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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

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 $R^{3}$ , and let $μ_{0} > 0$ , $c > 0$ , $ε_{0} = (μ_{0} c^{2})^{- 1}$ be the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Define, for $i \in {1, 2, 3}$ and $x \in R^{3}$ , $A_{i} (x) : = \frac{μ_{0}}{4 π} \int_{R^{3}} \frac{J_{i} (y)}{∣ x - y ∣} d y = μ_{0} N (J_{i}) (x), A : = (A_{1}, A_{2}, A_{3}),$ the componentwise Newtonian potentials of Newtonian potential of smooth compact data is smooth and solves the Poisson equation pointwise. Then:

(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.

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

Statement

Facts & Assumptions

Proof

Source notes

Depends on

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Sources