polynomials with nonvanishing denominator $x_{1}^{2} + x_{2}^{2}$ , so by [F6] the partial derivatives exist and satisfy $\partial_{1} f_{1} = \frac{2 x_{1} x_{2}}{ρ^{4}}$ , $\partial_{2} f_{2} = - \frac{2 x_{1} x_{2}}{ρ^{4}}$ , $\partial_{1} f_{2} = \frac{x_{2}^{2} - x_{1}^{2}}{ρ^{4}}$ , $\partial_{2} f_{1} = \frac{x_{2}^{2} - x_{1}^{2}}{ρ^{4}}$ and $\partial_{3} f = 0$ . Summing by [F7] gives $div f = \partial_{1} f_{1} + \partial_{2} f_{2} = 0$ , and the curl coordinates are $\partial_{1} f_{2} - \partial_{2} f_{1} = 0$ , $\partial_{3} f_{1} - \partial_{1} f_{3} = 0$ and $\partial_{2} f_{3} - \partial_{3} f_{2} = 0$ ; hence $div f = curl f = 0$ pointwise on $R^{3} ∖ Z$ .

1.3givenF8F12

Fix $r > 0$ and a height $h \in R$ , and let $γ (t) = (r \cos t, r \sin t, h)$ , $t \in [0, 2 π]$ , a closed $C^{1}$ path whose velocity is $γ^{'} (t) = r e_{φ} (γ (t))$ by [F8]. On the trace, $f (γ (t)) = r^{- 1} e_{φ} (γ (t))$ , so the integrand of the vector line integral of [F12] is the constant $f (γ (t)) \cdot γ^{'} (t) = r^{- 1} \cdot r = 1$ , and the circulation is $\int_{0}^{2 π} 1 d t = 2 π$ .

2.1givenstep 1.1F1F2F3F4F5

Let $U = R^{3} ∖ Z$ and define $\tilde{f} : R^{3} \to R^{3}$ by $\tilde{f} = f$ on $U$ and $\tilde{f} = 0$ on $Z$ . For every open $O \subseteq R^{3}$ , continuity of $f$ on $U$ makes $f^{- 1} (O)$ open in $U$ , hence open in $R^{3}$ because $U$ is open. The inverse image ${\tilde{f}}^{- 1} (O)$ is $f^{- 1} (O)$ if $0 \notin O$ , and $f^{- 1} (O) \cup Z$ if $0 \in O$ ; both are Borel, since $Z$ is closed. The class $C = {B \subseteq R^{3} : {\tilde{f}}^{- 1} (B) is Borel}$ is a sigma-algebra: inverse images preserve complements and countable unions. It contains every open set, hence every Borel set by [F5]. Thus $\tilde{f}$ is Borel measurable without applying the cited full-space continuity result to $f$ . For its local integrability, let $K \subset R^{3}$ be compact and choose $R > 0$ , $H > 0$ with $K \subset {∣ x_{3} ∣ \leq H, ρ \leq R}$ . By Tonelli [F3] and the polar formula [F4] in dimension two, applied to the nonnegative Borel function $u (x) = ρ^{- 1} 1_{{∣ x_{3} ∣ \leq H, ρ \leq R}} (x)$ off $Z$ , with $u = 0$ on $Z$ (the same open-inverse-image argument proves measurability of $ρ^{- 1}$ extended by zero, and restriction to this closed cylinder preserves it), $\int_{R^{3}} u d λ_{3} = \int_{- H}^{H} (\int_{ρ \leq R} ρ^{- 1} d λ_{2}) d x_{3} = \int_{- H}^{H} R σ (S^{1}) d x_{3} = 2 H R σ (S^{1})$ , and this is finite because $σ$ is a finite measure [F4]. Since $∣ \tilde{f} ∣ = ρ^{- 1}$ off $Z$ and $Z$ is null by step 1.1, $1_{K} ∣ \tilde{f} ∣ \leq u$ off a null set, so $\int_{K} ∣ \tilde{f} ∣ d λ_{3} < + \infty$ . Hence $f \in L_{l o c}^{1} (R^{3})$ and $T_{f}$ is a well-defined distribution by [F1], [F2].

3.1givenstep 1.1step 2.1F1F2F11

The regular-distribution pairings of [F1] and [F2] give $⟨ curl T_{f}, ψ ⟩ = \int_{R^{3}} f \cdot curl ψ d λ_{3}$ and $⟨ div T_{f}, φ ⟩ = - \int_{R^{3}} f \cdot \nabla φ d λ_{3}$ , the integrands being dominated by $∥ curl ψ ∥_{\infty} ρ^{- 1} 1_{supp ψ}$ and $∥ \nabla φ ∥_{\infty} ρ^{- 1} 1_{supp φ}$ , which are integrable by step 2.1. Fix $R > 0$ with $supp ψ \cup supp φ \subset {ρ < R, ∣ x_{3} ∣ < R}$ and put $V_{ε} : = {ε < ρ < R, ∣ x_{3} ∣ < R}$ for $0 < ε < R$ . The functions $1_{V_{ε}} f \cdot curl ψ$ are dominated by $∣ f \cdot curl ψ ∣$ and converge pointwise off the null axis $Z$ of step 1.1 to $f \cdot curl ψ$ ; by [F11], $\int_{R^{3}} f \cdot curl ψ d λ_{3} = \lim_{ε \to 0} \int_{V_{ε}} f \cdot curl ψ d λ_{3}$ , and the same argument gives $\int_{R^{3}} f \cdot \nabla φ d λ_{3} = \lim_{ε \to 0} \int_{V_{ε}} f \cdot \nabla φ d λ_{3}$ .

4.1givenstep 1.2step 3.1F7F8F9F10

On the neighborhood ${ρ > ε / 2}$ of the compact closure $\overline{V_{ε}}$ the field $ψ \times f$ is $C^{1}$ , since $f$ is smooth there, and by [F7] its divergence is $div (ψ \times f) = f \cdot curl ψ - ψ \cdot curl f$ ; here $curl f = 0$ on $V_{ε}$ by step 1.2. Applying the divergence theorem [F9] on $V_{ε}$ gives $\int_{V_{ε}} f \cdot curl ψ d λ_{3} = \int_{\partial V_{ε}} (ψ \times f) \cdot ν d S$ . The faces at $ρ = R$ and at $∣ x_{3} ∣ = R$ contribute $0$ because $supp ψ \subset {ρ < R, ∣ x_{3} ∣ < R}$ ; on the inner cylinder the outward normal is $- e_{ρ}$ and $f = e_{φ} / ε$ , so by [F8] the remaining term is $(ψ \times e_{φ}) \cdot (- e_{ρ}) / ε = - ψ \cdot (e_{φ} \times e_{ρ}) / ε = ψ_{3} / ε$ , and the surface integral of [F10] over the chart $X (θ, z) = (ε \cos θ, ε \sin θ, z)$ gives $\int_{ρ = ε, ∣ x_{3} ∣ < R} \frac{ψ_{3}}{ε} d S = \int_{0}^{2 π} \int_{- R}^{R} ψ_{3} (ε \cos θ, ε \sin θ, z) d z d θ .$

4.2givenstep 1.2step 3.1F7F9

On the same neighborhood the field $φ f$ is $C^{1}$ , and $div (φ f) = f \cdot \nabla φ + φ div f$ by [F7]; here $div f = 0$ on $V_{ε}$ by step 1.2. The divergence theorem [F9] gives $\int_{V_{ε}} f \cdot \nabla φ d λ_{3} = \int_{\partial V_{ε}} φ f \cdot ν d S$ . On both cylinder faces $f \cdot ν = 0$ because $f ⊥ e_{ρ}$ , and on the two cap annuli $φ = 0$ because $supp φ \subset {∣ x_{3} ∣ < R}$ ; hence $\int_{V_{ε}} f \cdot \nabla φ d λ_{3} = 0$ for every $ε$ .

5.1step 3.1step 4.2

With step 4.2, the identity $\int_{R^{3}} f \cdot \nabla φ d λ_{3} = \lim_{ε \to 0} \int_{V_{ε}} f \cdot \nabla φ d λ_{3}$ of step 3.1 gives $\int_{R^{3}} f \cdot \nabla φ d λ_{3} = 0$ , and therefore $⟨ div T_{f}, φ ⟩ = 0$ for every scalar test $φ$ by step 3.1.

5.2step 4.1F11

The double integral of step 4.1 has the limit $\int_{0}^{2 π} \int_{- R}^{R} ψ_{3} (0, 0, z) d z d θ = 2 π A (ψ)$ as $ε \to 0$ . Indeed, $ψ_{3}$ is continuous on $R^{3}$ , hence uniformly continuous on the compact set ${x : ρ \leq R, ∣ x_{3} ∣ \leq R}$ by [F11], so for $η > 0$ there is $δ > 0$ with $∣ ψ_{3} (ε \cos θ, ε \sin θ, z) - ψ_{3} (0, 0, z) ∣ \leq η$ whenever $0 < ε < δ$ and $(θ, z) \in [0, 2 π] \times [- R, R]$ ; dominated convergence [F11] on the finite-measure rectangle, or equivalently the estimate with the constant majorant $η$ , gives convergence of the double integrals; the limit integrand does not depend on $θ$ , and finite linearity [F11] gives the iterated integral $(\int_{0}^{2 π} 1 d θ) (\int_{- R}^{R} ψ_{3} (0, 0, z) d z) = 2 π A (ψ)$ .

6.1step 3.1step 4.1step 5.2

Combining steps 3.1, 4.1 and 5.2, $⟨ curl T_{f}, ψ ⟩ = \int_{R^{3}} f \cdot curl ψ d λ_{3} = \lim_{ε \to 0} \int_{V_{ε}} f \cdot curl ψ d λ_{3} = \lim_{ε \to 0} \int_{0}^{2 π} \int_{- R}^{R} ψ_{3} (ε \cos θ, ε \sin θ, z) d z d θ = 2 π A (ψ)$ for every vector test $ψ$ , which is the claimed identification $curl T_{f} = 2 π e_{3} δ_{Z}$ .

7.1step 1.2step 1.3step 2.1step 5.1step 6.1F2F13∎

Steps 5.1 and 6.1 prove the two distributional identities of (c), step 1.2 proves the pointwise vanishing (b), and step 1.3 proves the circulation (d); step 2.1 is (a). Every pairing above is an integral of a locally integrable function against a smooth test, so no product of two distributions is formed. Countable Choice is used only through the measure and distribution interfaces [F13], namely Tonelli, polar coordinates, the nullity of countable sets and the regular-distribution embedding; no stronger choice principle occurs.

Source notes

Tong, Section 3.1.1, printed pp. 42-43, and Griffiths, Section 5.3.1, printed pp. 221-222, supply the wire field and its circulation; the distributional identity $curl T_{f} = 2 π e_{3} δ_{Z}$ is the exact mathematical content behind the distributional divergence computation of Griffiths, Section 1.5.1, printed pp. 45-46, and the sheet bookkeeping of Chew, Lecture 4 Section 4.3.3, printed pp. 46-48. The lemma is a pure computation on the declared explicit field.

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

Divergence and curl of the Newtonian potential of a smooth compact vector field

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $f \in C_{c}^{\infty} (R^{3}; R^{3})$ and define componentwise $A_{i} (x) : = \frac{1}{4 π} \int_{R^{3}} \frac{f_{i} (y)}{∣ x - y ∣} d y, A : = (A_{1}, A_{2}, A_{3}), x \in R^{3},$ so that $A = N (f)$ componentwise with the Newtonian-potential transform $N$ of Newtonian potential of smooth compact data is smooth and solves the Poisson equation pointwise and Newtonian potential of compactly supported data. Then $A \in C^{\infty} (R^{3}; R^{3})$ , both integrals below are absolutely convergent for every $x$ , and for every $x \in R^{3}$ $div A (x) = N (div f) (x) = \frac{1}{4 π} \int_{R^{3}} \frac{(div f) (y)}{∣ x - y ∣} d y,$ $curl A (x) = \frac{1}{4 π} \int_{R^{3}} \frac{f (y) \times (x - y)}{∣ x - y ∣^{3}} d y .$ The curl identity needs no divergence-free hypothesis and holds pointwise at every point of $R^{3}$ , including points of the support of $f$ ; the kernel is differentiated, its smoothness at the pole is not assumed. Countable Choice is inherited from the smooth Newtonian supplier and the polar-coordinate and surface-measure interfaces of the estimates.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , let $f \in C_{c}^{\infty} (R^{3}; R^{3})$ , and let $A = N (f)$ componentwise be as in the Statement.

[F1]

In dimension three the fundamental solution of $- Δ$ is $Φ (z) = 1 / (4 π ∣ z ∣)$ for $z \neq 0$ , with an arbitrary finite value at the pole, and the Newtonian potential of compactly supported data is $N h (x) = \int_{R^{3}} Φ (x - y) h (y) d y$ at every point where the integral is absolutely finite (Fundamental solution for the positive operator minus Laplacian, Newtonian potential of compactly supported data).

[F2]

For $h \in C_{c}^{\infty} (R^{3}; R)$ the potential satisfies $N h \in C^{\infty} (R^{3})$ , $- Δ N h = h$ pointwise, and $D^{α} N h = N (D^{α} h)$ for every multi-index $α$ (Newtonian potential of smooth compact data is smooth and solves the Poisson equation pointwise).

[F3]

Partial derivatives on open subsets of $R^{3}$ are the ordinary one-variable derivatives along coordinate lines, and $div F = \partial_{1} F_{1} + \partial_{2} F_{2} + \partial_{3} F_{3}$ , $curl F = (\partial_{2} F_{3} - \partial_{3} F_{2}, \partial_{3} F_{1} - \partial_{1} F_{3}, \partial_{1} F_{2} - \partial_{2} F_{1})$ for $C^{1}$ fields; a map is $C^{k}$ when all components are (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Divergence and curl of a $C^{1}$ vector field, $C^{k}$ Euclidean maps and diffeomorphisms).

[F4]

On each coordinate line the ordinary sum and product rules hold, so for smooth $g, h$ one has $\partial_{j} (g h) = (\partial_{j} g) h + g \partial_{j} h$ pointwise (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 $g (c) \neq 0$ ).

[F5]

The spherical shell $Ω_{ε} = B_{R} (x) ∖ \overline{B_{ε} (x)}$ , $0 < ε < R$ , carries the specified finite piecewise $C^{1}$ presentation of Specified finite piecewise C1 boundary presentations given by coordinate-maximal faces of its two sphere boundary components, with outward normals $ν_{R} = (y - x) / ∣ y - x ∣$ on the outer sphere and $ν_{ε} = - (y - x) / ∣ y - x ∣$ on the inner sphere; the divergence theorem of Divergence for finite piecewise C1 presentations then holds for every $C^{1}$ field on a neighborhood of the compact closure.

[F6]

The surface integral is that of Surface integration on compact C1 hypersurfaces; under ${A C}_{ω}$ the sphere $\partial B_{ε} (x)$ has surface measure $ε^{2} ∣ S^{2} ∣$ , and $∣ S^{2} ∣ = 3 ∣ B_{1} ∣$ is finite (Agreement with the existing polar sphere measure).

[F7]

The polar formula gives $\int_{B_{R} (x)} ∣ x - y ∣^{- 1} d λ_{3} = \frac{R^{2}}{2} ∣ S^{2} ∣$ and $\int_{B_{R} (x)} ∣ x - y ∣^{- 2} d λ_{3} = R ∣ S^{2} ∣$ , both finite because the polar surface measure is finite (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F8]

The integral triangle inequality $∣ \int g ∣ \leq \int ∣ g ∣$ , monotonicity and positive homogeneity of the nonnegative integral, and finite additivity of the integral on integrable functions hold (The modulus of an integral is bounded by the integral of the modulus, Monotonicity and nonnegative homogeneity of the nonnegative integral, The Lebesgue integral is linear on $L^{1} (μ)$ , Integrable real and complex functions, and their integrals, Integral over a measurable subset).

[F9]

Dominated convergence: if $g_{ε} \to g$ pointwise and $∣ g_{ε} ∣ \leq G$ with $G$ integrable on the region, then $\int g_{ε} \to \int g$ (Dominated convergence).

[F10]

For $u, v, w \in R^{3}$ , $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ and $(u \times v)_{i} = ε_{i j k} u_{j} v_{k}$ , so $- ε_{i j k} a_{j} b_{k} = (b \times a)_{i}$ ; the Euclidean inner product is $⟨ u, v ⟩ = \sum_{i} u_{i} v_{i}$ (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F11]

${A C}_{ω}$ is the countable-choice principle invoked by the polar and surface-measure interfaces above (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · the componentwise derivative identity of the smooth Newtonian upgrade, then a finite excision (spherical shell) integration by parts and one dominated limit

1.1givenF1F2F3F7F8

Since each $f_{i}$ is smooth and compactly supported, [F2] gives $A_{i} = N (f_{i}) \in C^{\infty} (R^{3})$ with $D^{α} A_{i} = N (D^{α} f_{i})$ for every multi-index $α$ ; in particular $A \in C^{\infty} (R^{3}; R^{3})$ by [F3], and $\partial_{i} A_{i} = N (\partial_{i} f_{i})$ for each $i$ . Summing over $i$ and using finite additivity of the integral [F8], $div A = \sum_{i} \partial_{i} A_{i} = N (\sum_{i} \partial_{i} f_{i}) = N (div f)$ pointwise on $R^{3}$ , which is the first displayed identity; its integral is absolutely convergent because $∣ div f ∣ \leq ∥ div f ∥_{\infty}$ is bounded and compactly supported and $\int_{B_{R} (x)} ∣ x - y ∣^{- 1} d λ_{3} < + \infty$ by [F7].

1.2givenF4F5F6F8

Fix $x \in R^{3}$ and $j, k \in {1, 2, 3}$ , let $Φ (z) = 1 / (4 π ∣ z ∣)$ , and choose $R > 0$ with $supp f \subset B_{R} (x)$ . For $0 < ε < R$ let $Ω_{ε} : = B_{R} (x) ∖ \overline{B_{ε} (x)}$ , presented as in [F5]. On a neighborhood of the compact closure the scalar product $y \mapsto f_{k} (y) Φ (x - y)$ is $C^{1}$ , and by [F4] its partial derivative is $\partial_{y_{j}} (f_{k} Φ (x - y)) = (\partial_{j} f_{k}) (y) Φ (x - y) + f_{k} (y) \partial_{y_{j}} Φ (x - y)$ . Applying the divergence theorem [F5] to the field $f_{k} Φ (x - \cdot) e_{j}$ on $Ω_{ε}$ gives $\int_{Ω_{ε}} (\partial_{j} f_{k}) Φ (x - y) d λ_{3} (y) = \int_{\partial B_{R} (x)} f_{k} Φ (x - y) (ν_{R})_{j} d S + \int_{\partial B_{ε} (x)} f_{k} Φ (x - y) (ν_{ε})_{j} d S - \int_{Ω_{ε}} f_{k} (y) \partial_{y_{j}} Φ (x - y) d λ_{3} (y)$ . The outer sphere contributes $0$ because $f = 0$ there; the inner sphere term is bounded in absolute value, by [F8] and the bound $∣ Φ ∣ = 1 / (4 π ∣ y - x ∣) = 1 / (4 π ε)$ there, by $∥ f_{k} ∥_{\infty} \frac{1}{4 π ε} ∣ \partial B_{ε} (x) ∣ = ∥ f_{k} ∥_{\infty} \frac{∣ S^{2} ∣}{4 π} ε \to 0$ as $ε \to 0$ by [F6].

2.1step 1.1step 1.2F4F7F9

Let $ε \to 0$ in the identity of step 1.2. The shell integrands $(\partial_{j} f_{k}) Φ (x - y) 1_{Ω_{ε}}$ are dominated on $B_{R} (x)$ by $∥ \partial_{j} f_{k} ∥_{\infty} ∣ Φ ∣$ , and the shell integrands $f_{k} \partial_{y_{j}} Φ (x - y) 1_{Ω_{ε}}$ by $∥ f_{k} ∥_{\infty} ∣ \partial_{y_{j}} Φ ∣ \leq ∥ f_{k} ∥_{\infty} \frac{1}{4 π} ∣ x - y ∣^{- 2}$ ; both majorants are integrable on $B_{R} (x)$ by [F7], and both families converge pointwise off the null singleton ${x}$ to the corresponding unrestricted integrands. Dominated convergence [F9] therefore gives $\int_{B_{R} (x)} (\partial_{j} f_{k}) Φ (x - y) d λ_{3} (y) = - \int_{B_{R} (x)} f_{k} (y) \partial_{y_{j}} Φ (x - y) d λ_{3} (y)$ ; the left side is $N (\partial_{j} f_{k}) (x) = \partial_{j} A_{k} (x)$ by step 1.1, and since $\partial_{y_{j}} Φ (x - y) = \frac{x_{j} - y_{j}}{4 π ∣ x - y ∣^{3}}$ for $y \neq x$ , this reads $\partial_{j} A_{k} (x) = - \frac{1}{4 π} \int_{R^{3}} \frac{f_{k} (y) (x_{j} - y_{j})}{∣ x - y ∣^{3}} d y$ , the integral being absolutely convergent because $∣ f_{k} (y) (x_{j} - y_{j}) ∣ \leq ∥ f_{k} ∥_{\infty} ∣ x - y ∣$ on $B_{R} (x)$ and $\int_{B_{R} (x)} ∣ x - y ∣^{- 2} d λ_{3} < + \infty$ by [F7].

3.1step 1.1step 2.1F7F8F10F11∎

Summing the identity of step 2.1 with the alternating symbol, $(curl A)_{i} = ε_{i j k} \partial_{j} A_{k} = - \frac{1}{4 π} \int_{R^{3}} \frac{ε_{i j k} f_{k} (y) (x_{j} - y_{j})}{∣ x - y ∣^{3}} d y = \frac{1}{4 π} \int_{R^{3}} \frac{(f (y) \times (x - y))_{i}}{∣ x - y ∣^{3}} d y$ by the component identity of [F10] and finite linearity of the integral [F8]; as this holds for each $i$ , $curl A (x) = \frac{1}{4 π} \int_{R^{3}} \frac{f (y) \times (x - y)}{∣ x - y ∣^{3}} d y$ , and the integral is absolutely convergent because $∣ f (y) \times (x - y) ∣ \leq ∥ f ∥_{\infty} ∣ x - y ∣$ and $\int_{B_{R} (x)} ∣ x - y ∣^{- 2} < + \infty$ by [F7]. With step 1.1 this proves both displayed identities at the arbitrary point $x$ ; the point was allowed to lie in $supp f$ , since the excision argument only removed a ball around it and used the bound on the differentiated kernel, so no smoothness of the kernel at the pole is assumed. Countable Choice is used only through the polar and surface-measure interfaces [F11] and the stated choice assumption of the smooth-newtonian supplier, and no stronger principle occurs.

Source notes

Griffiths, Section 5.3.2, printed pp. 222-225, carries out exactly this product-rule-and-integration-by-parts computation with $\nabla \to - \nabla^{'}$ ; the excision above replaces the formal treatment of the singular kernel by a rigorous shell argument, and step 2.1 records that the differentiated kernel, not the kernel itself, is what the estimates require. Chew, Lecture 4 Section 4.1, printed pp. 39-41, and Tong, Section 3.2.3, printed pp. 49-52, present the same Newtonian construction of $A$ and the differentiated-kernel computation of $B = curl A$ .

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models

Physical scope

A model restriction inside the classical electromagnetic framework: the steady-current regime is the time-independent reduction of the adopted Maxwell equations in one inertial frame, and the compact smooth current class and the three declared ideal models are hypotheses, not measurements or empirical findings. The ideal noncompact models carry declared symmetry and decay data and have no finite-energy or compact-support property; no experiment, apparatus, uncertainty or statistical claim is involved, and no claim is made that any real current is exactly steady, compactly supported or divergence-free.

Definition

For the Lebesgue/surface/distributional source and admissibility branches (c)-(d), assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The smooth formulation in (a) and the smooth conserved-current predicate in (b) do not themselves use Choice. Let the laboratory region and its chart, the field and source objects and the units be those 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, Charge parameter, frame charge and current densities, volume charge and flux, and the four-current and Framework-local SI units, the vacuum constants and the exact/measured status of their values, and fix $μ_{0} > 0$ , $c > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ .

(a) [Steady-current regime]. A steady-current (magnetostatic) configuration on $Ω \times I$ consists of a field pair $E, B : Ω \times I \to R^{3}$ and source fields $ρ : Ω \times I \to R$ , $J : Ω \times I \to R^{3}$ that are independent of $t$ and satisfy the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches with the operators of Divergence and curl of a $C^{1}$ vector field and Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ . For such a configuration the formal distributional branch of Charge continuity as a compatibility condition of the vacuum Maxwell equations gives $div J = 0$ distributionally; this is pointwise when $J$ is $C^{1}$ . In the jointly $C^{2}$ field/ $C^{1}$ source branch its classical proof applies directly. The locally integrable distributional reading here inherits Countable Choice, whereas the stated classical $C^{2} / C^{1}$ compatibility calculation is choice-free, and the four adopted equations are equivalent to the electrostatic pair and the magnetostatic pair $div E = ρ / ε_{0}, curl E = 0, div B = 0, curl B = μ_{0} J .$

(b) [Compact smooth current class]. The compact smooth current class of this framework consists of the currents $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero to $R^{3}$ . A magnetostatic field of such a current is a field $B$ of class $C^{1}$ on $R^{3}$ satisfying $div B = 0$ and $curl B = μ_{0} J$ on $R^{3}$ with the extended $J$ . This is the hypothesis class of the vector-potential, Biot-Savart and dipole theorems of this page; this definition asserts neither existence nor uniqueness of $B$ in the class.

(c) [Ideal source objects]. Define three ideal current source objects outside the compact class, together with the data selecting their fields. Curve parameters $s$ and $L$ and the radius $a$ carry metres; the signed line-current amplitude $I$ carries amperes and the signed sheet-current amplitude $K$ carries A m $^{- 1}$ . The field $B$ carries tesla, as in Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(c1) [Closed thin loop]. For a closed embedded $C^{2}$ curve $γ : R / L Z \to R^{3}$ with $∣ γ^{'} ∣ \equiv 1$ ( $L > 0$ ) and $I \in R$ , the ideal loop current is the distribution $⟨ J_{γ}, ψ ⟩ : = I \int_{0}^{L} γ^{'} (s) \cdot ψ (γ (s)) d s, ψ \in C_{c}^{\infty} (R^{3}; R^{3}),$ with $ρ_{γ} : = 0$ .

(c2) [Infinite straight wire]. For $γ (s) = (0, 0, s)$ , $s \in R$ , and $I \in R$ , the ideal wire current is $⟨ J_{w}, ψ ⟩ : = I \int_{R} ψ_{3} (0, 0, s) d s,$ with $ρ_{w} : = 0$ .

(c3) [Infinite cylindrical sheet]. For $a > 0$ , $S = {ρ = a}$ , $ρ = \sqrt{x_{1}^{2} + x_{2}^{2}}$ , $e_{ρ} : = ρ^{- 1} (x_{1}, x_{2}, 0)$ and $e_{φ} : = ρ^{- 1} (- x_{2}, x_{1}, 0)$ on ${ρ > 0}$ , and $K \in R$ , the ideal solenoid sheet current is $⟨ J_{S}, ψ ⟩ : = K \int_{S} e_{φ} \cdot ψ d S$ (Surface integration on compact C1 hypersurfaces, read on compact cylinder segments), with $ρ_{S} : = 0$ . All three pairings are finite because $ψ$ has compact support.

(c4) [Admissibility]. A field $B$ is admitted for one of these declared models when: $B$ is locally integrable on $R^{3}$ (hence a distribution by Regular distribution from a locally integrable function, Locally integrable functions embed in distributions) and $C^{1}$ off the support of the current; $div B = 0$ and $curl B = 0$ hold pointwise off that support; the distributional identities $div B = 0$ and $curl B = μ_{0} J$ hold against every scalar divergence test $φ \in C_{c}^{\infty} (R^{3})$ and vector curl test $ψ \in C_{c}^{\infty} (R^{3}; R^{3})$ in the sense of Distributional derivative; $B$ has the model's declared symmetry: in (c2) and (c3), $B (x + h e_{3}) = B (x)$ for every $h \in R$ , and $B (R_{θ} x) = R_{θ} B (x)$ for every $θ \in R$ , at off-source points, where $R_{θ} (x_{1}, x_{2}, x_{3}) = (x_{1} \cos θ - x_{2} \sin θ, x_{1} \sin θ + x_{2} \cos θ, x_{3})$ ; no symmetry is required in (c1); and $B$ obeys the decay requirement: $∣ B (x) ∣ \to 0$ as $∣ x ∣ \to \infty$ in (c1), and in (c2), (c3), $\sup_{(θ, z) \in C} ∣ B (r \cos θ, r \sin θ, z) ∣ \to 0$ as $r \to \infty$ for every nonempty compact $C \subset R^{2}$ . The companion examples exhibit and verify the fields of the two axisymmetric models in this class; the definition itself asserts no existence, uniqueness, finite energy or limiting relation for the declared models, and no compact-class theorem of this page applies to them.

(d) [Scope]. This is a model restriction: the steady-current regime for compact smooth currents, and three explicitly declared ideal noncompact or singular models. It excludes point charges, weaker-regularity volume sources, moving sheets, normal sheet currents and any claim of empirical exactness; the equations in (c4) define a mathematical admissibility predicate; its physical adoption for these source objects is a separate postulate, and no product of distributions is formed.

Remarks

Well-definedness of the clauses. In (c1)-(c3) the pairings are order-zero continuous linear functionals on the test spaces of Test function space d of an open set: on a fixed compact test support their absolute values are bounded by $∣ I ∣ L ∥ ψ ∥_{\infty}$ , by $∣ I ∣$ times the finite length of the axis projection times $∥ ψ ∥_{\infty}$ , and by $∣ K ∣$ times the finite area of a containing compact cylindrical face times $∥ ψ ∥_{\infty}$ , respectively. The cylinder chart $X (θ, z) = (a \cos θ, a \sin θ, z)$ has Gram density $a$ , and finitely many angular charts cover each compact face, so the surface pairing is the chart-defined integral of Surface integration on compact C1 hypersurfaces; the scalar triple product and the direction $e_{φ}$ are those of The cross product in $R^{3}$ . The divergence pairing uses scalar tests and the curl pairing vector tests under Distributional derivative, a vector distribution being a triple of scalar distributions in $D^{'} (R^{3})$ of Distribution; a locally integrable field is a distribution in the sense of the regular-distribution embedding Regular distribution from a locally integrable function, Locally integrable functions embed in distributions. Smoothness statements use $C^{k}$ Euclidean maps and diffeomorphisms. The finite area of a compact angular face is the Riemann integral of the constant function one over its rectangular chart domain, with the properties of Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ . The declared ideal objects lie outside the source classes of Distributional Maxwell equations with point and stationary-sheet sources: that definition excludes line and noncompact sheet currents, so no equation for these singular objects is inherited from it, and Countable Choice is inherited only through the compact-face surface and regular-distribution interfaces.

Source notes

Griffiths, Section 5.2.1, printed pp. 215-216, and the opening of Tong, Chapter 3, printed pp. 41-42, describe steady currents and the reduction to $\nabla \cdot J = 0$ and the static field equations; Griffiths, Section 5.1.3, printed pp. 208-214, supplies the line, surface and volume current-density conventions reproduced in (c). Landau-Lifshitz, Section 43, printed pp. 101-103, and Chew, Lecture 4 Section 4.1, printed pp. 39-41, give the constant-field equations and the vector-potential formulation in the same regime. The ideal noncompact models are declared modelling hypotheses of this framework, not measurements or exact descriptions of any apparatus.

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Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models

Physical scope

Additional classical physical adoption in the named inertial laboratory for the exact ideal closed loop, infinite straight line and infinite cylindrical sheet models, with $E = 0$ and $ρ = 0$ . No experiment, matter dynamics or self-force.

Postulate

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). In the named inertial laboratory model of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the positive constants and SI conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values, adopt the following additional classical physical idealization: a loop, infinite straight line or infinite cylindrical sheet source defined by The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1)-(c3), with zero charge density, is modeled by a time-independent $B$ satisfying exactly the admissibility predicate (c4), together with $E = 0$ and $ρ = 0$ . Thus $div B = 0$ and $curl B = μ_{0} J$ hold distributionally on $R^{3}$ , with the stated off-source regularity, symmetry and decay; the electrostatic and temporal Maxwell identities are zero. This physical adoption extends to these explicitly declared source types only. It is separate from the earlier $L^{1}$ -source and point/compact-sheet postulates, which do not include these line and noncompact sheet sources. No existence, uniqueness, finite self-energy, force product, self-force, apparatus accuracy, matter dynamics or empirical certainty is asserted.

Status and scope

This is an adopted physical assumption, not a theorem: it states which ideal models this framework elects to use, and it is not deduced from the vacuum equations or from any experiment. The mathematical content it refers to — the exact source functionals of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1)-(c3) and the admissibility predicate (c4) — is fixed by that definition; the postulate adds the physical decision to treat those declared objects as sources of fields in the named laboratory. Existence and distributional consistency of the fields for the line and sheet models are established separately by the mathematical lemmas of this page, and the physical reading of the compact thin-loop limit remains a declared idealization rather than a measured property of any coil. The assumption carries no empirical uncertainty record because it reports no measurement; any later empirical comparison must state its own preparation, apparatus and sampling assumptions, and agreement would support the model under those conditions rather than establish it.

Source notes

Tong, Chapter 3, printed pp. 41-46 and 51, adopts neutral steady currents, the infinite wire, the infinite cylindrical solenoid and the thin-wire model as idealizations of a long straight conductor and a tightly wound solenoid. Those bounded passages are the source context for the adopted models. The exact source functionals, the admissibility predicate and the field objects are this framework's own precise formulation, not claims reproduced verbatim from the source; no source is claimed to prove the idealization exact for any apparatus.

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The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop

Physical scope

The magnetic moment defined here is a source moment of two declared current models (the compact smooth conserved current and the ideal closed thin loop). It is not a rest mass, not an intrinsic spin and not an empirically measured quantity; the definition asserts no far-field property, no equivalence between the two clauses and no moment for any other source. No measurement, apparatus or uncertainty is involved.

Definition

Let the compact smooth current class and the declared ideal models be those of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models, with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Compact smooth currents]. Let $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero to $R^{3}$ . The magnetic dipole moment of $J$ is the vector $m : = \frac{1}{2} \int_{R^{3}} y \times J (y) d y \in R^{3},$ of unit A m $^{2}$ . The integral is computed componentwise over any nondegenerate closed bounding rectangle $Q$ containing $supp J$ , using the continuous functions $y \mapsto (y \times J (y))_{i}$ on $Q$ and their zero extension outside $Q$ . Its value is independent of the chosen larger bounding rectangle by finite rectangular subdivision and vanishing of the integrand off its support; the support itself need not be Jordan measurable. Thus this is a Riemann integral of continuous compactly supported functions (The Riemann integral of a bounded function over a bounded Jordan measurable set, A continuous real function on a compact Jordan measurable set is Riemann integrable over that set, The cross product in $R^{3}$ ); it is absolutely finite because $J$ is bounded with compact support.

(b) [Declared ideal thin loop]. For a closed embedded $C^{2}$ curve $γ : R / L Z \to R^{3}$ with $∣ γ^{'} ∣ \equiv 1$ and $I \in R$ , the magnetic moment of the ideal loop current of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1) is $m : = \frac{I}{2} \int_{0}^{L} γ (s) \times γ^{'} (s) d s,$ again a componentwise Riemann integral over the compact interval $[0, L]$ of continuous functions; for the circle $γ (s) = a (\cos (s / a), \sin (s / a), 0)$ traversed in the positive orientation this is $m = I π a^{2} e_{3}$ .

(c) [Status and scope]. In both clauses $m$ is a source moment of the declared current model: it is not a rest mass, not an intrinsic spin and not an observed quantity. The definition asserts no far-field property of $m$ (that is the content of Far-field dipole expansion of a compact conserved current with a uniform explicit remainder under its hypotheses), no agreement between the two clauses in a common limit, and no moment for currents outside the two classes; no higher multipole is defined here. The finite Riemann moment definitions and circular-loop evaluation themselves are choice-free; no distribution-embedding or surface-measure theorem is used to compute these moments.

Remarks

Well-definedness and units. In (a), $y \mapsto y \times J (y)$ is continuous on the compact rectangle $Q$ and vanishes off $supp J$ , so each component is Riemann integrable on $Q$ by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set and bounded there, hence the integral of The Riemann integral of a bounded function over a bounded Jordan measurable set is finite. Two bounding rectangles are compared through a common larger rectangle subdivided at their face coordinates: every added cell lies outside $supp J$ , where the integrand vanishes, so finite additivity of the integral (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ ) identifies both values without integrating over the support itself. The cross product is that of The cross product in $R^{3}$ . Since $y$ carries the metre and $J$ the ampere per square metre, the product $y \times J$ carries A m $^{- 1}$ and the volume integral A m $^{2}$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values). In (b), $γ$ and $γ^{'}$ are continuous on the compact circle and periodic, so each component of $γ \times γ^{'}$ is a continuous periodic function on $[0, L]$ and the componentwise Riemann integral exists; $∣ γ^{'} ∣ \equiv 1$ makes the parametrization unit-speed so that the value is independent of an orientation-preserving change of unit-speed parameter; reversing orientation reverses the tangent current unless the signed current parameter $I$ is also reversed. The circular evaluation is the componentwise computation $γ \times γ^{'} = a e_{3}$ and $\int_{0}^{2 π a} a e_{3} d s = 2 π a^{2} e_{3}$ . No measurement, apparatus or uncertainty enters either clause.

Source notes

Griffiths, Section 5.4.3, printed pp. 242-246, introduces the line-current and volume-current magnetic moments for volume currents (eqs. (5.84) and (5.91)); Tong, Sections 3.3.1-3.3.2, printed pp. 54-57, gives $m = \frac{1}{2} \int d^{3} r^{'} r^{'} \times J (r^{'})$ (eq. (3.24)); Landau-Lifshitz, Section 44, printed pp. 103-105, gives the corresponding charged-particle moment in Gaussian units, re-expressed here in SI. The definition keeps the moment a declared source quantity of the two explicit models and defers every far-field interpretation to the dipole theorem.

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Coulomb-gauge Newtonian vector potential for a compact smooth conserved current

Physical scope

A conditional construction theorem for the compact smooth current class of the framework's steady-current regime (adopted Maxwell equations, one inertial frame, SI units). The Coulomb gauge is a construction choice; no uniqueness, no noncompact-current or singular-source statement and no empirical claim about any apparatus or observed field is made.

Statement

(a) $A \in C^{\infty} (R^{3}; R^{3})$ 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) $div A = 0$ pointwise on $R^{3}$ : the Coulomb-gauge condition holds;

(c) the field $B : = curl A$ is of class $C^{\infty}$ and satisfies the magnetostatic pair of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(a) pointwise on $R^{3}$ , $div B = 0, curl B = μ_{0} J;$ consequently $E = 0$ , $ρ = 0$ , with $B, J$ independent of time, gives an explicit full smooth-Maxwell completion on $R^{3} \times I$ for every open time interval $I$ under The microscopic vacuum Maxwell equations: smooth and weak branches. More generally any compatible steady electrostatic pair satisfying $curl E = 0$ and $div E = ρ / ε_{0}$ can be combined with $B, J$ ; no electrostatic existence theorem is implied;

(d) [Scope] $A$ is a vector potential for $B$ in the Coulomb gauge; the gauge condition is a construction choice and no uniqueness of $A$ or of $B$ is asserted. The construction is carried out for the compact smooth class only: nothing is claimed for the declared ideal models of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c), for noncompact currents or for currents of weaker regularity.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero to $R^{3}$ ; let $μ_{0}, c, ε_{0}$ be as in Framework-local SI units, the vacuum constants and the exact/measured status of their values and $A$ be as in the Statement.

[F1]

For $h \in C_{c}^{\infty} (R^{3}; R)$ the Newtonian potential satisfies $N h \in C^{\infty} (R^{3})$ , $- Δ N h = h$ pointwise, and $D^{α} N h = N (D^{α} h)$ for every multi-index $α$ (Newtonian potential of smooth compact data is smooth and solves the Poisson equation pointwise).

[F2]

The Laplacian is $Δ F = (Δ F_{1}, Δ F_{2}, Δ F_{3})$ componentwise, and $Δ f = div \nabla f = \sum_{i} \partial_{i} \partial_{i} f$ for $C^{2}$ scalars (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field); divergence and curl are the coordinate expressions $div G = \sum_{i} \partial_{i} G_{i}$ and $curl G = (\partial_{2} G_{3} - \partial_{3} G_{2}, \partial_{3} G_{1} - \partial_{1} G_{3}, \partial_{1} G_{2} - \partial_{2} G_{1})$ (Divergence and curl of a $C^{1}$ vector field), with partial derivatives as in Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ .

[F3]

For a $C^{2}$ field $G$ on an open subset of $R^{3}$ , $div (curl G) = 0$ (The divergence of the curl of a $C^{2}$ field vanishes).

[F4]

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

[F5]

The compact smooth current class is the class of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b): $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero; the magnetostatic pair $div B = 0$ , $curl B = μ_{0} J$ is the time-independent reduction of clause (a) there, and the adopted smooth equations with these constants are The microscopic vacuum Maxwell equations: smooth and weak branches(a).

[F6]

${A C}_{ω}$ is the countable-choice principle assumed here and inherited by the Newtonian-potential supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct; componentwise reduction to the smooth Newtonian upgrade and the standard second-derivative vector identities

1.1givenF1F2

For each $i$ the datum $J_{i}$ is smooth and compactly supported, so [F1] gives $A_{i} = μ_{0} N (J_{i}) \in C^{\infty} (R^{3})$ , the derivative identity $D^{α} A_{i} = μ_{0} N (D^{α} J_{i})$ for every multi-index $α$ , and $- Δ A_{i} = μ_{0} J_{i}$ pointwise on $R^{3}$ ; since all components are smooth, $A \in C^{\infty} (R^{3}; R^{3})$ by [F2], and $- Δ A = μ_{0} J$ holds pointwise componentwise. This is (a).

2.1step 1.1F2F5

Taking $α = e_{i}$ in the derivative identity of step 1.1 and summing, $div A = \sum_{i} \partial_{i} A_{i} = μ_{0} \sum_{i} N (\partial_{i} J_{i})$ ; $N$ is componentwise linear by finite additivity of the integral over the compactly supported data, so this equals $μ_{0} N (\sum_{i} \partial_{i} J_{i}) = μ_{0} N (div J)$ pointwise, and $div J = 0$ gives $div A = 0$ on $R^{3}$ . This is (b), and the hypothesis $div J = 0$ is used in no other step.

3.1step 1.1step 2.1F2F3F4F5

Since $A$ is $C^{\infty}$ , the field $B : = curl A$ is $C^{\infty}$ by [F2]. Then $div B = div curl A = 0$ by [F3], and $curl B = curl curl A = \nabla (div A) - Δ A$ by [F4]; substituting step 2.1 and step 1.1 gives $curl B = 0 - (- μ_{0} J) = μ_{0} J$ . Hence the magnetostatic pair of [F5] holds pointwise on $R^{3}$ .

4.1step 3.1F5

Read step 3.1 inside the framework: [F5] identifies the pair $div B = 0$ , $curl B = μ_{0} J$ with the time-independent reduction of the adopted equations, and the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches(a) then holds for the fields $E : = 0$ , $ρ : = 0$ on $Ω \times I$ and the time-independent $B, J$ restricted to $Ω \times I$ : indeed $div E = 0 = ρ / ε_{0}$ , $curl E = 0$ and $\partial_{t} B = 0$ give $curl E = - \partial_{t} B$ , $div B = 0$ , and $\partial_{t} E = 0$ with $curl B = μ_{0} J$ gives $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ . More generally a compatible steady electrostatic pair satisfying $curl E = 0$ and $div E = ρ / ε_{0}$ may be combined with $B, J$ ; no existence theorem for such a pair is asserted here.

5.1step 1.1step 2.1step 3.1step 4.1F5F6∎

Clauses (a)-(d) are steps 1.1, 2.1, 3.1, 4.1 together with the scope statement: $A$ is a vector potential for $B$ in the Coulomb gauge, the gauge condition $div A = 0$ is a construction choice, and no uniqueness of $A$ , no uniqueness of $B$ and no statement for the ideal noncompact models of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c) is claimed. Countable Choice is used only through the Newtonian-potential supplier [F1] and the measure interfaces it inherits [F6]; no stronger choice principle occurs, and the vector identities of [F3], [F4] are choice-free.

Source notes

Griffiths, Section 5.4.1, printed pp. 234-240, derives the Poisson equation $\nabla^{2} A = - μ_{0} J$ in the Coulomb gauge and gives the integral solution (5.63); Tong, Sections 3.2-3.2.3, printed pp. 46-52, and Chew, Lecture 4 Section 4.1, printed pp. 39-41, give the same componentwise construction; Landau-Lifshitz, Section 43, printed pp. 101-103, gives the corresponding constant-field formulation in Gaussian units, re-expressed here in SI. The proof above is the exact componentwise version: smoothness and the Poisson identity come from the smooth Newtonian upgrade, and the divergence-free hypothesis enters only through step 2.1.

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Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit

Physical scope

Biot-Savart representation of the magnetostatic field of a compact smooth conserved current, together with the declared thin-loop limit of that representation. Both legs are conditional on the framework's adopted equations and on the declared ideal models; the smooth-current formula holds even on its support, whereas no pointwise value on the ideal loop is asserted; nothing is claimed about self-energy, radiation, forces or noncompact currents, and no measurement, apparatus or uncertainty is involved.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ be a compact smooth current of The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b), extended by zero to $R^{3}$ , let $A$ be the vector potential of Coulomb-gauge Newtonian vector potential for a compact smooth conserved current and $B = curl A$ its magnetostatic field, with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Then for every $x \in R^{3}$ the absolutely convergent Biot-Savart integral represents the field, $B (x) = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J (y) \times (x - y)}{∣ x - y ∣^{3}} d y .$ Moreover the same formula holds in the declared thin-loop limit, constructed in the all-space laboratory $Ω = R^{3}$ : if $γ : R / L Z \to R^{3}$ is a closed embedded $C^{2}$ curve parametrized by arc length ( $∣ γ^{'} ∣ \equiv 1$ ), $I \in R$ , and $j_{ε} = j * η_{ε}$ are the mollified divergence-free compact currents of Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields associated with the tangent line current of $γ$ , then their fields (given by the displayed formula with $J : = j_{ε}$ ) converge, with every derivative uniformly on every compact $K$ disjoint from $γ$ , to $B_{γ} (x) = \frac{μ_{0} I}{4 π} \int_{0}^{L} \frac{γ^{'} (s) \times (x - γ (s))}{∣ x - γ (s) ∣^{3}} d s,$ a locally integrable global distributional field with $div B_{γ} = 0$ , $curl B_{γ} = μ_{0} J_{γ}$ , smoothness off the curve and decay at spatial infinity, hence the admitted field of the declared ideal thin loop under Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models. Its almost-everywhere representative on $R^{3}$ is supplied by Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields, independently of the away-from-curve convergence. Nothing is asserted on the wire, about line self-energy, about the infinite-wire or sheet models, or for currents outside the compact smooth class and its declared thin-loop limit.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ , extended by zero to $R^{3}$ ; let $A$ , $B = curl A$ be as in Coulomb-gauge Newtonian vector potential for a compact smooth conserved current with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

[F1]

The vector potential satisfies $A_{i} = μ_{0} N (J_{i})$ componentwise and $B = curl A$ is $C^{\infty}$ with $div B = 0$ and $curl B = μ_{0} J$ pointwise on $R^{3}$ (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current, clauses (a)-(c)).

[F2]

For $f \in C_{c}^{\infty} (R^{3}; R^{3})$ the curl of the Newtonian potential is $curl N (f) (x) = \frac{1}{4 π} \int_{R^{3}} \frac{f (y) \times (x - y)}{∣ x - y ∣^{3}} d y$ at every $x \in R^{3}$ , the integral being absolutely convergent, and the estimate $∣ f (y) \times (x - y) ∣ \leq ∥ f ∥_{\infty} ∣ x - y ∣$ with $\int_{B_{R} (x)} ∣ x - y ∣^{- 2} d λ_{3} < + \infty$ underlies it (Divergence and curl of the Newtonian potential of a smooth compact vector field).

[F3]

For a closed embedded $C^{2}$ arc-length curve $γ$ and $I \in R$ , the tangent line current $j$ has $div j = 0$ distributionally; for any family of nonnegative smooth radial unit-mass kernels supported in $B_{ε} (0)$ the mollified fields $j_{ε} = j * η_{ε}$ are smooth, compactly supported and divergence-free; their Biot-Savart fields converge with every derivative uniformly on every compact $K$ disjoint from $γ$ to $B_{γ} (x) = \frac{μ_{0} I}{4 π} \int_{0}^{L} \frac{γ^{'} (s) \times (x - γ (s))}{∣ x - γ (s) ∣^{3}} d s$ ; and the same field has a locally integrable representative on $R^{3}$ , is smooth off $γ$ , satisfies $div T_{B_{γ}} = 0$ and $curl T_{B_{γ}} = μ_{0} j$ globally, and decays with $∣ B_{γ} (x) ∣ \leq \frac{μ_{0} ∣ I ∣ L}{4 π (∣ x ∣ - a)^{2}}$ for $∣ x ∣ > a$ when $γ \subset B_{a} (0)$ (Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields).

[F4]

The compact smooth current class is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b); the ideal closed thin loop is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1); and a field is admitted for it exactly when the predicate (c4) holds, which includes local integrability, $C^{1}$ regularity and vanishing divergence and curl off the source, the two global distributional identities, the model symmetry and the decay requirement (The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models).

[F5]

The physical reading of the declared ideal models is the separate adoption Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models; it asserts no existence, uniqueness, finite self-energy, force, self-force or apparatus accuracy, and it is an adopted assumption rather than a deduction.

[F6]

The cross product is that of The cross product in $R^{3}$ , volume integrals are over measurable sets in the sense of Integral over a measurable subset, and the modulus of an integral is bounded by the integral of the modulus (The modulus of an integral is bounded by the integral of the modulus).

[F7]

${A C}_{ω}$ is the countable-choice principle assumed here and inherited by the cited measure, regular-distribution and Newtonian interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct; the differentiated-kernel identity for the compact class, then the declared thin-current limit with its supplier-supplied global properties

1.1givenF1F2F6

By [F1], $B = curl A$ with $A_{i} = μ_{0} N (J_{i})$ . Applying [F2] to $f : = J$ and multiplying by $μ_{0}$ gives, at every $x \in R^{3}$ , $B (x) = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J (y) \times (x - y)}{∣ x - y ∣^{3}} d y$ , with the integral absolutely convergent by the estimate $∣ J (y) \times (x - y) ∣ \leq ∥ J ∥_{\infty} ∣ x - y ∣$ and $\int_{B_{R} (x)} ∣ x - y ∣^{- 2} d λ_{3} < + \infty$ for a ball containing $supp J$ ; this proves the first assertion at every point, including points of the support of $J$ .

2.1step 1.1F3F4

By [F3], the tangent line current $j$ of the declared loop is divergence-free, its mollified currents $j_{ε}$ are smooth, compactly supported and divergence-free, so, taking $Ω = R^{3}$ for this loop construction, each $j_{ε}$ is a compact smooth conserved current of the class F4; step 1.1 applied with $J : = j_{ε}$ identifies the field of $j_{ε}$ with its displayed Biot-Savart integral. The supplier's every-derivative uniform convergence on every compact $K$ disjoint from $γ$ therefore identifies the limit field with $B_{γ} (x) = \frac{μ_{0} I}{4 π} \int_{0}^{L} \frac{γ^{'} (s) \times (x - γ (s))}{∣ x - γ (s) ∣^{3}} d s$ on the complement of $γ$ .

3.1step 2.1F3F4F5

The same supplier [F3] supplies, independently of the away-from-curve convergence, the globally locally integrable representative of $B_{γ}$ on $R^{3}$ , its smoothness off $γ$ , the global distributional identities $div T_{B_{γ}} = 0$ and $curl T_{B_{γ}} = μ_{0} j$ , and the decay bound. Together with the symmetry requirement (vacuous for the loop clause) these are exactly the clauses of the admissibility predicate F4, so $B_{γ}$ is an admitted field of the declared ideal thin loop; the physical reading of that declaration is the separate adopted assumption [F5], which asserts no existence, uniqueness, self-energy or force content.

4.1step 1.1step 2.1step 3.1F3F7∎

Clauses (a) and (b) of the Statement are steps 1.1 and 2.1-3.1; the scope clause records that no formula is asserted for noncompact currents (the infinite-wire model is verified separately on the companion examples page), and that no near-wire, self-energy, radiation or force claim is made. Countable Choice is inherited only through the cited measure, regular-distribution and Newtonian supplier interfaces [F7]; no stronger choice principle occurs.

Source notes

Griffiths, Section 5.2.2, printed pp. 215-219, states the Biot-Savart law for line, surface and volume currents and evaluates the straight wire (5.36) and circular loop (5.38); Section 5.3.2, printed pp. 222-225, derives $curl B = μ_{0} J$ from the volume integral. Tong, Section 3.2.3, printed pp. 49-52, gives (3.15) and its thin-wire form (3.16); Chew, Lecture 5 Section 5.1, printed pp. 49-52, derives the law from the vector potential, and Landau-Lifshitz, Sections 43-44, printed pp. 101-104, give the corresponding constant-field and Biot-Savart formulas, re-expressed here in SI. The thin-loop leg of this item is conditional on its mathematical supplier Thin closed current: divergence-free distribution and convergence of its mollified Biot-Savart fields for the global distributional identities and the local integrability, which are not consequences of away-from-curve convergence alone.

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Far-field dipole expansion of a compact conserved current with a uniform explicit remainder

Physical scope

A far-field dipole estimate for the magnetostatic field of a compact smooth conserved current, uniform for $r \geq 2 a$ , with an explicit remainder and absolute constants. The leading term is not an exact field; nothing is asserted inside $r < 2 a$ or at the support, and no experiment, apparatus or uncertainty is involved.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ), with the exact meaning of The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $J \in C_{c}^{\infty} (Ω; R^{3})$ with $div J = 0$ and $supp J \subset B_{a} (0)$ for some $a > 0$ , let $A$ be the vector potential of Coulomb-gauge Newtonian vector potential for a compact smooth conserved current and $B = curl A$ its magnetostatic field, and let $m = \frac{1}{2} \int_{R^{3}} y \times J (y) d y$ be the magnetic moment of The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(a). Then for $r = ∣ x ∣ \geq 2 a$ , uniformly in the unit vector $n = x / r$ , $A (x) = \frac{μ_{0}}{4 π} \frac{m \times x}{r^{3}} + O ⁣ (\frac{μ_{0} a^{2} ∥ J ∥_{1}}{r^{3}}), B (x) = \frac{μ_{0}}{4 π} \frac{3 n (m \cdot n) - m}{r^{3}} + O ⁣ (\frac{μ_{0} a^{2} ∥ J ∥_{1}}{r^{4}}),$ where $∥ J ∥_{1} = \int_{R^{3}} ∣ J ∣ d λ_{3}$ and the constants implicit in the $O$ terms are absolute: one constant works for all $r \geq 2 a$ and all unit $n$ . This is a controlled two-term far-field estimate: nothing is asserted for $r < 2 a$ or at the support, no higher multipole is defined here, and the leading dipole term is not claimed to be exact.

Facts & Assumptions

Given: Assume ${A C}_{ω}$ ; let $J \in C_{c}^{\infty} (R^{3}; R^{3})$ with $div J = 0$ and $supp J \subset B_{a} (0)$ ; let $A$ , $B = curl A$ be as in Coulomb-gauge Newtonian vector potential for a compact smooth conserved current and $m$ be as in The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(a).

[F1]

The vector potential satisfies $A_{i} = μ_{0} N (J_{i}) = (μ_{0} / 4 π) \int_{R^{3}} J_{i} (y) ∣ x - y ∣^{- 1} d y$ componentwise and $B = curl A$ is its magnetostatic field (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current, clauses (a)-(c)).

[F2]

The magnetic moment of the compact smooth current is the source moment $m = \frac{1}{2} \int_{R^{3}} y \times J (y) d y$ of The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(a), computed componentwise as a Riemann integral over a nondegenerate closed bounding rectangle $Q \supset supp J$ of the continuous compactly supported integrand, with the cross product of The cross product in $R^{3}$ ; clause (c) records that $m$ is a model quantity and that its far-field interpretation is the content of this theorem under its hypotheses.

[F3]

Under the stated hypotheses, with $m = \frac{1}{2} \int_{R^{3}} y \times J (y) d y$ (Lebesgue), $r = ∣ x ∣ \geq 2 a$ and $n = x / r$ , one has uniformly in $n$ : $A = \frac{μ_{0}}{4 π} \frac{m \times x}{r^{3}} + O (μ_{0} a^{2} ∥ J ∥_{1} r^{- 3})$ and $B = \frac{μ_{0}}{4 π} \frac{3 n (m \cdot n) - m}{r^{3}} + O (μ_{0} a^{2} ∥ J ∥_{1} r^{- 4})$ , with absolute constants and $∥ J ∥_{1} = \int ∣ J ∣$ (Magnetic dipole expansion with an explicit uniform remainder).

[F4]

If $f$ is a bounded Borel real function on a nondegenerate closed rectangle $Q$ and $f$ is Riemann integrable there, then $f$ is Lebesgue integrable with the same integral; equivalently, for smooth compactly supported integrands the Riemann integral over a bounding rectangle equals the Lebesgue integral of their zero extension (Riemann–Lebesgue comparison for distribution test integrands).

[F5]

${A C}_{ω}$ is the countable-choice principle assumed here and inherited by the cited comparison and measure interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct transport of the explicit uniform-remainder lemma to the framework's objects, with a Riemann-Lebesgue comparison for the two displayed moments

1.1givenF1F2F3F4

The hypotheses of [F3] are exactly the hypotheses stated here for $J$ (smooth, compactly supported, divergence-free, contained in $B_{a} (0)$ ), and the objects of [F3] are the objects of this theorem: $A = \frac{μ_{0}}{4 π} \int J (y) ∣ x - y ∣^{- 1} d y$ componentwise and $B = curl A$ by [F1]. The moment of [F2] agrees with the Lebesgue moment of [F3]: each component $y \mapsto (y \times J (y))_{i}$ is smooth and compactly supported, hence bounded Borel and Riemann integrable on any nondegenerate closed rectangle $Q$ containing $supp J$ ; by [F4] its Riemann integral over $Q$ equals its Lebesgue integral over $Q$ , and since the integrand vanishes off $supp J \subset Q$ this equals the Lebesgue integral over $R^{3}$ . Hence $\frac{1}{2} \int_{R^{3}} y \times J (y) d y$ is the same vector in both items.

2.1step 1.1F3

Substituting the identifications of step 1.1 into the conclusion of [F3] gives, for $r = ∣ x ∣ \geq 2 a$ and uniformly in the unit vector $n = x / r$ , the displayed estimates $A (x) = \frac{μ_{0}}{4 π} \frac{m \times x}{r^{3}} + O (μ_{0} a^{2} ∥ J ∥_{1} r^{- 3})$ and $B (x) = \frac{μ_{0}}{4 π} \frac{3 n (m \cdot n) - m}{r^{3}} + O (μ_{0} a^{2} ∥ J ∥_{1} r^{- 4})$ for the framework's $A$ and $B$ , with the lemma's uniformity and absolute constants and with $∥ J ∥_{1} = \int_{R^{3}} ∣ J ∣ d λ_{3}$ as in the Statement.

3.1step 1.1step 2.1F3F4F5∎

The estimates of step 2.1 are the content of the Statement; no assertion is made for $r < 2 a$ or at the support, no higher multipole is defined, and the leading dipole term is not claimed to be exact. Countable Choice is used only through the Riemann-Lebesgue comparison [F4] and the measure and Newtonian interfaces [F5] of the cited suppliers; no stronger choice principle occurs.

Source notes

Griffiths, Section 5.4.3, printed pp. 242-247, expands the vector potential and gives the dipole field (5.85)-(5.87) with the coordinate-free form and its delta term (5.90); Tong, Sections 3.3.1-3.3.2, printed pp. 54-57, gives the loop far field and the general moment (3.24); Landau-Lifshitz, Section 44, printed pp. 103-105, gives the first-order average field in Gaussian units, re-expressed here in SI. The uniform remainder is carried by the explicit mathematical supplier Magnetic dipole expansion with an explicit uniform remainder, not by the textbook expansions, and the Riemann moment of the definition is reconciled with the supplier's Lebesgue moment in step 1.1.

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Force and torque on a declared ideal thin-wire loop in a prescribed external field

Physical scope

Force and torque on a declared ideal thin-wire loop in a prescribed external magnetic field, under an explicitly adopted neutral-wire Lorentz coupling. Only the external field acts: self-fields, radiation reaction, mechanical support, deformation and material failure are outside the statement; no dynamics, radiation or empirical measurement is claimed, and the result is conditional on the adopted coupling (W).

Statement

Let $γ : [0, L] \to R^{3}$ be a $C^{2}$ curve parametrized by arc length ( $∣ γ^{'} ∣ \equiv 1$ ) that is either closed ( $γ (0) = γ (L)$ with matching one-sided derivatives) or a wire segment with free endpoints, let $I \in R$ , let $m = \frac{I}{2} \int_{0}^{L} γ (s) \times γ^{'} (s) d s$ be the finite Riemann source moment defined by this displayed integral in the closed case. For an embedded closed loop this agrees with The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(b); embeddedness is not required for the present parametrized-curve calculation. Let $B$ be a prescribed external magnetic field, of class $C^{0}$ on a neighbourhood of the trace of $γ$ in part (a) and of class $C^{2}$ on an open set containing the closed ball $\overline{B_{a} (0)}$ in parts (b)-(d). Adopt the thin-wire coupling hypothesis (W): for a general closed parametrized curve use the stated direct neutral-wire Riemann coupling (with multiplicity of traversal retained); in the embedded closed-loop specialization its source interpretation is exactly The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1); for an open curve use a compact segment of a fed or closed neutral circuit, including a segment of the declared infinite wire of (c2). Adopt separately for that parametrized wire or local segment the direct external magnetic coupling $d F = I d γ \times B$ and $d N = γ \times d F$ , as an ideal extension motivated by the smooth continuum Lorentz law The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii); that smooth-source postulate does not itself include singular currents. An isolated static neutral open segment is not asserted: its standalone current satisfies $div J_{s e g} = I (δ_{γ (0)} - δ_{γ (L)})$ , so its endpoint feed/return cannot be suppressed in a global model. Force and torque here concern only the specified local segment under this adopted coupling. Then:

(a) [Force and torque under (W)] $F : = I \int_{0}^{L} γ^{'} (s) \times B (γ (s)) d s, N : = I \int_{0}^{L} γ (s) \times (γ^{'} (s) \times B (γ (s))) d s,$ the componentwise Riemann integrals of continuous functions on the compact interval $[0, L]$ (The Riemann integral of a bounded function over a bounded Jordan measurable set, A continuous real function on a compact Jordan measurable set is Riemann integrable over that set).

(b) [Constant field] If $γ$ is closed and $B$ is constant on a neighbourhood of the loop, then $F = 0$ and $N = m \times B$ exactly.

(c) [Slowly varying field] Let $γ$ be closed with $∣ γ (s) ∣ \leq a$ for all $s$ , let $B$ be $C^{2}$ on an open set containing $\overline{B_{a} (0)}$ with constants $M_{1}, M_{2}$ such that $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ and $∣ B (x) - B (0) - D B (0) x ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ for $∣ x ∣, ∣ y ∣ \leq a$ , and suppose $div B = 0$ on $B_{a} (0)$ (as holds for the external part of any smooth-branch vacuum field by The microscopic vacuum Maxwell equations: smooth and weak branches, Divergence and curl of a $C^{1}$ vector field). Then $F = \nabla (m \cdot B) (0) + R_{F}, N = m \times B (0) + R_{N},$ with $∣ R_{F} ∣ \leq \frac{1}{2} ∣ I ∣ L a^{2} M_{2}$ and $∣ R_{N} ∣ \leq ∣ I ∣ L a^{2} M_{1}$ , where $\nabla (m \cdot B) (0)$ is the gradient at $0$ of $x \mapsto m \cdot B (x)$ for the fixed vector $m$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

(d) [Sufficient condition] If the external field is the magnetostatic field of a compact smooth steady current $J_{e x t} \in C_{c}^{\infty} (Ω; R^{3})$ with $div J_{e x t} = 0$ (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current, Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit), then it is $C^{\infty}$ on $R^{3}$ , satisfies $div B = 0$ , and meets the hypotheses of (c) on every ball $\overline{B_{a} (0)}$ with finite constants $M_{1}, M_{2}$ .

(e) [Scope] Only the external field acts on the loop: the loop's own field, self-force, radiation reaction, mechanical support and material response are outside the statement, and no claim is made about the loop's dynamics, deformation or failure. The coupling (W) is an explicit idealisation of a neutral thin wire at rest in the laboratory frame; it is adopted, not derived, and the theorem is conditional on it. No measurement or uncertainty is involved. Parts (a)-(c), conditional on the explicit coupling (W) and the supplied curve and external field, are finite Riemann-integral calculations and use no Choice axiom. Part (d), transporting the Newtonian-potential construction, assumes Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Facts & Assumptions

Given: The $C^{2}$ arc-length curve $γ$ , the real $I$ , the prescribed field $B$ , the coupling (W) of the Statement, and, in parts (b)-(d), the ball containment and regularity hypotheses stated there.

[F1]

Under the coupling (W), the force and torque on the parametrized wire are the componentwise Riemann integrals $F = I \int_{0}^{L} γ^{'} \times B (γ) d s$ and $N = I \int_{0}^{L} γ \times (γ^{'} \times B (γ)) d s$ of the Statement; continuous integrands on the compact interval are Riemann integrable (The Riemann integral of a bounded function over a bounded Jordan measurable set, A continuous real function on a compact Jordan measurable set is Riemann integrable over that set, $C^{k}$ Euclidean maps and diffeomorphisms, The cross product in $R^{3}$ ).

[F2]

The cross product is $u \times v = (u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1})$ with $(u \times v)_{i} = ε_{i j k} u_{j} v_{k}$ , the inner product is $⟨ u, v ⟩ = \sum_{i} u_{i} v_{i}$ , and the identities $a \times (b \times c) = b (a \cdot c) - c (a \cdot b)$ and $ε_{i j k} ε_{l j p} = δ_{i l} δ_{k p} - δ_{i p} δ_{k l}$ hold (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F3]

The Riemann integral on a compact rectangle is linear, monotone, satisfies $∣ \int f ∣ \leq \int ∣ f ∣$ , and the integral of a constant $c$ over $[0, L]$ is $c L$ ; these properties are used for the remainder estimates (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ , The Riemann integral of a bounded function over a bounded Jordan measurable set).

[F4]

The bounds $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ and $∣ B (x) - B (0) - D B (0) x ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ on $\overline{B_{a} (0)}$ are the explicit hypotheses of (c). For (d), the coordinate FTC of [F9] applied twice to $s \mapsto B (s x)$ gives the componentwise finite-Riemann identity $B (x) - B (0) - D B (0) x = \int_{0}^{1} (1 - s) D^{2} B (s x) [x, x] d s$ . It does not require a common Lagrange point for all vector components.

[F5]

The moment of The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop(b) for a closed embedded arc-length loop is exactly $m = \frac{I}{2} \int_{0}^{L} γ \times γ^{'} d s$ ; the declared ideal loop source is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c1) and the declared infinite wire is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(c2) (The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop, The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models).

[F6]

The compact smooth current class is The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models(b); its magnetostatic field construction is Coulomb-gauge Newtonian vector potential for a compact smooth conserved current and its Biot-Savart representation is Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit: the external field of a compact smooth conserved current is $C^{\infty}$ on $R^{3}$ with $div B = 0$ (Coulomb-gauge Newtonian vector potential for a compact smooth conserved current, Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit); the smooth-branch vacuum equations are The microscopic vacuum Maxwell equations: smooth and weak branches and the smooth continuum Lorentz coupling that motivates (W) is The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii).

[F7]

Distributional derivatives act by signed test differentiation, $⟨ \partial^{α} u, χ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} χ ⟩$ (Distributional derivative); a $C^{1}$ composition $φ \circ γ$ has derivative $(φ \circ γ)^{'} = (\nabla φ) (γ) \cdot γ^{'}$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ) and satisfies the fundamental theorem $\int_{0}^{L} (φ \circ γ)^{'} d s = φ (γ (L)) - φ (γ (0))$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ); the Dirac distribution at a point $p$ acts by $δ_{p} (φ) = φ (p)$ (Dirac delta and its derivatives).

[F8]

${A C}_{ω}$ is the countable-choice principle assumed in part (d) and inherited by the Newtonian-potential suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F9]

Compact closed Euclidean balls are compact (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); continuous real functions on a nonempty compact metric set attain finite extrema (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value). The fundamental theorem integrates continuous derivatives on compact intervals (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ). Finite-vector Cauchy-Schwarz gives $∣ u \cdot v ∣ \leq ∣ u ∣ ∣ v ∣$ (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation); testing a vector integral against its unit direction gives the corresponding integral norm bound.

Proof

technique · direct evaluation of the pairings under the coupling, the antisymmetric first-moment identities for a closed curve, one Taylor expansion with an explicit remainder, and the compact-current sufficient condition

1.1givenF1F2

Under the adopted coupling (W) the total force and torque of the parametrized wire are the integrals of the Statement: at arc-length parameter $s$ the element $d γ = γ^{'} (s) d s$ carries current $I$ , so $d F = I γ^{'} (s) \times B (γ (s)) d s$ and $d N = γ (s) \times d F$ . The integrands are compositions and products of continuous maps on the compact interval $[0, L]$ , hence continuous and Riemann integrable, so the componentwise integrals of [F1] are finite. This is (a).

1.2givenF1F2F5

Let $B$ be constant on a neighbourhood of the closed curve. Then $F = I (\int_{0}^{L} γ^{'} (s) d s) \times B = I (γ (L) - γ (0)) \times B = 0$ . For the torque, $F$ 's definition and [F2] give $γ \times (γ^{'} \times B) = (γ \cdot B) γ^{'} - (γ \cdot γ^{'}) B$ , so with the matrix $M_{j l} : = \int_{0}^{L} γ_{j} γ_{l}^{'} d s$ one has $N = I \int_{0}^{L} (γ \cdot B) γ^{'} d s - I (\int_{0}^{L} γ \cdot γ^{'} d s) B = I B_{j} M_{j l} e_{l}$ . The scalar $γ \cdot γ^{'} = \frac{1}{2} \frac{d}{d s} ∣ γ ∣^{2}$ integrates to $0$ over the closed curve, and integrating $\frac{d}{d s} (γ_{j} γ_{l})$ over the closed curve gives $M_{j l} + M_{l j} = 0$ , so $M$ is antisymmetric. For antisymmetric $M$ , the moment $m_{p} = \frac{I}{2} ε_{p a b} M_{a b}$ satisfies $I M_{j l} = ε_{j l p} m_{p}$ because $ε_{a b p} ε_{p c d} = δ_{a c} δ_{b d} - δ_{a d} δ_{b c}$ gives $ε_{a b p} m_{p} = \frac{I}{2} (M_{a b} - M_{b a}) = I M_{a b}$ . Substituting, $N = ε_{j l p} m_{p} B_{j} e_{l} = m \times B$ by the cyclic identity of the alternating symbol, and this is (b).

1.3givenF7

Let $J_{s e g}$ be the standalone open-segment current defined by $⟨ J_{s e g}, ψ ⟩ : = I \int_{0}^{L} γ^{'} (s) \cdot ψ (γ (s)) d s$ for vector tests $ψ$ . With the componentwise distributional divergence pairing of [F7], $⟨ div J_{s e g}, φ ⟩ = - I \int_{0}^{L} γ^{'} (s) \cdot \nabla φ (γ (s)) d s = - I \int_{0}^{L} (φ \circ γ)^{'} (s) d s = I (φ (γ (0)) - φ (γ (L))) = I (δ_{γ (0)} - δ_{γ (L)}) (φ)$ for every scalar test $φ$ , the chain rule and the fundamental theorem of [F7] giving the middle equality. This is the endpoint identity asserted in the Statement, and it shows why the segment is treated as a local part of a fed or closed circuit.

2.1step 1.1F4

Under the hypotheses of (c), step 1.1's formulas apply, and [F4] gives the decomposition $B (x) = B (0) + D B (0) x + R (x)$ with $∣ R (x) ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ for $∣ x ∣ \leq a$ , together with the Lipschitz bound $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ on the ball.

3.1step 1.2step 2.1F2F3

For the force in (c), the constant term vanishes as in step 1.2, and the linear term is $I ε_{i j k} \partial_{l} B_{k} (0) M_{l j} e_{i}$ with the matrix $M$ of step 1.2. Substituting $I M_{l j} = ε_{l j p} m_{p}$ from step 1.2 and contracting by [F2], $I ε_{i j k} \partial_{l} B_{k} (0) M_{l j} e_{i} = (δ_{i l} δ_{k p} - δ_{i p} δ_{k l}) \partial_{l} B_{k} (0) m_{p} e_{i} = (\partial_{i} (m \cdot B) (0) - m_{i} div B (0)) e_{i} = \nabla (m \cdot B) (0)$ , since $m$ is constant and $div B (0) = 0$ . By [F3] and step 2.1 the remainder obeys $∣ I \int_{0}^{L} γ^{'} \times R (γ) d s ∣ \leq ∣ I ∣ \int_{0}^{L} ∣ γ^{'} ∣ ∣ R (γ) ∣ d s \leq \frac{1}{2} ∣ I ∣ L a^{2} M_{2}$ , because $∣ γ^{'} ∣ \equiv 1$ and $∣ γ (s) ∣ \leq a$ . This proves the force assertion of (c) with $∣ R_{F} ∣ \leq \frac{1}{2} ∣ I ∣ L a^{2} M_{2}$ .

3.2step 1.2step 2.1F2F3

For the torque in (c), split $B (γ (s)) = B (0) + (B (γ (s)) - B (0))$ . The first part is the constant-field torque of step 1.2 with the constant field $B (0)$ , that is $m \times B (0)$ . For the remainder, [F2] gives $γ \times (γ^{'} \times (B (γ) - B (0))) = γ^{'} (γ \cdot (B (γ) - B (0))) - (B (γ) - B (0)) (γ \cdot γ^{'})$ and hence, by [F3] and step 2.1, $∣ I \int_{0}^{L} γ \times (γ^{'} \times (B (γ) - B (0))) d s ∣ \leq ∣ I ∣ \int_{0}^{L} ∣ γ ∣ ∣ γ^{'} ∣ ∣ B (γ) - B (0) ∣ d s \leq ∣ I ∣ a \cdot 1 \cdot M_{1} a \cdot L$ . This proves the torque assertion of (c) with $∣ R_{N} ∣ \leq ∣ I ∣ L a^{2} M_{1}$ .

3.3step 2.1F4F6F9

For (d): by [F6] the external field of a compact smooth conserved current satisfies $B = curl A$ , hence is $C^{\infty}$ on $R^{3}$ with $div B = 0$ ; on the compact ball $\overline{B_{a} (0)}$ let $C_{1}, C_{2}$ be the maxima of the absolute coordinate first and second partial derivatives, finite by [F9], and put $M_{1} : = 9 C_{1}$ , $M_{2} : = 27 C_{2}$ . The coordinate sum bounds give $∣ D B (x) v ∣ \leq M_{1} ∣ v ∣$ and $∣ D^{2} B (x) [v, w] ∣ \leq M_{2} ∣ v ∣ ∣ w ∣$ . Integrating the first derivative along the segment from $x$ to $y$ gives $∣ B (x) - B (y) ∣ \leq M_{1} ∣ x - y ∣$ ; integrating the identity of [F4] gives $∣ B (x) - B (0) - D B (0) x ∣ \leq \frac{1}{2} M_{2} ∣ x ∣^{2}$ . All segments lie in the convex closed ball Hence the hypotheses of (c) are met on every such ball with finite constants.

4.1step 1.1step 1.2step 1.3step 3.1step 3.2step 3.3F5F8∎

Parts (a), (b), (c), (d) are steps 1.1, 1.2, 3.1, 3.2 and 3.3; the endpoint identity quoted in the Statement is step 1.3. The scope clause (e) records that only the external field acts, that the coupling (W) is an adopted idealisation motivated by the smooth continuum law The Lorentz coupling: external force on a test charge and the continuum force and power densities(ii) rather than derived from it, and that no self-field, self-force, radiation reaction, mechanical support, deformation or dynamical claim is made. Parts (a)-(c) are finite Riemann-integral calculations and use no Choice axiom; part (d) inherits Countable Choice only through the Newtonian-potential suppliers [F8]. The moment identity is contracted without dividing by $I$ ; for $I = 0$ the moment, force, torque and both remainders vanish, and the algebra applies to every stated closed parametrized $C^{2}$ curve, including multiply traversed and self-intersecting ones, since only closure, unit speed and the finite moment integral are used.

Source notes

Griffiths, Section 5.1.3, printed pp. 208-214, gives the wire force $F = \int I d l \times B$ (5.17), and Section 6.1.2, printed pp. 255-260, gives $N = m \times B$ in a uniform field (6.1), the vanishing net force in a uniform field, and $F = \nabla (m \cdot B)$ for a small loop in a nonuniform field (6.3), with Problems 6.2 and 6.4; Tong, Sections 3.4.1-3.4.2, printed pp. 57-62, performs the Taylor expansion over a small localised current and the moment manipulations; Landau-Lifshitz, Section 45, printed pp. 105-106, gives the average moment $K = m \times H$ (45.1), re-expressed here in SI. The theorem above is the exact conditional version for the declared parametrized wire under the explicitly adopted coupling (W); the texts do not supply the neutral-wire hypothesis, which is stated here as an idealisation.

5 · Examples, experiments and counterexamples

None yet.

Magnetostatics in vacuum

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The piecewise constant cylinder field e31{ρ<a}e3​1{ρ<a}​: distributional curl on the cylinder

Statement

Facts & Assumptions

Proof

Source notes

The azimuthal line field ρ−1eφρ−1eφ​: local integrability and distributional curl on the axis

Statement

Facts & Assumptions

Proof

Source notes

Divergence and curl of the Newtonian potential of a smooth compact vector field

Statement

Facts & Assumptions

Proof

Source notes

The steady-current (magnetostatic) regime, the compact smooth current class and the declared ideal noncompact models

Definition

Remarks

Source notes

Physical adoption of the declared ideal magnetostatic loop, line and cylindrical-sheet models

Postulate

Status and scope

Source notes

The magnetic dipole moment of a compact conserved current and of the declared ideal thin loop

Definition

Remarks

Source notes

Coulomb-gauge Newtonian vector potential for a compact smooth conserved current

Statement

Facts & Assumptions

Proof

Source notes

Biot-Savart representation for a compact smooth conserved current and its declared thin-loop limit

Statement

Facts & Assumptions

Proof

Source notes

Far-field dipole expansion of a compact conserved current with a uniform explicit remainder

Statement

Facts & Assumptions

Proof

Source notes

Force and torque on a declared ideal thin-wire loop in a prescribed external field

Statement

Facts & Assumptions

Proof

Source notes

5 · Examples, experiments and counterexamples

Sources

The piecewise constant cylinder field $e_{3} 1_{{ρ < a}}$ : distributional curl on the cylinder

The azimuthal line field $ρ^{- 1} e_{φ}$ : local integrability and distributional curl on the axis