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How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Electrostatics in vacuum

1 · Prerequisites

2 · Summary

This page fixes the static and pure electrostatic reading of the vacuum Maxwell equations in one named inertial laboratory frame and develops the standard solutions from it. The static and pure electrostatic regimes and their fixed-time reading states what it means for the fields and sources to be time independent, reduces the smooth Maxwell equations to $div E = ρ / ε_{0}$ and $curl E = 0$ in the pure case, fixes the fixed-time spatial reading of every object, and records the static-to-distributional dictionary, including the stationary point charge as the fixed-time reading of the framework's weak point source. Scalar potential and the Poisson equation in a star-shaped electrostatic region constructs the scalar potential on a star-shaped region, proves the Poisson equation and the uniqueness up to an additive constant. The localized Newtonian potential and the ideal point-charge Coulomb field solves the problem for compact smooth and compact $L^{1}$ sources with the localised Newtonian potential, proves the decaying uniqueness statement, and states the distributional identities of the idealised point charge. Work of the electrostatic force equals charge times the potential drop and is path independent gives the work identity and path independence for a prescribed path in a prescribed field, with the external-coupling reading; Finite-source assembly energy equals the electric field energy proves that the finite field energy, the assembly integral and the symmetric double-kernel energy of a compact $C^{1}$ source agree; and Divergence of the point-charge field energy and the limits of mass inference records the divergence of the truncated point-charge field energy together with the explicit limits of the mass inference. Three mathematical prerequisites local to this page certify the distributional gradient of a locally integrable function smooth off one point (The distributional gradient of a locally integrable function smooth off one point), the far-field gradient and decay of the compactly supported Newtonian potential (The far-field gradient of a compactly supported Newtonian potential), and the membership of Euclidean balls, annuli and spheres in the bounded $C^{1}$ domain and compact embedded hypersurface classes (Euclidean balls, annuli and spheres as bounded smooth domains and hypersurfaces).

The companion page em-electrostatics-in-vacuum-examples carries the worked infinite-line and uniform-ball examples, including the logarithmic potential and energy divergence of the line and the finite energy of the ball.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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

The far-field gradient of a compactly supported Newtonian potential

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), the convention of the Lebesgue interface used below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 3$ , let $Φ$ be the fundamental solution of $- Δ$ on $R^{n}$ normalised by $- Δ T_{Φ} = δ_{0}$ (Fundamental solution for the positive operator minus Laplacian), so that for $z \neq 0$ $Φ (z) = \frac{∣ z ∣^{2 - n}}{(n - 2) ω_{n - 1}}, \nabla Φ (z) = - \frac{z}{ω_{n - 1} ∣ z ∣^{n}}, ∣ \nabla Φ (z) ∣ = \frac{1}{ω_{n - 1} ∣ z ∣^{n - 1}},$ let $f \in L^{1} (R^{n})$ be real, measurable and zero $λ_{n}$ -almost everywhere outside a closed ball ${\overline{B}}_{R} (0) = {x : ∣ x ∣ \leq R}$ of finite radius $R > 0$ (Open ball, closed ball and sphere in a metric space), and let $N f (x) = \int_{R^{n}} Φ (x - y) f (y) d λ_{n} (y)$ be the Newtonian potential (Newtonian potential of compactly supported data). Then:

(a) [Representation and differentiability off the far field] for every $x \in R^{n}$ with $∣ x ∣ > 2 R$ the integral $\int_{R^{n}} \nabla Φ (x - y) f (y) d λ_{n} (y)$ is absolutely convergent, the function $N f$ is differentiable at $x$ with $\nabla (N f) (x) = \int_{R^{n}} \nabla Φ (x - y) f (y) d λ_{n} (y),$ and $x \mapsto \nabla (N f) (x)$ is continuous on the open set ${x : ∣ x ∣ > 2 R}$ ;

(b) [Decay bound] for every $x$ with $∣ x ∣ > 2 R$ , $∣ \nabla (N f) (x) ∣ \leq \frac{2^{n - 1} ∥ f ∥_{1}}{ω_{n - 1} ∣ x ∣^{n - 1}}, ∥ f ∥_{1} : = \int_{R^{n}} ∣ f ∣ d λ_{n};$

(c) [Scope] the hypothesis is compact support together with $L^{1}$ -integrability; no boundedness of $f$ is assumed, and nothing is asserted at points with $∣ x ∣ \leq 2 R$ (where the kernel singularity may meet the support of $f$ ).

Facts & Assumptions

Given: The Axiom of Countable Choice, $n \geq 3$ , the fundamental solution $Φ$ of $- Δ$ with the displayed form off the origin, and a real $f \in L^{1} (R^{n})$ vanishing $λ_{n}$ -almost everywhere outside ${\overline{B}}_{R} (0)$ , $R > 0$ .

[F1]

The fundamental solution is $Φ (z) = ∣ z ∣^{2 - n} / ((n - 2) ω_{n - 1})$ for $z \neq 0$ (Fundamental solution for the positive operator minus Laplacian), and the Newtonian potential is $N f (x) = \int Φ (x - y) f (y) d λ_{n} (y)$ , defined where the integral is absolutely finite and agreeing with its almost-everywhere class wherever that holds (Newtonian potential of compactly supported data).

[F2]

Real powers are smooth where defined; the chain, product and algebra rules hold for derivatives, and a map with continuous partial derivatives is totally differentiable with the gradient as derivative (Continuity and derivatives of positive-base real powers, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, $C^{k}$ Euclidean maps and diffeomorphisms).

[F3]

The Euclidean norm satisfies the triangle inequality and the reverse triangle inequality, and is continuous and nonnegative (The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Open ball, closed ball and sphere in a metric space).

[F4]

Differentiation under the integral sign: if $f (x, t)$ is integrable in $x$ for every $t$ , differentiable in $t$ for almost every $x$ , with measurable $t$ -derivative dominated by one integrable $g$ uniformly in $t$ , then $t \mapsto \int f (x, t) d μ (x)$ is differentiable with the derivative integrated (Differentiation under the integral sign).

[F5]

Dominated convergence: pointwise almost-everywhere convergence under one integrable majorant passes to the integrals (Dominated convergence).

[F6]

Continuous maps on Euclidean spaces are Borel measurable; integrals are unchanged by almost-everywhere modification, and the nonnegative integral is monotone and homogeneous (Continuous functions on Euclidean spaces are Borel measurable, Monotonicity and nonnegative homogeneity of the nonnegative integral, Integral over a measurable subset, Integrable real and complex functions, and their integrals, Measure-null sets and almost-everywhere statements relative to a measure, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree). Sums, scalar multiples and absolute values of measurable real functions are measurable (Closure properties of measurable functions used by the integral); product measurability used here is justified in step 1.1.

Proof

technique · direct

1.1givenF1F2F6

Fix $x_{0}$ with $∣ x_{0} ∣ > 2 R$ , choose $0 < δ < ∣ x_{0} ∣ - 2 R$ , put $U : = B_{δ} (x_{0})$ and $c : = ∣ x_{0} ∣ - δ - R > 0$ . For measurable real $a$ , the function $a^{2}$ is measurable because ${a^{2} > b}$ is the whole space for $b < 0$ and ${∣ a ∣ > \sqrt{b}}$ for $b \geq 0$ ; for measurable real $a, b$ , the identity $a b = ((a + b)^{2} - (a - b)^{2}) / 4$ therefore gives measurable products by [F6]. Since $f$ vanishes almost everywhere on the complement of ${\overline{B}}_{R}$ , the function $f_{0} : = f 1_{{\overline{B}}_{R}}$ is measurable, integrable, agrees with $f$ almost everywhere and is pointwise zero outside ${\overline{B}}_{R}$ ; replacing $f$ by $f_{0}$ changes no integral of the form $\int Φ (x - y) f (y) d y$ or $\int \partial_{i} Φ (x - y) f (y) d y$ ([F6]), so we may assume $f$ vanishes pointwise off ${\overline{B}}_{R}$ . Differentiating the displayed formula of [F1] with the chain and algebra rules gives, for every $z \neq 0$ , $\nabla Φ (z) = \frac{1}{(n - 2) ω_{n - 1}} (2 - n) ∣ z ∣^{- n} z = - \frac{z}{ω_{n - 1} ∣ z ∣^{n}}$ and hence $∣ \nabla Φ (z) ∣ = \frac{1}{ω_{n - 1} ∣ z ∣^{n - 1}}$ ([F2]).

2.1step 1.1F3F6

For $x \in U$ and $y \in {\overline{B}}_{R}$ the triangle inequality of [F3] gives $∣ x - y ∣ \geq ∣ x_{0} - y ∣ - ∣ x - x_{0} ∣ \geq ∣ x_{0} ∣ - R - δ = c,$ so with $f$ vanishing off ${\overline{B}}_{R}$ the bounds $∣ Φ (x - y) f (y) ∣ \leq \frac{c^{2 - n} ∣ f (y) ∣}{(n - 2) ω_{n - 1}}, ∣ \partial_{i} Φ (x - y) f (y) ∣ \leq \frac{c^{1 - n} ∣ f (y) ∣}{ω_{n - 1}}$ hold for all $x \in U$ and $y \in R^{n}$ , the exponents being nonpositive because $n \geq 3$ and $c > 0$ ([F3]). Both majorants are integrable: $f \in L^{1}$ and the constants are finite. Hence $N f (x)$ and each integral $\int \partial_{i} Φ (x - y) f (y) d λ_{n} (y)$ exist absolutely for every $x \in U$ , and $g : = c^{1 - n} ∣ f ∣ / ω_{n - 1}$ is a single integrable majorant for all partial-derivative integrands on $U$ . Moreover each integrand $y \mapsto \partial_{i} Φ (x - y) f (y)$ is measurable, as follows: the kernel restricted to ${\overline{B}}_{R}$ is continuous and hence Borel measurable there; its extension by zero outside that closed ball is Borel (preimages of open sets are a relative Borel preimage, possibly with the complement of the ball added). Its product with $f$ is measurable by step 1.1. The same reasoning applies to the potential integrand.

3.1step 2.1F2F4

Fix $i \in {0, \dots, n - 1}$ and $x \in U$ ; since $U$ is open there is $ε > 0$ with $x + t e_{i} \in U$ for $∣ t ∣ < ε$ . Put $J : = (- ε, ε)$ and $h (y, t) : = Φ (x + t e_{i} - y) f (y)$ on $R^{n} \times J$ . For every $t \in J$ the section $y \mapsto h (y, t)$ is integrable by step 2.1; for $y \in {\overline{B}}_{R}$ the map $t \mapsto h (y, t)$ is differentiable on $J$ because $∣ x + t e_{i} - y ∣ \geq c$ there, with derivative $\partial_{i} Φ (x + t e_{i} - y) f (y)$ ; outside ${\overline{B}}_{R}$ set $h (y, t) = 0$ , whose derivative is zero; this derivative is measurable in $y$ by step 2.1; and it is dominated uniformly on $J$ by the integrable $g$ of step 2.1. The differentiation-under-the-integral-sign theorem [F4] therefore gives that $t \mapsto N f (x + t e_{i})$ is differentiable at $t = 0$ with derivative $\int \partial_{i} Φ (x - y) f (y) d λ_{n} (y)$ ; by the definition of the partial derivative [F2] this says that $\partial_{i} N f (x)$ exists and equals that integral.

4.1step 2.1step 3.1F1F2F5

Let $x_{m} \to x$ be a sequence in $U$ . For every $y$ with $f (y) \neq 0$ (so $y \in {\overline{B}}_{R}$ ) the points $x_{m} - y$ stay at distance at least $c$ from the origin by step 2.1, so $\partial_{i} Φ (x_{m} - y) f (y) \to \partial_{i} Φ (x - y) f (y)$ by continuity of the kernel off the origin ([F1], [F2]); the convergence also holds where $f (y) = 0$ , both sides being $0$ . All these integrands are dominated by the integrable $g$ of step 2.1, so dominated convergence [F5] gives $\int \partial_{i} Φ (x_{m} - y) f (y) d λ_{n} (y) \to \int \partial_{i} Φ (x - y) f (y) d λ_{n} (y) .$ By step 3.1 the left-hand side is $\partial_{i} N f (x_{m})$ , so each partial derivative of $N f$ is sequentially continuous on $U$ , hence continuous by Metric continuity characterisations, with countable choice for the sequential converse under the assumed ${A C}_{ω}$ .

5.1step 4.1F2

By steps 3.1 and 4.1 the function $N f$ has all first partial derivatives at every point of $U$ and each of them is continuous on $U$ ; the characterization of total differentiability by continuous partial derivatives [F2] therefore makes $N f$ differentiable at every $x \in U$ , with $\nabla (N f) (x)$ the vector of the partial derivatives, namely $\nabla (N f) (x) = \int_{R^{n}} \nabla Φ (x - y) f (y) d λ_{n} (y),$ the vector integral being absolutely convergent by step 2.1. As $x_{0}$ with $∣ x_{0} ∣ > 2 R$ was arbitrary, this proves (a) throughout the open far field ${∣ x ∣ > 2 R}$ .

6.1step 1.1step 5.1F3F6∎

For (b), let $∣ x ∣ > 2 R$ and let $y$ satisfy $f (y) \neq 0$ , so $∣ y ∣ \leq R$ ; the reverse triangle inequality gives $∣ x - y ∣ \geq ∣ x ∣ - R > ∣ x ∣ / 2$ ([F3]), hence by step 1.1 $∣ \nabla Φ (x - y) ∣ = \frac{1}{ω_{n - 1} ∣ x - y ∣^{n - 1}} \leq \frac{2^{n - 1}}{ω_{n - 1} ∣ x ∣^{n - 1}} .$ Integrating against $∣ f ∣$ and using the estimate of step 5.1 gives $∣ \nabla (N f) (x) ∣ \leq \int ∣ \nabla Φ (x - y) f (y) ∣ d λ_{n} (y) \leq \frac{2^{n - 1} ∥ f ∥_{1}}{ω_{n - 1} ∣ x ∣^{n - 1}}$ , which is (b). The argument used only the $L^{1}$ hypothesis and the a.e. compact support, never boundedness of $f$ , and it makes no assertion at points with $∣ x ∣ \leq 2 R$ , where the singularity of $Φ$ can meet the support of $f$ ; this is exactly the scope recorded in (c).

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

The distributional gradient of a locally integrable function smooth off one point

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) for the Lebesgue and product-measure interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Use the explicit one-based labels $x_{k} : = x (k - 1)$ for $k = 1, \dots, n$ ; here $\partial_{k}$ denotes differentiation in $x (k - 1)$ , corresponding to $\partial_{k - 1}$ in the zero-based supplier convention of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ . Let $n \geq 2$ , let $p \in R^{n}$ , and let $u \in L_{l o c}^{1} (R^{n})$ be continuous on $R^{n} ∖ {p}$ and of class $C^{1}$ there, with each classical partial derivative $\partial_{i} u : R^{n} ∖ {p} \to R$ (extended by $0$ at $p$ ) again in $L_{l o c}^{1} (R^{n})$ (A locally integrable function on $R^{n}$ ). Then for every $i = 1, \dots, n$ , with $T_{u} \in D^{'} (R^{n})$ the regular distribution of $u$ (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions) and $\partial_{i}$ the distributional derivative (Distributional derivative), $\partial_{i} T_{u} = T_{\partial_{i} u} in D^{'} (R^{n}),$ that is, $⟨ \partial_{i} T_{u}, ψ ⟩ = \int_{R^{n}} \partial_{i} u ψ d λ_{n}$ for every $ψ \in C_{c}^{\infty} (R^{n})$ (Test function space d of an open set). The value assigned to $\partial_{i} u$ at the single point $p$ is immaterial, because a singleton is Lebesgue null (A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ , A nonnegative integral over a null set vanishes, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree). Both hypotheses are used: $u$ must be locally integrable and continuous away from $p$ , and each classical partial must be locally integrable; the lemma is asserted for one exceptional point, not for an exceptional set of positive surface measure.

Facts & Assumptions

Given: The Axiom of Countable Choice, $n \geq 2$ , $p \in R^{n}$ , and a function $u \in L_{l o c}^{1} (R^{n})$ that is continuous on $R^{n} ∖ {p}$ and $C^{1}$ there, with every classical partial $\partial_{i} u$ (extended by $0$ at $p$ ) in $L_{l o c}^{1} (R^{n})$ . Fix an index $i \in {1, \dots, n}$ and a test function $ψ \in C_{c}^{\infty} (R^{n})$ .

[F1]

For $f \in L_{l o c}^{1} (R^{n})$ the regular distribution is $⟨ T_{f}, φ ⟩ = \int_{R^{n}} f φ d λ_{n}$ , and the embedding $f \mapsto T_{f}$ is complex-linear and injective modulo almost-everywhere equality (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions).

[F2]

The distributional derivative is defined by $⟨ \partial_{i} T, φ ⟩ = - ⟨ T, \partial_{i} φ ⟩$ , and the pairing is bilinear with no conjugation (Distributional derivative).

[F3]

The test space $C_{c}^{\infty} (R^{n})$ consists of smooth compactly supported functions; each such $ψ$ is bounded and its support is a compact subset of $R^{n}$ (Test function space d of an open set, $C^{k}$ Euclidean maps and diffeomorphisms).

[F4]

A singleton ${p}$ is Lebesgue null, the integral over a measurable null set vanishes, and changing an integrand on a null set does not change its integral or its $L_{l o c}^{1}$ class (A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ , A nonnegative integral over a null set vanishes, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree, Measure-null sets and almost-everywhere statements relative to a measure).

[F5]

Measurable real functions are closed under pointwise-defined sums, scalar multiples and absolute values; a nonnegative measurable function may be restricted by a measurable-set indicator, and increasing limits are measurable (Closure properties of measurable functions used by the integral). Integrability means finiteness of the integral of the absolute value, and the nonnegative integral is monotone and homogeneous (Integral over a measurable subset, Integrable real and complex functions, and their integrals, Monotonicity and nonnegative homogeneity of the nonnegative integral). Measurability of the particular products in this proof is established locally in step 1.1.

[F6]

$f \in L_{l o c}^{1} (R^{n})$ means that $f$ is measurable and $\int_{B (x, r)} ∣ f ∣ d λ_{n} < \infty$ for every Euclidean ball (A locally integrable function on $R^{n}$ , Open ball, closed ball and sphere in a metric space); the Lebesgue and product-measure interfaces used below carry the declared Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F7]

Under the standard identification $R^{n} = R^{n - 1} \times R$ , the Borel sigma-algebras satisfy $B (R^{n - 1}) \otimes B (R) = B (R^{n})$ , and $λ_{n}$ is the completion of the product measure $λ_{n - 1} \times λ_{1}$ (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, The Euclidean Lebesgue measure is the completion of the product of the factor Lebesgue measures).

[F8]

Fubini and Tonelli hold for the completed product of two sigma-finite measure spaces: for $G \in L^{1} (λ_{n})$ the sections are integrable almost everywhere and the repeated integrals agree with the integral against $λ_{n}$ (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

[F9]

A coordinate permutation of $R^{n}$ is a linear isometry for the Euclidean norm, hence an orthogonal operator, and every orthogonal operator has $∣ \det T ∣ = 1$ ; for a $C^{1}$ diffeomorphism $T$ and $F \in L^{1} (λ_{n})$ , $\int F d λ_{n} = \int F \circ T ∣ \det D T ∣ d λ_{n}$ (Linear map between vector spaces over the same field, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces, Orthogonal and unitary operators form groups, and their determinants have modulus one, A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F10]

Integration by parts holds on a compact interval for functions differentiable at every point whose derivatives are integrable, continuous functions on a compact interval are Darboux integrable, and a bounded Borel Riemann integrable function on a box has equal Riemann and Lebesgue integrals (If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ , A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, Riemann–Lebesgue comparison for distribution test integrands).

[F11]

The classical partial derivative $\partial_{i} u$ is the derivative of the one-variable function obtained by freezing all other coordinates (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

Proof

technique · direct

1.1givenF3F4F5F6

Set $F : = \partial_{i} u \cdot ψ$ and $H : = u \cdot \partial_{i} ψ$ on $R^{n}$ . By [F4] replace $u$ and each $\partial_{i} u$ at $p$ by $0$ without changing their local integrability classes or integrals. On $O : = R^{n} ∖ {p}$ these functions, and the real and imaginary parts of $F, H$ , are continuous. For any one of these real functions $v$ extended by $0$ at $p$ and every real $a$ , the set ${v > a} \cap O$ is open in $O$ , hence open in $R^{n}$ ; the full superlevel set is this open set with ${p}$ added when $0 > a$ . It is therefore Borel, so the real and imaginary parts of $F, H$ are Borel measurable. Choose a ball containing the compact support of $ψ$ and its derivatives. On it $∣ F ∣ \leq ∥ ψ ∥_{\infty} ∣ \partial_{i} u ∣$ and $∣ H ∣ \leq ∥ \partial_{i} ψ ∥_{\infty} ∣ u ∣$ , and both products vanish outside it; monotonicity and homogeneity yield $F, H \in L^{1} (λ_{n})$ .

2.1step 1.1F5F9

Let $π : R^{n} \to R^{n}$ be the coordinate permutation moving the $i$ -th coordinate to the last position, $π (x) = (x_{1}, \dots, x_{i - 1}, x_{i + 1}, \dots, x_{n}, x_{i})$ . Then $∥ π (x) ∥^{2} = \sum_{j} x_{j}^{2} = ∥ x ∥^{2}$ , so $π$ is a linear isometry, hence an orthogonal operator with $∣ \det D π ∣ = ∣ \det π ∣ = 1$ ([F9]); $π$ is a $C^{1}$ diffeomorphism of $R^{n}$ onto itself. Applying the change-of-variables identity of [F9] to $F$ and to $H$ , and using $∣ \det D π ∣ = 1$ , gives $\int_{R^{n}} F d λ_{n} = \int_{R^{n}} F \circ π^{- 1} d λ_{n}, \int_{R^{n}} H d λ_{n} = \int_{R^{n}} H \circ π^{- 1} d λ_{n} .$ The compositions $F \circ π^{- 1}$ and $H \circ π^{- 1}$ are Borel and in $L^{1} (λ_{n})$ because $\int ∣ F \circ π^{- 1} ∣ d λ_{n} = \int ∣ F ∣ d λ_{n} < \infty$ and likewise for $H$ .

3.1step 2.1F7F8

Identify $R^{n}$ with $R^{n - 1} \times R$ by the standard ordering, so that a point with last coordinate $t$ and remaining coordinates $y$ is written $(y, t)$ ; the permutation of step 2.1 then satisfies $π (x) = (y (x), x_{i})$ where $y (x)$ lists the coordinates other than the $i$ -th. Define $G (y, t) : = F (x (y, t))$ and $K (y, t) : = H (x (y, t))$ , where $x (y, t) \in R^{n}$ is the point with $i$ -th coordinate $t$ and remaining coordinates $y$ . Then $G = F \circ π^{- 1}$ and $K = H \circ π^{- 1}$ under the identification, so step 2.1 reads $\int F d λ_{n} = \int G d λ_{n}$ and $\int H d λ_{n} = \int K d λ_{n}$ ; by [F7] and [F8] these equal the repeated integrals $\int_{R^{n - 1}} (\int_{R} G (y, t) d t) d y$ and $\int_{R^{n - 1}} (\int_{R} K (y, t) d t) d y$ , the inner sections being integrable for almost every $y$ .

4.1step 3.1F3F4F10F11

Fix $y \in R^{n - 1}$ such that the line $L_{y} = {x (y, t) : t \in R}$ does not pass through $p$ ; the excluded set of $y$ is the single point $y^{*}$ listing the coordinates of $p$ other than $p_{i}$ , which is a degenerate box and hence $λ_{n - 1}$ -null ([F4]). For such $y$ , the map $t \mapsto u (x (y, t))$ is $C^{1}$ on $R$ with derivative $t \mapsto \partial_{i} u (x (y, t))$ ([F11]), and $t \mapsto ψ (x (y, t))$ is smooth with compact support ([F3]). Choose $a < b$ with the support of $t \mapsto ψ (x (y, t))$ contained in $(a, b)$ . Then $v (t) : = u (x (y, t)) ψ (x (y, t))$ is differentiable at every point of $[a, b]$ with continuous derivative $v^{'} = \partial_{i} u ψ + u \partial_{i} ψ$ there, so $v^{'}$ is integrable in the Darboux sense ([F10]); applying the real integration-by-parts formula of [F10] to the real and imaginary parts of both factors on $[a, b]$ , combining the resulting identities bilinearly and using that the boundary terms vanish gives $\int_{R} G (y, t) d t = - \int_{R} K (y, t) d t,$ where the Darboux integrals have been replaced by the equal Lebesgue integrals of the continuous compactly supported integrands ([F10]).

5.1step 3.1step 4.1F1F2F5∎

The equality of step 4.1 holds for every $y$ outside the null set ${y^{*}}$ ([F4]), hence almost everywhere, and the section functions are integrable there by step 3.1. Integrating the identity over $R^{n - 1}$ and using the linearity of the integral gives $\int F d λ_{n} = \int G d λ_{n} = - \int K d λ_{n} = - \int H d λ_{n}$ . Therefore, by [F1] and [F2], $⟨ \partial_{i} T_{u}, ψ ⟩ = - ⟨ T_{u}, \partial_{i} ψ ⟩ = - \int_{R^{n}} u \partial_{i} ψ d λ_{n} = - \int_{R^{n}} H d λ_{n} = \int_{R^{n}} F d λ_{n} = \int_{R^{n}} \partial_{i} u ψ d λ_{n} = ⟨ T_{\partial_{i} u}, ψ ⟩ .$ Since $ψ \in C_{c}^{\infty} (R^{n})$ was arbitrary, $\partial_{i} T_{u} = T_{\partial_{i} u}$ in $D^{'} (R^{n})$ , as claimed.

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

Euclidean balls, annuli and spheres as bounded smooth domains and hypersurfaces

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), the convention of the boundary-chart and surface-measure interfaces used below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Use the explicit coordinate labels $x_{k} : = x (k - 1)$ for $k = 1, 2, 3$ ; these relabel the zero-based coordinates of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . On $R^{3}$ with its Euclidean structure (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Open ball, closed ball and sphere in a metric space) let $0 < ε < R$ and put $B_{R} : = {x : ∣ x ∣ < R}, {\overline{B}}_{R} : = {x : ∣ x ∣ \leq R}, A_{ε, R} : = {x : ε < ∣ x ∣ < R}, S_{r} : = {x : ∣ x ∣ = r} .$ Then:

(a) [Domains] $B_{R}$ and $A_{ε, R}$ are bounded $C^{1}$ domains in the sense of Bounded C1 domains and their outward normals; their boundaries are $S_{R}$ , respectively $S_{ε} \cup S_{R}$ , and the corresponding outward unit normals are $ν (x) = \frac{x}{∣ x ∣} (x \in S_{R}), ν (x) = - \frac{x}{∣ x ∣} (x \in S_{ε}, boundary of A_{ε, R});$

(b) [Spheres] each $S_{r}$ is a compact embedded $C^{1}$ (indeed $C^{\infty}$ ) hypersurface in the sense of Surface integration on compact C1 hypersurfaces with two-sided unit normals $\pm x / ∣ x ∣$ , and the boundary components $S_{ε}, S_{R}$ , each subdivided into six compact signed-maximal-coordinate graph faces $F_{r, k, σ} : = {x \in S_{r} : σ x_{k} \geq ∣ x_{j} ∣ for all j}$ ( $k = 1, 2, 3$ , $σ = \pm 1$ ), with the compact edge set consisting of their seams $σ x_{k} = ∣ x_{j} ∣$ for $j \neq k$ , give a finite piecewise $C^{1}$ presentation of $A_{ε, R}$ in the sense of Specified finite piecewise C1 boundary presentations;

(c) [Consequences used downstream] the divergence theorem and Green's first identity of Divergence on a bounded C1 Euclidean domain and First Green identity apply on $B_{R}$ and on $A_{ε, R}$ with the outward normals of (a), and $S_{r}$ carries a finite Borel surface measure with the standard graph charts.

Facts & Assumptions

Given: The Axiom of Countable Choice, reals $0 < ε < R$ , the sets $B_{R}, {\overline{B}}_{R}, A_{ε, R}, S_{r}$ above, and the Euclidean norm $N (x) = ∣ x ∣$ on $R^{3}$ .

[F1]

The Euclidean norm is continuous, and a map between metric spaces is continuous exactly when preimages of open sets are open (The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , Metric continuity characterisations, with countable choice for the sequential converse, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form).

[F2]

The interior, closure, and boundary of a subset of a metric space are the sets determined by its open balls; $B_{R}$ is the open ball of radius $R$ and ${\overline{B}}_{R}$ its closed ball (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, Open ball, closed ball and sphere in a metric space).

[F3]

Real powers are smooth where defined, with the chain, product and algebra rules for derivatives and the converse characterization of differentiable maps by continuous partial derivatives (Continuity and derivatives of positive-base real powers, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, $C^{k}$ Euclidean maps and diffeomorphisms).

[F4]

A bounded $C^{1}$ domain is a nonempty bounded open set whose boundary is locally, after a rigid coordinate change, a graph $z = h (y)$ of a $C^{1}$ function with the set locally the subgraph; the outward normal in these coordinates is $ν = (- D h, 1) / \sqrt{1 + ∣ D h ∣^{2}}$ , transported by the orthogonal coordinate map, and $∣ \cdot ∣$ enters through the Euclidean inner product (Bounded C1 domains and their outward normals, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F5]

A subset of $R^{3}$ is compact exactly when it is closed and bounded (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, Open cover, subcover, compact metric space, and compact subset of a metric space).

[F6]

The chart surface integral on a compact embedded $C^{1}$ hypersurface is a finite Borel measure independent of charts and partitions; in graph coordinates its density is $\sqrt{1 + ∣ D h ∣^{2}}$ , and outward unit normals agree on chart overlaps (Surface integration on compact C1 hypersurfaces, Chart and partition independence of surface measure).

[F7]

A finite piecewise $C^{1}$ presentation of a bounded open set consists of compact faces covering its boundary, each inside a regular $C^{1}$ patch and carrying the actual outward normal off the compact edge set $E$ , with $S_{j} \cap E$ of surface measure zero, and $E$ containing the relative face boundaries and all overlaps (Specified finite piecewise C1 boundary presentations).

[F8]

For a bounded Borel function on a compact parameter rectangle the iterated planar integral of its indicator vanishes when almost every section is a null set, and coordinate hyperplanes and degenerate boxes are Lebesgue null; the Borel product sigma-algebra of $R^{2}$ is the Euclidean one and the product measure agrees with $λ_{2}$ there (Fubini's theorem for L^1 functions on a sigma-finite product, A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ , The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}).

[F9]

The nonnegative integral is monotone and homogeneous and the integral of an indicator is the measure of the set (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F10]

On a bounded $C^{1}$ domain the divergence theorem holds for $C^{1}$ fields up to the closure and Green's first identity holds for real $u \in C^{2} (\overline{Ω})$ , $v \in C^{1} (\overline{Ω})$ , both with the outward normal on each boundary component (Divergence on a bounded C1 Euclidean domain, First Green identity).

Proof

technique · direct

1.1givenF1F2

Because $N$ is continuous and $(0, R)$ and $(ε, R)$ are open, $B_{R} = N^{- 1} ((- \infty, R))$ and $A_{ε, R} = N^{- 1} ((ε, R))$ are open, nonempty (they contain the origin, respectively a point of norm $(ε + R) / 2$ ) and bounded, being contained in the closed ball of radius $R$ ([F1], [F2]). If $∣ x ∣ > R$ , continuity of $N$ gives a ball about $x$ on which $∣ y ∣ > R$ , so no point outside ${\overline{B}}_{R}$ is a limit of points of $B_{R}$ ; if $∣ x ∣ = R$ , the points $(1 - 1 / n) x$ lie in $B_{R}$ and converge to $x$ ; hence the closure of $B_{R}$ is ${\overline{B}}_{R}$ and, since $B_{R}$ is open, its boundary is $S_{R}$ ([F1], [F2]). The same argument with $N^{- 1} ((ε, R))$ , together with the identification of its closure with ${x : ε \leq ∣ x ∣ \leq R}$ , gives that the boundary of $A_{ε, R}$ is $S_{ε} \cup S_{R}$ , and both sets have nonempty interior equal to themselves because they are open.

2.1step 1.1F3F4

Fix $k \in {1, 2, 3}$ and $σ \in {\pm 1}$ , let $z = σ x_{k}$ and let $y$ collect the two remaining coordinates in a fixed order, labelled $y = (y_{1}, y_{2})$ ; the map $x \mapsto (y, z)$ is a rigid change of coordinates (a permutation with one sign change). For $r > 0$ put $U_{r, k, σ} : = {x : σ x_{k} > r / (2 \sqrt{3})}$ and $h (y) : = \sqrt{R^{2} - ∣ y ∣^{2}}$ , which is $C^{\infty}$ on the open set $∣ y ∣ < R$ with $D h (y) = - y / h (y)$ ([F3]). On $U_{R, k, σ} \cap {∣ y ∣ < R}$ one has $z > 0$ and $∣ x ∣^{2} < R^{2} ⟺ z^{2} < R^{2} - ∣ y ∣^{2} ⟺ z < h (y),$ so $S_{R} \cap U_{R, k, σ}$ is the graph $z = h (y)$ over the open parameter set $V_{R} : = {y : ∣ y ∣ < R, h (y) > R / (2 \sqrt{3})}$ and $B_{R} \cap U_{R, k, σ} \cap {∣ y ∣ < R}$ is its subgraph; this is the local presentation required of a bounded $C^{1}$ domain ([F4]). The graph normal of [F4] is $\frac{(- D h (y), 1)}{\sqrt{1 + ∣ D h (y) ∣^{2}}} = \frac{(y, h (y))}{R},$ because $- D h = y / h$ and $1 + ∣ D h ∣^{2} = R^{2} / h^{2}$ ; transporting back through the rigid coordinate map gives $ν (x) = x / R = x / ∣ x ∣$ on $S_{R}$ . The six coordinate neighbourhoods $U_{R, k, σ}$ cover the sphere $S_{R}$ : for $x \neq 0$ some coordinate satisfies $∣ x_{k} ∣ \geq ∣ x ∣ / \sqrt{3}$ , so with $σ : = sign (x_{k})$ one has $σ x_{k} = ∣ x_{k} ∣ \geq R / \sqrt{3} > R / (2 \sqrt{3})$ whenever $∣ x ∣ \geq R$ , and in particular the sphere $S_{R}$ is covered by the six graph patches.

3.1step 2.1F3F4

Fix again $k, σ$ and put $h_{ε} (y) = \sqrt{ε^{2} - ∣ y ∣^{2}}$ on $∣ y ∣ < ε$ . In the coordinates $(y, z)$ of step 2.1 the complement of the closed ball satisfies $z > h_{ε} (y)$ inside the patch $U_{ε, k, σ}$ with $∣ y ∣ < ε$ , and the second rigid change $w = - z$ turns this into the subgraph $w < - h_{ε} (y)$ of the smooth function $g (y) : = - h_{ε} (y)$ with $D g (y) = y / h_{ε} (y)$ ([F3]). The graph normal of [F4] in the $(y, w)$ coordinates is $\frac{(- D g (y), 1)}{\sqrt{1 + ∣ D g (y) ∣^{2}}} = \frac{(- y, h_{ε} (y))}{ε},$ and transporting back through $w = - σ x_{k}$ gives $ν (x) = - x / ε = - x / ∣ x ∣$ on $S_{ε}$ . Near a point of $S_{ε}$ the ball $B_{R}$ occupies a full neighborhood because $∣ x ∣ = ε < R$ , so $A_{ε, R} = B_{R} ∖ {\overline{B}}_{ε}$ is locally exactly the subgraph just described, while near a point of $S_{R}$ it is locally the subgraph of $h$ from step 2.1 with normal $x / ∣ x ∣$ . Hence $A_{ε, R}$ is a bounded $C^{1}$ domain with boundary $S_{ε} \cup S_{R}$ and the stated outward normals ([F4]).

4.1step 2.1step 3.1F1F3F5F6

Each $S_{r}$ is closed, being the preimage of the closed set ${r}$ under the continuous norm, and bounded, hence compact ([F1], [F5]). The restrictions of the six maps $y \mapsto (y, h_{r} (y))$ from steps 2.1 and 3.1, composed with the inverse rigid coordinate maps, are regular injective $C^{\infty}$ parametrizations of the open patches $S_{r} \cap U_{r, k, σ}$ with $C^{\infty}$ transitions (each transition is the coordinate projection from one graph chart composed with the other smooth graph parametrization) ([F3]). Therefore each $S_{r}$ is a compact embedded $C^{1}$ hypersurface in the sense of [F6], and the two graph normal computations of steps 2.1 and 3.1 exhibit both unit normals $\pm x / ∣ x ∣$ : the positive sign from the ball chart and the negative sign from the complement chart, continuously across overlaps.

5.1step 4.1F5

For $r = ε, R$ define the twelve compact faces $F_{r, k, σ} = {x \in S_{r} : σ x_{k} \geq ∣ x_{j} ∣ for all j}$ , $k = 1, 2, 3$ , $σ = \pm 1$ . They cover $S_{r}$ : choose $k$ with $∣ x_{k} ∣ = \max_{j} ∣ x_{j} ∣$ ; then $x_{k} \neq 0$ and $σ = sign (x_{k})$ gives $σ x_{k} \geq ∣ x_{j} ∣$ for all $j$ . Each $F_{r, k, σ}$ is closed in the compact $S_{r}$ , hence compact, and on it $σ x_{k} \geq r / \sqrt{3} > r / (2 \sqrt{3})$ , so it is contained in the single regular patch $U_{r, k, σ}$ of steps 2.1-3.1 with parameter set contained in the compact set ${∣ y ∣ \leq r \sqrt{2 / 3}} \subset {∣ y ∣ < r}$ ([F5]).

6.1step 5.1F6F7F8F9

Let $E$ be the union over $r = ε, R$ of the seams ${x \in S_{r} : σ x_{k} = ∣ x_{j} ∣ for some j \neq k, σ = \pm 1}$ , a finite union of closed subsets of the compact spheres, hence compact. Every relative boundary point of a face $F_{r, k, σ}$ has $σ x_{k} = ∣ x_{j} ∣$ for some $j$ , and every overlap $F_{r, k, σ} \cap F_{r, k^{'}, σ^{'}}$ with $(k, σ) \neq (k^{'}, σ^{'})$ lies in such a seam, so $E$ contains all relative face boundaries and all overlaps ([F7]). To see that $F_{r, k, σ} \cap E$ has zero surface measure, work in the chart $z = σ x_{k} = h_{r} (y)$ of $F_{r, k, σ}$ : a seam involving the maximal coordinate reads either $∣ y_{1} ∣ = h_{r} (y)$ or $∣ y_{2} ∣ = h_{r} (y)$ . The first is $y_{1}^{2} = (r^{2} - y_{2}^{2}) / 2$ , the second $y_{2}^{2} = (r^{2} - y_{1}^{2}) / 2$ . For either case take the other coordinate as the fixed section parameter: the seam meets the vertical line in at most two points, a finite union of singletons and hence a $λ_{1}$ -null set ([F8]); Fubini applied to the bounded Borel indicator on a containing rectangle gives the planar measure of these parameter sets as zero ([F8]). The other seams in $E$ equate the absolute values of the two nonmaximal coordinates; in this face chart they lie on $y_{1} = \pm y_{2}$ , whose sections are again finite. The same argument gives planar measure zero for them. Thus the full finite union of seams meeting this face is planar null. The graph density $\sqrt{1 + ∣ D h_{r} ∣^{2}} = r / h_{r}$ is continuous and bounded on the compact parameter set ${∣ y ∣ \leq r \sqrt{2 / 3}}$ , so integrating it against the indicator of a planar null set gives zero ([F9]); by the chart-independence of the surface measure this says exactly that $S_{j} \cap E$ has zero surface measure in each face, as required by [F7].

7.1step 2.1step 3.1step 4.1step 6.1F6F7F10∎

Statement (a) is steps 2.1 and 3.1; statement (b) is steps 4.1, 5.1 and 6.1, the twelve compact faces with their null compact seam set satisfying exactly the requirements of [F7] for the annulus $A_{ε, R}$ , whose exposed faces carry the outward normals computed in steps 2.1 and 3.1 off $E$ . Finally, (c): the divergence theorem and Green's first identity of [F10] apply on $B_{R}$ and on $A_{ε, R}$ because both are bounded $C^{1}$ domains with the outward normals of (a), and $S_{r}$ carries a finite Borel surface measure with the standard graph charts by [F6].

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

The static and pure electrostatic regimes and their fixed-time reading

Physical scope

The static and pure electrostatic idealizations of classical electromagnetism in one named inertial frame of the laboratory region $U ≅ I \times Ω$ : fields and sources whose time derivatives vanish there, with the reduction of the smooth Maxwell equations to $div E = ρ / ε_{0}$ , $curl E = 0$ in the pure case, together with the fixed-time spatial reading and the static-to-distributional dictionary for locally integrable time-independent objects, including the stationary point-charge reading. The conditions are frame-relative and are imposed, not derived; no dynamical stability, equilibrium support, boost behaviour or empirical content is asserted.

Definition

Let $U ≅ I \times Ω$ be the laboratory region of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the units, constants and symbols of Framework-local SI units, the vacuum constants and the exact/measured status of their values, let $E : Ω \times I \to R^{3}$ and $B : Ω \times I \to R^{3}$ be fields of The electric and magnetic fields as fixed-time spatial vector fields and let $(ρ, J)$ be source fields of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current on $Ω \times I$ , all of class $C^{1}$ (with every additional derivative appearing below existing and continuous); $\partial_{t}$ is the partial time derivative at fixed position of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ .

(a) [Static regime] The static (time-independent) electromagnetic regime on $Ω \times I$ is the requirement $\partial_{t} E = 0, \partial_{t} B = 0, \partial_{t} ρ = 0, \partial_{t} J = 0 on Ω \times I .$

(b) [Pure electrostatics] The pure electrostatic regime is the static regime together with $J = 0$ and $B = 0$ on $Ω \times I$ .

(c) [Reduced equations] If the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches holds in the static regime, then the four Maxwell equations reduce to $div E = ρ / ε_{0}, curl E = 0, div B = 0, curl B = μ_{0} J$ pointwise on $Ω \times I$ , with $div$ and $curl$ of Divergence and curl of a $C^{1}$ vector field applied at each fixed time; in the pure electrostatic regime the reduction is exactly the pair $div E = ρ / ε_{0}$ , $curl E = 0$ .

(d) [Fixed-time reading] In the static regime $E, B, ρ, J$ do not depend on $t$ , so each is the pullback along $(y, t) \mapsto y$ of a single map on $Ω$ ; this fixed-time reading (the spatial field or source) is the object analysed by the electrostatic statements of this page, and no other time dependence is introduced. The equations of (c) are equations between those spatial maps.

(e) [Static-to-distributional reading] Let fields $E, B$ and volume densities $ρ, J$ be locally integrable and time independent on $Ω \times I$ , and let $Ω$ be open. Then, under the ${A C}_{ω}$ convention of the cited measure and distribution interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the weak equations of the solution predicate of Distributional Maxwell equations with point and stationary-sheet sources(e) for these time-independent objects hold if and only if the four spatial equations $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = 0$ , $curl B = μ_{0} J$ hold in $D^{'} (Ω)$ ; the passage between them is the integration in $t$ of the pairings against $C_{c}^{\infty} (I \times Ω)$ tests, justified by Fubini's theorem for L^1 functions on a sigma-finite product after identifying fields and densities with their regular distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions, Distributional derivative). In particular a stationary point charge at $x_{0} \in Ω$ with charge parameter $q \in R$ is the point source of Distributional Maxwell equations with point and stationary-sheet sources(b) with constant trajectory $z (t) \equiv x_{0}$ , whose pairing is $ψ \mapsto q \int_{I} ψ (x_{0}, t) d t$ and whose fixed-time reading is the spatial point source $η \mapsto q η (x_{0})$ (Dirac delta and its derivatives).

(f) [Conventions and scope] The regime is imposed in the one inertial frame of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields; no claim is made that time independence survives a change of inertial description, and where a frame change is used the transformed fields must be rechecked. The static regime is an adopted idealization, not a dynamical stability theorem: static charged matter requires external support or an adopted equilibrium model. No equation is asserted here for singular sources beyond the dictionary of (e), and no empirical claim is made.

Facts & Assumptions

Given: The Axiom of Countable Choice, the laboratory region $U ≅ I \times Ω$ with $I$ a nonempty open interval and $Ω \subseteq R^{3}$ nonempty open, jointly $C^{1}$ fields $E, B$ and $C^{1}$ sources $(ρ, J)$ as in The electric and magnetic fields as fixed-time spatial vector fields and Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, the smooth branch of the vacuum Maxwell equations, and for clause (e) locally integrable time-independent fields and densities on $Ω \times I$ together with $ψ \in C_{c}^{\infty} (U)$ .

[F1]

The laboratory region, its fixed-time slices, its fields and its units are those of the primitives: the chart restriction $x (U) = {(c t, y)}$ , the fixed-time readings, and the identification $U ≅ I \times Ω$ (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields); the SI constants with $ε_{0} = (μ_{0} c^{2})^{- 1}$ , $c > 0$ , $μ_{0} > 0$ and the units of $E, B, ρ, J$ (Framework-local SI units, the vacuum constants and the exact/measured status of their values); a $C^{k}$ field on $Ω \times I$ with its polar/axial conventions and no asserted values at singular support (The electric and magnetic fields as fixed-time spatial vector fields); smooth source fields $(ρ, J)$ and their branches (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current).

[F2]

The smooth Maxwell branch asserts $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ pointwise for jointly $C^{1}$ fields with the additional derivatives continuous, and the weak branch asserts the same four pairings against $C_{c}^{\infty} (U)$ tests for locally integrable fields (The microscopic vacuum Maxwell equations: smooth and weak branches).

[F3]

The weak solution predicate of the singular-source formulation is: for locally integrable $E, B$ and a source pair with volume part $(ρ_{v}, J_{v})$ and singular part of point or stationary-sheet type, $E, B$ satisfy the weak Maxwell equations when the four distributional pairings $⟨ div E - ρ / ε_{0}, ψ ⟩$ , $⟨ div B, ψ ⟩$ , $⟨ curl E + \partial_{t} B, ψ ⟩$ , $⟨ curl B - μ_{0} J - c^{- 2} \partial_{t} E, ψ ⟩$ vanish for every $ψ \in C_{c}^{\infty} (U)$ ; point sources with $C^{1}$ trajectory $z$ proper into $U$ have $⟨ ρ_{p t}, ψ ⟩ = q \int ψ (z (t), t) d t$ and $⟨ J_{p t}, ψ ⟩ = q \int z^{'} (t) ψ (z (t), t) d t$ (Distributional Maxwell equations with point and stationary-sheet sources(b),(e)).

[F4]

Divergence and curl are the coordinate expressions of Divergence and curl of a $C^{1}$ vector field evaluated at fixed time, and $\partial_{t}$ is the partial derivative at fixed position (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

A real function on an interval with everywhere-vanishing derivative is constant (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ), and for a differentiable function with integrable derivative on $[a, b]$ , $\int_{a}^{b} G^{'} = G (b) - G (a)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[F6]

Locally integrable functions are identified with their regular distributions, $⟨ \partial_{μ} T, ψ ⟩ = - ⟨ T, \partial_{μ} ψ ⟩$ , and test functions are smooth with compact support in $U$ (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions, Distribution, Distributional derivative, Test function space d of an open set).

[F7]

Under Countable Choice every measurable function has a Borel representative modulo null sets, the Euclidean Borel sigma-algebra of $R^{4}$ is the product of those of $R^{3}$ and $R$ , and the corresponding product measure agrees with Lebesgue measure on Borel sets; Fubini applies to integrable functions on such products, and a continuous compactly supported one-variable integrand has equal Riemann and Lebesgue integrals (A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra, $L (R^{n})$ is exactly the completion of the restriction of $λ_{n}$ to the Borel sets, The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Fubini's theorem for L^1 functions on a sigma-finite product, Riemann–Lebesgue comparison for distribution test integrands, Integral over a measurable subset, Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F8]

The spatial point source $δ_{x_{0}}$ pairs a compactly supported smooth test $η$ as $η \mapsto η (x_{0})$ (Dirac delta and its derivatives).

[F9]

Choose $t_{c} \in I$ and $0 < r < R$ with $[t_{c} - R, t_{c} + R] \subset I$ . The fixed scaled bump of Compactly supported scaled Euclidean bumps in dimension one is nonnegative, equals one on $[t_{c} - r, t_{c} + r]$ and has compact support inside $(t_{c} - R, t_{c} + R)$ . Its finite integral $c_{0}$ is at least $2 r > 0$ , by [F7] and the interval measure formula of A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included. Dividing this bump by $c_{0}$ gives $χ \in C_{c}^{\infty} (I)$ of integral one.

Verification

technique · direct

1.1givenF1F2F4

Clause (c): in the static regime $\partial_{t} E = \partial_{t} B = 0$ ; substituting into the two curl equations of the smooth branch of [F2] leaves $curl E = 0$ and $curl B = μ_{0} J$ at every point and every fixed time, while the two divergence equations are unchanged, giving exactly the displayed four equations; the operators are those of [F4] applied to the fixed-time fields of [F1]. In the pure electrostatic regime $B = 0$ and $J = 0$ , so $div B = 0$ and $curl B = 0$ are identities and the reduction is the pair $div E = ρ / ε_{0}$ , $curl E = 0$ .

2.1step 1.1F4F5

Clause (d): in the static regime each of $E, B, ρ, J$ has vanishing partial $t$ -derivative at every point of $Ω \times I$ . For fixed $y$ and two times $t_{1} < t_{2}$ in $I$ , the mean value theorem [F5] applied to $t \mapsto F (y, t)$ on $[t_{1}, t_{2}]$ gives $F (y, t_{2}) - F (y, t_{1}) = 0$ , so $F (y, t)$ depends only on $y$ ; writing $F_{0} (y) : = F (y, t_{0})$ for a fixed $t_{0} \in I$ we get $F = F_{0} \circ π$ with $π (y, t) : = y$ , and the four reduced equations of clause (c), which are pointwise conditions at each fixed time [F4], are therefore equations between the spatial maps $E_{0}, B_{0}, ρ_{0}, J_{0}$ . The same reading is available for the locally integrable time-independent objects of clause (e), for which time independence is an explicit hypothesis rather than a consequence of differentiability.

3.1step 2.1F4F6F7

Clause (e), spatial pairings as time integrals: let the time-independent locally integrable data be read as spatial maps $E_{0}, B_{0}, ρ_{0}, J_{0}$ by the fixed-time reading of step 2.1, and fix $ψ \in C_{c}^{\infty} (U)$ . For a spatial operator expression $S$ that is linear in the data (for instance $S = div E_{0} - ρ_{0} / ε_{0}$ ), choosing Borel representatives of the coefficients [F7] and moving every distributional spatial derivative onto the test gives the pairing identity $⟨ S_{U}, ψ ⟩ = \int_{I} ⟨ S_{Ω}, ψ (\cdot, t) ⟩_{Ω} d t,$ where $S_{U}$ denotes the distributional expression on $U$ evaluated with distributional derivatives on $U$ and $S_{Ω}$ the same expression on $Ω$ evaluated with distributional spatial derivatives; in the displayed instance both sides equal the same absolutely convergent iterated integrals, computed for $S_{Ω}$ by moving the spatial derivative onto the test, $- \int_{I} \int_{Ω} E_{0} \cdot \nabla_{y} ψ d λ_{3} d t - \frac{1}{ε_{0}} \int_{I} \int_{Ω} ρ_{0} ψ d λ_{3} d t,$ products of locally integrable coefficients with derivatives of the compactly supported test being integrable by Fubini [F7], and for data that are additionally $C^{1}$ the inner pairing agrees with the classical integral $\int_{Ω} (div E_{0} - ρ_{0} / ε_{0}) ψ (\cdot, t) d λ_{3}$ ([F4]); each $t$ -slice $ψ (\cdot, t)$ is smooth with compact support in $Ω$ because the section at $t$ of the compact support of $ψ$ is compact and lies in $Ω$ ([F6]). The same computation applies to $div B$ , $curl E$ and $curl B - μ_{0} J$ .

4.1step 3.1F5F6F7

Clause (e), vanishing of the time derivatives: let $F$ be a time-independent locally integrable function on $U$ (one component of $E$ or $B$ , or $ρ$ ). Choosing a Borel representative of $F$ [F7] and using the product structure and the fixed-time reading of step 3.1, $⟨ T_{F}, \partial_{t} ψ ⟩ = \int_{U} F (y) \partial_{t} ψ (y, t) d λ_{4} = \int_{Ω} F (y) (\int_{I} \partial_{t} ψ (y, t) d t) d λ_{3} (y) = 0,$ because for each $y$ the function $t \mapsto ψ (y, t)$ is smooth with compact support in $I$ , so its integral of derivative vanishes by the fundamental theorem [F5] (the inner integral is a Lebesgue integral of a continuous compactly supported integrand, equal to its Riemann value [F7]); the iterated integral is absolutely convergent since $ψ$ has compact support and $F$ is locally integrable on the bounded region of the support. By [F6] this says $\partial_{t} T_{F} = 0$ in $D^{'} (U)$ , so the two time-dependent terms of the weak predicate [F3] vanish identically for time-independent fields.

5.1step 3.1step 4.1F3F5F6F9

Clause (e), the equivalence: if the four spatial equations hold in $D^{'} (Ω)$ , then for every $t \in I$ the test $ψ (\cdot, t)$ of step 3.1 gives $⟨ S_{Ω}, ψ (\cdot, t) ⟩_{Ω} = 0$ , hence the integral of step 3.1 vanishes; together with the vanishing time-derivative pairings of step 4.1 this makes all four weak pairings of [F3] vanish, so the weak equations hold. Conversely, if the four weak pairings of [F3] vanish, choose $η \in C_{c}^{\infty} (Ω)$ and a normalized temporal bump $χ \in C_{c}^{\infty} (I)$ with $\int_{I} χ = 1$ from [F9] and test with $ψ (y, t) = η (y) χ (t)$ , which lies in $C_{c}^{\infty} (U)$ ; by steps 3.1 and 4.1, $0 = \int_{I} χ (t) ⟨ S_{Ω}, η ⟩_{Ω} d t = ⟨ S_{Ω}, η ⟩_{Ω}$ , so each spatial equation holds in $D^{'} (Ω)$ since $η$ was arbitrary. This is the asserted if-and-only-if, and it uses no product of distributions: all pairings are against a single test function.

6.1step 5.1F3F6F8

Clause (e), stationary point charge: take $I_{0} : = I$ , the constant $C^{1}$ trajectory $z (t) \equiv x_{0}$ with $x_{0} \in Ω$ , and $q \in R$ . Its graph is ${x_{0}} \times I$ , and the graph map is proper into $U = I \times Ω$ because the preimage of a compact subset of $U$ is the corresponding compact closed subset of $I$ ; the source of [F3] is then $⟨ ρ_{p t}, ψ ⟩ = q \int_{I} ψ (x_{0}, t) d t$ and $⟨ J_{p t}, ψ ⟩ = q \int_{I} z^{'} (t) ψ (z (t), t) d t = 0$ . For each fixed time the spatial reading is the point source pairing a compactly supported smooth spatial test $η$ as $η \mapsto q η (x_{0})$ , which is exactly the spatial point source of [F8]; each spatial slice $ψ (\cdot, t)$ belongs to $D (Ω)$ by step 3.1, and the spacetime pairing is the time integral of these spatial pairings: $⟨ ρ_{p t}, ψ ⟩ = \int_{I} ⟨ q δ_{x_{0}}, ψ (\cdot, t) ⟩_{Ω} d t = q \int_{I} ψ (x_{0}, t) d t$ . This integral is finite because $t \mapsto ψ (x_{0}, t)$ is smooth and compactly supported in $I$ .

7.1step 1.1step 2.1step 5.1step 6.1∎

Clauses (a) and (b) are the definitions of the static and pure electrostatic regimes; clause (c) is step 1.1, clause (d) is step 2.1, clause (e) is steps 3.1-6.1, and clause (f) records that the conditions were imposed in the one frame, that no stability, support, boost or singular-source equation beyond the dictionary of (e), and no empirical claim, is asserted. The verification used only the cited smooth branch, the definitions of the primitives, and the measure/distribution interfaces; no equation of motion and no empirical premise entered.

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Scalar potential and the Poisson equation in a star-shaped electrostatic region

Physical scope

Classical electrostatics in vacuum in a star-shaped spatial region $Ω$ of a named inertial frame: a static, curl-free $C^{1}$ electric field with a continuous fixed-time charge density in the pure electrostatic regime, for which the electrostatic potential exists as a $C^{2}$ function with $E = - \nabla φ$ and satisfies the Poisson equation $- Δ φ = ρ / ε_{0}$ , uniquely up to an additive constant. The region is star-shaped (no general topology), no boundary data or decay normalization is included, and no empirical claim, dynamical stability claim or material model is asserted.

Statement

Let $Ω \subseteq R^{3}$ be a nonempty open set that is star-shaped with respect to some $a \in Ω$ (Star-shaped open subsets of Euclidean space), let $E : Ω \to R^{3}$ be of class $C^{1}$ in the spatial vector-field sense of The electric and magnetic fields as fixed-time spatial vector fields ( $C^{k}$ Euclidean maps and diffeomorphisms), and let $ρ : Ω \to R$ be continuous — the fixed-time reading, in the sense of The static and pure electrostatic regimes and their fixed-time reading, of a pure electrostatic regime with the units and constant $ε_{0} > 0$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current). Suppose the reduced smooth-branch equations hold pointwise on $Ω$ : $div E = ρ / ε_{0}, curl E = 0 on Ω,$ with the divergence and curl of Divergence and curl of a $C^{1}$ vector field. Then:

(a) [Closed field and potential] $- E$ is closed in the sense of Exact and closed C1 vector fields — equivalently $curl (- E) = 0$ (A $C^{1}$ field on an open subset of $R^{3}$ is closed exactly when its curl vanishes) — and the function $φ (x) : = - \int_{0}^{1} E (a + s (x - a)) \cdot (x - a) d s$ (the Euclidean inner product of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ) is a $C^{2}$ function on $Ω$ with $\nabla φ = - E on Ω$ (Poincare's lemma on a star-shaped domain: every closed C1 field is exact applied to the closed $C^{1}$ field $- E$ ); continuity of the integrand and its first parameter derivatives and the proved identity $\nabla φ = - E$ with $E \in C^{1}$ give the stated regularity.

(b) [Poisson equation] with $Δ φ = div \nabla φ$ the Laplacian of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, $- Δ φ = div E = \frac{ρ}{ε_{0}} pointwise on Ω .$

(c) [Uniqueness up to constants] if $ψ \in C^{2} (Ω)$ satisfies $\nabla ψ = - E$ on $Ω$ , then $ψ - φ$ is constant on $Ω$ ; in particular $φ$ is uniquely determined by $\nabla φ = - E$ and one prescribed value. Star-shapedness gives path-connectedness through the centre $a$ , so the constant is the same on all of $Ω$ (Two potentials of the same field differ by a constant on each piecewise-C1 path component, Piecewise-C1 path-connected domains, potential functions, conservative fields, and path independence).

No normalisation of $φ$ at infinity, and no boundary condition, is asserted on a general star-shaped $Ω$ ; the additive constant remains free.

Facts & Assumptions

Given: A star-shaped open $Ω \subseteq R^{3}$ with centre $a \in Ω$ , a $C^{1}$ field $E : Ω \to R^{3}$ , a continuous $ρ : Ω \to R$ , and $ε_{0} = (μ_{0} c^{2})^{- 1} > 0$ , the fixed-time reading of a pure electrostatic regime in which $div E = ρ / ε_{0}$ and $curl E = 0$ hold pointwise.

[F1]

The fixed-time reading of a pure electrostatic regime, the field conventions and the source conventions are those of the primitives: in particular $curl E = 0$ and $div E = ρ / ε_{0}$ are the reduced smooth-branch equations, and the objects carry no time dependence (The static and pure electrostatic regimes and their fixed-time reading, 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, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

A field is closed exactly when its curl vanishes identically on its open domain $Ω \subseteq R^{3}$ ; closedness is the symmetry system $\partial_{j} F_{i} = \partial_{i} F_{j}$ (A $C^{1}$ field on an open subset of $R^{3}$ is closed exactly when its curl vanishes, Exact and closed C1 vector fields).

[F3]

Every closed $C^{1}$ field on an open star-shaped $U \subseteq R^{n}$ is exact, with the $C^{2}$ potential $ϕ (x) = \int_{0}^{1} ⟨ F (a + t (x - a)), x - a ⟩ d t$ satisfying $\nabla ϕ = F$ (Poincare's lemma on a star-shaped domain: every closed C1 field is exact).

[F4]

The Laplacian is $Δ φ = div \nabla φ$ , and divergence and curl are linear in the field with the partial-derivative convention of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Divergence and curl of a $C^{1}$ vector field, 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 $g (c) \neq 0$ , $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

If two $C^{1}$ functions have the same gradient, their difference is constant on each piecewise- $C^{1}$ path component; a star-shaped set is connected through its centre by the $C^{1}$ segments $s \mapsto a + s (x - a)$ , $s \in [0, 1]$ (Two potentials of the same field differ by a constant on each piecewise-C1 path component, Piecewise-C1 path-connected domains, potential functions, conservative fields, and path independence, Star-shaped open subsets of Euclidean space, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

technique · direct

1.1givenF1F2F4

The field $F : = - E$ is $C^{1}$ on the open set $Ω$ , being the componentwise negative of the $C^{1}$ field $E$ ([F1], [F4]), and it is closed: by [F2] closedness is equivalent to the vanishing of the curl, and $curl F = - curl E = 0$ by linearity of the curl in the field and the reduced equation $curl E = 0$ ([F2], [F4]).

2.1step 1.1F3

The Poincaré lemma [F3] applies to the closed $C^{1}$ field $F = - E$ on the star-shaped open set $Ω$ with centre $a$ , and yields the $C^{2}$ function $φ (x) = \int_{0}^{1} ⟨ F (a + s (x - a)), x - a ⟩ d s = - \int_{0}^{1} E (a + s (x - a)) \cdot (x - a) d s$ with $\nabla φ = F = - E$ on $Ω$ , the inner product being the Euclidean one; this is clause (a).

3.1step 2.1F1F4

For clause (b), the Laplacian of the $C^{2}$ potential is $Δ φ = div \nabla φ$ ([F4]); substituting $\nabla φ = - E$ from step 2.1 and using the linearity of the divergence in the field gives $Δ φ = div (- E) = - div E$ , so $- Δ φ = div E = ρ / ε_{0}$ by the reduced divergence equation ([F1], [F4]).

3.2step 2.1F4F5

For clause (c), let $ψ \in C^{2} (Ω)$ satisfy $\nabla ψ = - E$ . Then $\nabla (ψ - φ) = \nabla ψ - \nabla φ = 0$ on $Ω$ , because both gradients equal $- E$ and differentiation is linear ([F4]); by [F5] the difference $ψ - φ$ is constant on every piecewise- $C^{1}$ path component of $Ω$ . Every $x \in Ω$ is joined to the centre $a$ by the segment $s \mapsto a + s (x - a)$ , $s \in [0, 1]$ , which is $C^{1}$ , lies in $Ω$ by star-shapedness and has constant value $a$ at $s = 0$ ; hence all points lie in the single path component containing $a$ ([F5]), and $ψ - φ$ is constant on $Ω$ . Fixing the value of $φ$ at one point therefore determines $φ$ uniquely.

4.1step 2.1step 3.1step 3.2F1F5∎

Clauses (a), (b) and (c) are steps 2.1, 3.1 and 3.2. The argument used only the pointwise reduced equations, the $C^{1}$ regularity of the field, continuity of $ρ$ , star-shapedness of $Ω$ and the cited mathematical suppliers; no boundary condition, no normalisation at infinity, no uniqueness of the field equations, no existence of a physical equilibrium and no empirical assertion entered, which is exactly the scope stated.

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The localized Newtonian potential and the ideal point-charge Coulomb field

Physical scope

Classical electrostatics in vacuum for a compactly supported charge density (smooth $C^{1}$ or merely $L^{1}$ ), with the localised Newtonian potential $φ = ε_{0}^{- 1} N ρ$ and its field $E = - \nabla φ$ ; in the smooth case the field is the continuous Coulomb integral, the potential is the unique decaying $C^{2}$ solution of $- Δ φ = ρ / ε_{0}$ , and $(E, 0, ρ, 0)$ realises the pure electrostatic regime. The point-charge clause states the distributional identities for the idealised singular source $q δ_{x_{0}}$ in the framework's SI conventions, with no uniqueness, self-force, reaction or empirical claim.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ), inherited from the cited Lebesgue, polar, Newtonian and distribution interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Work in $R^{3}$ with its Euclidean structure (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Open ball, closed ball and sphere in a metric space) and the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values, $ε_{0} = (μ_{0} c^{2})^{- 1} > 0$ . Use one-based labels $v_{i} : = v (i - 1)$ for vectors $v \in R^{3}$ , $i = 1, 2, 3$ ; $\partial_{i}$ means the derivative in that coordinate (the supplier's $\partial_{i - 1}$ ), and $e_{i}$ below is the standard unit vector in that direction. Divergence, curl and gradient are the cited operators with these labels, so relabelling does not change them. Let $Φ$ be the $n = 3$ fundamental solution of $- Δ$ of Fundamental solution for the positive operator minus Laplacian, normalised by $- Δ T_{Φ} = δ_{0}$ ; then $ω_{2} = 4 π$ and $Φ (z) = \frac{1}{4 π ∣ z ∣} (z \neq 0)$ (The closed form for the volume of the unit $n$ -ball, Agreement with the existing polar sphere measure). Let $N f$ be the Newtonian potential of Newtonian potential of compactly supported data.

(a) [Smooth compact sources] Let $ρ : R^{3} \to R$ be of class $C^{1}$ with compact support contained in ${\overline{B}}_{R}$ , $R > 0$ ( $C^{k}$ Euclidean maps and diffeomorphisms, Open ball, closed ball and sphere in a metric space), put $φ : = ε_{0}^{- 1} N ρ$ and $E : = - \nabla φ$ . Then $φ \in C^{2} (R^{3})$ (Hölder data give a classical Newtonian solution), and pointwise on $R^{3}$ $- Δ φ = \frac{ρ}{ε_{0}}, div E = \frac{ρ}{ε_{0}}, curl E = 0$ with the operators of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field and Divergence and curl of a $C^{1}$ vector field. Moreover, with $Q : = \int ρ d λ_{3}$ , for every $x \in R^{3}$ the absolutely convergent Coulomb integral representation holds, and its far-field bound holds for $∣ x ∣ > 2 R$ , $E (x) = \frac{1}{4 π ε_{0}} \int_{R^{3}} \frac{x - y}{∣ x - y ∣^{3}} ρ (y) d λ_{3} (y) (x \in R^{3}), ∣ E (x) ∣ \leq \frac{∥ ρ ∥_{1}}{π ε_{0} ∣ x ∣^{2}} (∣ x ∣ > 2 R),$ and for $∣ x ∣ > 2 R$ the potential obeys the decay bound $∣ φ (x) ∣ \leq ∣ Q ∣ / (4 π ε_{0} ∣ x ∣) + R ∥ ρ ∥_{1} / (π ε_{0} ∣ x ∣^{2})$ (The far-field gradient of a compactly supported Newtonian potential, Far-field asymptotics of compact-source Newtonian potentials). The pair $(E, 0, ρ, 0)$ therefore realises the pure electrostatic regime of The static and pure electrostatic regimes and their fixed-time reading with source $ρ$ , and $E$ is the field of the localised Newtonian potential.

(b) [Uniqueness with vanishing at infinity] If $ψ \in C^{2} (R^{3})$ satisfies $- Δ ψ = ρ / ε_{0}$ pointwise on $R^{3}$ and $\lim_{∣ x ∣ \to \infty} ψ (x) = 0$ , then $ψ = φ$ (Weak maximum principle for the laplacian). In particular, among $C^{2}$ solutions of the Poisson equation of a compactly supported $C^{1}$ source decaying at infinity, the localised Newtonian potential is the unique one.

(c) [Compactly supported $L^{1}$ sources] If instead $ρ \in L^{1} (R^{3})$ is real, measurable and compactly supported, then $N ρ$ is finite $λ_{3}$ -almost everywhere, belongs to $L_{l o c}^{1} (R^{3})$ , is $C^{\infty}$ and harmonic off the support of $ρ$ , and its regular distribution satisfies $- Δ T_{ε_{0}^{- 1} N ρ} = T_{ρ / ε_{0}} in D^{'} (R^{3})$ (Newtonian potentials solve the distributional Poisson equation, scaled by $ε_{0}^{- 1}$ ; Regular distribution from a locally integrable function, Distributional derivative).

(d) [Point charge] Let $q \in R$ and $x_{0} \in R^{3}$ and define, with arbitrary values at $x_{0}$ , $φ_{q} (x) : = \frac{q}{4 π ε_{0} ∣ x - x_{0} ∣}, E_{q} (x) : = \frac{q (x - x_{0})}{4 π ε_{0} ∣ x - x_{0} ∣^{3}} (x \neq x_{0}) .$ Then $φ_{q} \in L_{l o c}^{1} (R^{3})$ , $E_{q}$ is locally integrable and $C^{\infty}$ off $x_{0}$ , $E_{q} = - \nabla φ_{q}$ on $R^{3} ∖ {x_{0}}$ , and, with $T_{φ_{q}}$ the regular distribution of $φ_{q}$ and $div E_{q} : = \sum_{i} \partial_{i} T_{E_{q, i}}$ , $curl E_{q} : = (\partial_{2} T_{E_{q, 3}} - \partial_{3} T_{E_{q, 2}}, \partial_{3} T_{E_{q, 1}} - \partial_{1} T_{E_{q, 3}}, \partial_{1} T_{E_{q, 2}} - \partial_{2} T_{E_{q, 1}})$ , $- Δ T_{φ_{q}} = \frac{q}{ε_{0}} δ_{x_{0}}, div E_{q} = \frac{q}{ε_{0}} δ_{x_{0}}, curl E_{q} = 0 in D^{'} (R^{3})$ (The negative Laplacian of the fundamental solution is the unit Dirac distribution for the first identity, The distributional gradient of a locally integrable function smooth off one point for the passage from the potential to the field, Distributional derivative, Dirac delta and its derivatives). The distribution $q δ_{x_{0}}$ is the fixed-time reading of the stationary point charge of Distributional Maxwell equations with point and stationary-sheet sources(b) with charge parameter $q$ and constant trajectory $z \equiv x_{0}$ (The static and pure electrostatic regimes and their fixed-time reading(e)); so the pair $(E_{q}, 0)$ satisfies the distributional pure electrostatic equations with that point source. This is an idealised point-source model: no claim of uniqueness for the point source, no self-force and no statement about the field at $x_{0}$ itself is made.

Facts & Assumptions

Given: The Axiom of Countable Choice, the $n = 3$ kernel $Φ$ with $Φ (z) = 1 / (4 π ∣ z ∣)$ off the origin and $- Δ T_{Φ} = δ_{0}$ , the SI constant $ε_{0} = (μ_{0} c^{2})^{- 1} > 0$ , and the compactly supported data and point data of clauses (a)-(d).

[F1]

The $n = 3$ kernel is $Φ (z) = ∣ z ∣^{- 1} / ω_{2}$ with $ω_{2} = 4 π$ , since the chart/polar sphere measure satisfies $∣ S^{n - 1} ∣ = n ∣ B_{1} ∣$ and the unit ball in dimension three has volume $4 π / 3$ ; the kernel is locally integrable, smooth off the pole and harmonic there, and $- Δ T_{Φ} = δ_{0}$ , with the same identity for every translate (Fundamental solution for the positive operator minus Laplacian, Agreement with the existing polar sphere measure, The closed form for the volume of the unit $n$ -ball, The polar surface set function on the unit sphere, The Laplace fundamental solution is harmonic off its pole, The negative Laplacian of the fundamental solution is the unit Dirac distribution).

[F2]

The Newtonian potential is $N f (x) = \int Φ (x - y) f (y) d λ_{3} (y)$ , defined where the integral is absolutely finite; for continuous compactly supported $f$ with finite global Hölder seminorm of exponent $α \in (0, 1)$ it is $C^{2}$ on all of $R^{3}$ with $- Δ N f = f$ pointwise; for compactly supported $f \in L^{1}$ it is finite almost everywhere, locally integrable, $C^{\infty}$ and harmonic off the support, and satisfies $- Δ T_{N f} = T_{f}$ in $D^{'}$ (Newtonian potential of compactly supported data, Hölder data give a classical Newtonian solution, Newtonian potentials solve the distributional Poisson equation).

[F3]

On the far field the gradient representation and the decay of the potential hold: for compactly supported $L^{1}$ data $f$ and $∣ x ∣ > 2 R$ one has $\nabla (N f) (x) = \int \nabla Φ (x - y) f (y) d λ_{3} (y)$ , $∣ \nabla (N f) (x) ∣ \leq 2^{n - 1} ∥ f ∥_{1} / (ω_{n - 1} ∣ x ∣^{n - 1})$ , and for $n = 3$ , $∣ N f (x) - Q / (4 π ∣ x ∣) ∣ \leq R ∥ f ∥_{1} / (π ∣ x ∣^{2})$ whenever $f$ vanishes off the open ball $B_{R}$ , with $Q = \int f$ and $f$ real (The far-field gradient of a compactly supported Newtonian potential, Far-field asymptotics of compact-source Newtonian potentials).

[F4]

The Laplacian and the divergence and curl are the classical operators of The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field and Divergence and curl of a $C^{1}$ vector field; the curl of a gradient of a $C^{2}$ function vanishes, divergence is linear in the field with coordinate partial derivatives of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , and the algebra and chain rules for derivatives apply to real powers (The curl of the gradient of a $C^{2}$ function vanishes, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Continuity and derivatives of positive-base real powers, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, $C^{k}$ Euclidean maps and diffeomorphisms).

[F5]

A $C^{1}$ compactly supported function on $R^{3}$ is bounded and Lipschitz: the continuous gradient norm attains its maximum on a nonempty compact closed ball containing the support, and the gradient vanishes outside that ball, and the mean value theorem bounds increments; consequently it has finite global $α$ -Hölder seminorm for every $α \in (0, 1]$ (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , Open ball, closed ball and sphere in a metric space, The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F6]

Differentiation under the integral sign and $C^{1}$ change of variables for $L^{1}$ functions are available: a continuously differentiable family of integrable sections with one integrable majorant may be differentiated under the integral, and integrals are invariant under translations and reflections of bounded measurable sets (Differentiation under the integral sign, A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

[F7]

Locally integrable functions are identified with their regular distributions; distributional derivatives satisfy $⟨ \partial^{α} T, φ ⟩ = (- 1)^{∣ α ∣} ⟨ T, \partial^{α} φ ⟩$ , commute with one another and with constant multiples, and agree with classical derivatives for $C^{k}$ functions; the regular pairing depends only on the almost-everywhere class, and a singleton is Lebesgue null (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions, Distribution, Distributional derivative, Distributional differentiation is continuous and commutes, Clairaut--Schwarz theorem for continuous second partial derivatives, Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree, A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ , A nonnegative integral over a null set vanishes, Measure-null sets and almost-everywhere statements relative to a measure).

[F8]

The Dirac distribution is $δ_{x_{0}} (φ) = φ (x_{0})$ ; test functions are smooth with compact support, and the polar formula computes radial integrals: $\int_{∣ z ∣ < b} ∣ z ∣^{- 1} d λ_{3} = 2 π b^{2}$ and $\int_{∣ z ∣ < b} ∣ z ∣^{- 2} d λ_{3} = 4 π b$ for $b > 0$ (Dirac delta and its derivatives, Test function space d of an open set, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Integral over a measurable subset, Monotonicity and nonnegative homogeneity of the nonnegative integral).

[F9]

The pure electrostatic regime and its fixed-time reading are those of The static and pure electrostatic regimes and their fixed-time reading, the field and source conventions are those of The electric and magnetic fields as fixed-time spatial vector fields and Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, the laboratory region is that of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, the units are those of Framework-local SI units, the vacuum constants and the exact/measured status of their values, and the weak solution predicate and stationary point source are those of Distributional Maxwell equations with point and stationary-sheet sources(b),(e).

[F10]

The weak maximum principle: for nonempty bounded open $Ω$ and $u \in C^{2} (Ω) \cap C (\overline{Ω})$ with $Δ u \geq 0$ , $\max_{\overline{Ω}} u = \max_{\partial Ω} u$ (Weak maximum principle for the laplacian).

[F11]

For a nonnegative unit-mass smooth radial mollifier supported in $B_{δ} (0)$ (A radial mollifier family in Rn), $h_{δ} = h * η_{δ}$ is smooth for locally integrable $h$ (Convolution with a mollifier is smooth, and derivatives pass under the integral sign). For $h \in C_{c}^{1}$ , $h_{δ}$ has support in a fixed compact set for $0 < δ \leq 1$ , and $h_{δ} \to h$ , $\partial_{i} h_{δ} \to \partial_{i} h$ uniformly: differentiating $h_{δ} (z) = \int η_{δ} (w) h (z - w) d w$ gives $\partial_{i} h_{δ} = η_{δ} * \partial_{i} h$ by [F6], and each uniform error is at most the corresponding uniform-continuity modulus at $δ$ . Dominated convergence passes these uniformly bounded, compactly supported approximants through integrals against the locally integrable kernels $Φ$ and $\partial_{i} Φ$ (Dominated convergence).

Proof

technique · direct

1.1givenF1F4F5F8

By [F1], $ω_{2} = ∣ S^{2} ∣ = 4 π$ and $Φ (z) = 1 / (4 π ∣ z ∣)$ for $z \neq 0$ , so $\nabla Φ (z) = - z / (4 π ∣ z ∣^{3})$ and $∣ \nabla Φ (z) ∣ = 1 / (4 π ∣ z ∣^{2})$ by the chain and power rules ([F4]); $Φ$ and $\nabla Φ$ are locally integrable on $R^{3}$ , since $\int_{∣ z ∣ < b} ∣ z ∣^{- k} d λ_{3} = 4 π b^{3 - k} / (3 - k)$ is finite for $k = 1, 2$ , while away from the pole the kernels are continuous and hence bounded on compacta ([F8]). Let $ρ \in C_{c}^{1}$ with support in ${\overline{B}}_{R}$ ; by [F5] it is bounded, Lipschitz and has finite global $α$ -Hölder seminorm for $α = 1 / 2$ , and it vanishes off the open ball $B_{R}$ because it is continuous and zero on and beyond $S_{R}$ .

2.1step 1.1F2F4F9

The Hölder theorem [F2] applied with $α = 1 / 2$ gives $N ρ \in C^{2} (R^{3})$ with $- Δ (N ρ) = ρ$ pointwise; scaling $φ = ε_{0}^{- 1} N ρ$ and using linearity of the second coordinate derivatives gives $- Δ φ = ρ / ε_{0}$ pointwise, because $ε_{0} > 0$ is a constant ([F4]). With $E : = - \nabla φ$ one has $div E = - div \nabla φ = - Δ φ = ρ / ε_{0}$ on $R^{3}$ and $curl E = - curl \nabla φ = 0$ by [F4], since $φ$ is $C^{2}$ .

3.1step 2.1F1F5F6F7F8F11

Coulomb representation: since Lebesgue measure is translation invariant, $N ρ (x) = \int Φ (x - y) ρ (y) d λ_{3} (y) = \int Φ (z) ρ (x - z) d λ_{3} (z)$ ([F6]); fix $i \in {1, 2, 3}$ , $x_{0} \in R^{3}$ , $δ > 0$ and the interval $J = (- δ, δ)$ of coordinate increments, so $x_{0} + s e_{i} \in B (x_{0}, δ)$ for $s \in J$ . For such $x$ the integrand $Φ (z) ρ (x - z)$ is supported in the compact ball $K : = {\overline{B}}_{∣ x_{0} ∣ + δ + R} (0)$ , and its $s$ -derivative $Φ (z) \partial_{i} ρ (x - z)$ is dominated on $J$ by $∥ \nabla ρ ∥_{\infty} ∣ Φ (z) ∣ 1_{K} (z)$ , which is integrable ([F5], [F8]); differentiation under the integral sign [F6] therefore gives $\partial_{i} N ρ (x) = \int Φ (z) \partial_{i} ρ (x - z) d λ_{3} (z)$ . For fixed $x$ put $h (z) : = ρ (x - z)$ , a $C^{1}$ compactly supported function; then $\partial_{i} ρ (x - z) = - \partial_{z_{i}} h (z)$ , so $\partial_{i} N ρ (x) = - \int Φ (z) \partial_{z_{i}} h (z) d λ_{3} (z) .$ The kernel $Φ$ is locally integrable, continuous and smooth off the single point $0$ , its classical partials are locally integrable, so The distributional gradient of a locally integrable function smooth off one point applies with $u = Φ$ and gives $\partial_{i} T_{Φ} = T_{\partial_{i} Φ}$ ; This identity is initially paired only with smooth tests: for $h_{δ}$ of [F11], $\int Φ \partial_{i} h_{δ} = - \int \partial_{i} Φ h_{δ}$ by [F7]. Since $h \in C_{c}^{1}$ , [F11] supplies uniform convergence of both $h_{δ}$ and $\partial_{i} h_{δ}$ on a fixed compact support; the two locally integrable kernels therefore allow the limit $\int Φ \partial_{i} h = - \int \partial_{i} Φ h$ , without pairing any distribution against a nonsmooth test. Substituting back and translating gives $\partial_{i} N ρ (x) = \int \partial_{i} Φ (x - y) ρ (y) d λ_{3} (y) = - \frac{1}{4 π} \int \frac{x_{i} - y_{i}}{∣ x - y ∣^{3}} ρ (y) d λ_{3} (y)$ for every $x \in R^{3}$ , the integral being absolutely convergent because $supp ρ \subseteq {\overline{B}}_{R}$ lies in the ball $B (x, ∣ x ∣ + 2 R)$ and $\int_{B (x, ∣ x ∣ + 2 R)} ∣ x - y ∣^{- 2} d λ_{3} (y) = 4 π (∣ x ∣ + 2 R) < \infty$ , while $∣ ρ ∣ \leq ∥ ρ ∥_{\infty}$ on the support ([F1], [F5], [F8]). Multiplying by $- ε_{0}^{- 1}$ gives exactly the displayed Coulomb representation of $E = - \nabla φ$ .

4.1step 3.1F3F9

Bounds and regime reading: for $∣ x ∣ > 2 R$ the far-field lemma [F3] with $n = 3$ , $ω_{2} = 4 π$ gives $∣ \nabla N ρ (x) ∣ \leq 4 ∥ ρ ∥_{1} / (4 π ∣ x ∣^{2}) = ∥ ρ ∥_{1} / (π ∣ x ∣^{2})$ , hence $∣ E (x) ∣ = ε_{0}^{- 1} ∣ \nabla N ρ (x) ∣ \leq ∥ ρ ∥_{1} / (π ε_{0} ∣ x ∣^{2})$ . Since $ρ$ is continuous, compactly supported in ${\overline{B}}_{R}$ and zero off $B_{R}$ , the decay theorem [F3] gives $∣ N ρ (x) - Q / (4 π ∣ x ∣) ∣ \leq R ∥ ρ ∥_{1} / (π ∣ x ∣^{2})$ for $∣ x ∣ > 2 R$ , so for $∣ x ∣ > 2 R$ we get $∣ φ (x) ∣ \leq ∣ Q ∣ / (4 π ε_{0} ∣ x ∣) + R ∥ ρ ∥_{1} / (π ε_{0} ∣ x ∣^{2})$ by the triangle inequality, as claimed. Both terms have units V: $Q$ and $∥ ρ ∥_{1}$ carry C, $R$ and $∣ x ∣$ carry m, and $ε_{0}$ carries F m $^{- 1}$ ; the bound also gives $φ (x) \to 0$ as $∣ x ∣ \to \infty$ . Finally the fields and density are time independent, and the pointwise identities of step 2.1 are exactly the reduced equations of the pure electrostatic regime, so $(E, 0, ρ, 0)$ realises that regime with source $ρ$ in the sense of the fixed-time reading ([F9]).

5.1step 4.1F4F10

Uniqueness (b): let $ψ \in C^{2} (R^{3})$ with $- Δ ψ = ρ / ε_{0}$ pointwise and $ψ (x) \to 0$ as $∣ x ∣ \to \infty$ , and put $h : = ψ - φ$ , where $φ$ is the potential of clause (a). Then $h \in C^{2} (R^{3})$ , $Δ h = Δ ψ - Δ φ = - ρ / ε_{0} + ρ / ε_{0} = 0$ and $h (x) \to 0$ ([F4]). Given $ε > 0$ choose $R_{ε}$ with $∣ h (x) ∣ \leq ε$ for $∣ x ∣ \geq R_{ε}$ ; for $R > R_{ε}$ the weak maximum principle [F10] applied on the bounded open ball $B_{R}$ to the harmonic $h$ gives $\max_{{\overline{B}}_{R}} h = \max_{\partial B_{R}} h \leq ε$ , and applied to the harmonic $- h$ gives $\min_{{\overline{B}}_{R}} h \geq - ε$ ; hence $∣ h ∣ \leq ε$ on ${\overline{B}}_{R}$ . Every point of $R^{3}$ lies in $B_{R}$ for large $R$ , so $∣ h (x) ∣ \leq ε$ for all $x$ and all $ε > 0$ , whence $h = 0$ and $ψ = φ$ ([F4]).

6.1step 5.1F2F7

Clause (c): let $ρ \in L^{1} (R^{3})$ be real, measurable and compactly supported. The distributional Poisson theorem [F2] gives that $N ρ$ is finite almost everywhere, belongs to $L_{l o c}^{1}$ , is $C^{\infty}$ and harmonic off the support of $ρ$ , and satisfies $- Δ T_{N ρ} = T_{ρ}$ in $D^{'} (R^{3})$ . Multiplying the regular distribution by the positive constant $ε_{0}^{- 1}$ and using linearity of distributional differentiation, $- Δ T_{ε_{0}^{- 1} N ρ} = ε_{0}^{- 1} (- Δ T_{N ρ}) = T_{ρ / ε_{0}}$ ([F7]); this is the displayed identity, with the same smoothness and harmonicity assertions carried over.

7.1step 6.1F1F4F7F8

Clause (d), first part: $φ_{q} = (q / ε_{0}) Φ (\cdot - x_{0})$ , so $φ_{q} \in L_{l o c}^{1} (R^{3})$ and its classical partials $\partial_{i} φ_{q}$ are locally integrable, by the local integrability of $∣ z ∣^{- 1}$ and $∣ z ∣^{- 2}$ in dimension three ([F1], [F8]); the value at $x_{0}$ is arbitrary and immaterial ([F7]). By [F1] the translated kernel satisfies $- Δ_{x} T_{Φ (\cdot - x_{0})} = δ_{x_{0}}$ ; multiplying by $q / ε_{0}$ and using linearity of distributional differentiation gives $- Δ T_{φ_{q}} = (q / ε_{0}) δ_{x_{0}}$ ([F7]). Moreover $E_{q} = - \nabla φ_{q}$ on $R^{3} ∖ {x_{0}}$ : differentiating $φ_{q} (x) = q / (4 π ε_{0} ∣ x - x_{0} ∣)$ by the chain and power rules gives $\nabla φ_{q} (x) = - \frac{q (x - x_{0})}{4 π ε_{0} ∣ x - x_{0} ∣^{3}}$ for $x \neq x_{0}$ ([F4]).

8.1step 7.1F7F8F9∎

Clause (d), field identities and reading: by The distributional gradient of a locally integrable function smooth off one point applied to $u = φ_{q}$ (locally integrable, continuous and smooth off the single point $x_{0}$ , classical partials locally integrable), $\partial_{i} T_{φ_{q}} = T_{\partial_{i} φ_{q}}$ ([F7]). Since $E_{q, i} = - \partial_{i} φ_{q}$ off $x_{0}$ and a singleton is null, the two functions agree almost everywhere, hence $T_{E_{q, i}} = - T_{\partial_{i} φ_{q}} = - \partial_{i} T_{φ_{q}}$ ([F7]); taking one more distributional derivative and summing over $i$ gives $div E_{q} = \sum_{i} \partial_{i} T_{E_{q, i}} = - \sum_{i} \partial_{i} \partial_{i} T_{φ_{q}} = - Δ T_{φ_{q}} = \frac{q}{ε_{0}} δ_{x_{0}},$ and the curl components vanish because distributional partial derivatives commute: $\partial_{2} T_{E_{q, 3}} - \partial_{3} T_{E_{q, 2}} = - (\partial_{2} \partial_{3} - \partial_{3} \partial_{2}) T_{φ_{q}} = 0,$ and likewise for the other two components ([F7]). By [F9] the distribution $q δ_{x_{0}}$ is the fixed-time reading of the stationary point source with charge $q$ at $x_{0}$ , so $(E_{q}, 0)$ satisfies the distributional pure electrostatic equations with that source. No product of distributions was formed, no value of the field at $x_{0}$ was used, and no uniqueness, self-force, reaction or empirical claim is made; this is exactly the scope stated.

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

Work of the electrostatic force equals charge times the potential drop and is path independent

Physical scope

The work done by the electric force of a prescribed static external field on a test charge of constant parameter $q$ carried along a prescribed piecewise $C^{1}$ path in a star-shaped pure electrostatic region: the work equals $q$ times the potential drop, depends only on the endpoints, vanishes on closed paths and is unchanged by the additive constant of the potential. No equation of motion, radiative loss, work on sources or empirical claim is included.

Statement

Let $Ω \subseteq R^{3}$ be open, star-shaped (Star-shaped open subsets of Euclidean space) and the spatial region of a pure electrostatic regime as in Scalar potential and the Poisson equation in a star-shaped electrostatic region, with fields of The electric and magnetic fields as fixed-time spatial vector fields, charge parameters of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current and the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values; thus $E : Ω \to R^{3}$ is $C^{1}$ , $B = 0$ and $J = 0$ (The static and pure electrostatic regimes and their fixed-time reading), and by Scalar potential and the Poisson equation in a star-shaped electrostatic region there is $φ \in C^{2} (Ω)$ with $E = - \nabla φ$ . Let $q \in R$ (coulombs) be the charge of a test particle and let $γ : [t_{0}, t_{1}] \to Ω$ be a piecewise $C^{1}$ path (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, $C^{k}$ Euclidean maps and diffeomorphisms). Define the work of the electric force along the prescribed path by the vector line integral $W (γ) : = \int_{γ} q E \cdot d r$ of Scalar line integrals with respect to arc length and vector-field line integrals; $q E$ is the laboratory three-force of The Lorentz coupling: external force on a test charge and the continuum force and power densities(i) when $B = 0$ , using the tensor components of The electromagnetic two-form, its observer decomposition and the four-current(a) and the four-force/three-force relation of Four-force identities and the work-energy relation in an inertial frame(c), as computed in [F5] below. Then:

(a) [Value] $W (γ) = q (φ (γ (t_{0})) - φ (γ (t_{1})))$ ;

(b) [Path independence] if $γ_{1}, γ_{2}$ are piecewise $C^{1}$ paths in $Ω$ with the same initial and the same terminal point, then $W (γ_{1}) = W (γ_{2})$ ; in particular $W (γ) = 0$ for every closed piecewise $C^{1}$ path;

(c) [Potential energy] with the potential energy $U : = q φ$ (joules) the identity reads $W (γ) = - (U (γ (t_{1})) - U (γ (t_{0})))$ ;

(d) [Gauge invariance of the work] if $φ^{'} \in C^{2} (Ω)$ is any other potential with $\nabla φ^{'} = - E$ , then $φ^{'} = φ + c$ with a constant $c \in R$ and the formula of (a) is unchanged (Two potentials of the same field differ by a constant on each piecewise-C1 path component).

The statement is a line-integral identity for a prescribed path in a prescribed static external field. It asserts no equation of motion, no radiation, no work done on the sources and no claim that the test particle actually follows $γ$ ; the interpretation of $W$ as work done by the adopted external coupling is the physical reading supplied by The Lorentz coupling: external force on a test charge and the continuum force and power densities, not a dynamical theorem.

Facts & Assumptions

Given: A star-shaped open $Ω \subseteq R^{3}$ carrying a pure electrostatic regime with $C^{1}$ field $E$ and $B = J = 0$ , a charge parameter $q \in R$ , and a piecewise $C^{1}$ path $γ : [t_{0}, t_{1}] \to Ω$ .

[F1]

The pure electrostatic regime has $B = 0$ , $J = 0$ , time-independent fields and the reduced equations; on a star-shaped region the potential theorem provides $φ \in C^{2} (Ω)$ with $\nabla φ = - E$ (The static and pure electrostatic regimes and their fixed-time reading, Scalar potential and the Poisson equation in a star-shaped electrostatic region, 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, Framework-local SI units, the vacuum constants and the exact/measured status of their values, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Star-shaped open subsets of Euclidean space).

[F2]

The vector line integral is $\int_{γ} F \cdot d r = \sum_{i} \int_{t_{i}}^{t_{i + 1}} ⟨ F (γ (t)), v_{i} (t) ⟩ d t$ over an admissible partition, the inner product is the Euclidean one and the integral is linear in the field (Scalar line integrals with respect to arc length and vector-field line integrals, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , $C^{k}$ Euclidean maps and diffeomorphisms, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, 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 $g (c) \neq 0$ ).

[F3]

The gradient theorem evaluates the line integral of a gradient of a $C^{1}$ function along a piecewise $C^{1}$ path as the endpoint increment (The gradient theorem: the line integral of a gradient is the endpoint increment).

[F4]

Two $C^{1}$ potentials with the same gradient differ by a constant on each piecewise- $C^{1}$ path component, and a star-shaped set has a single such component through its centre (Two potentials of the same field differ by a constant on each piecewise-C1 path component, Star-shaped open subsets of Euclidean space, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F5]

For a massive test worldline in the prescribed external field, The Lorentz coupling: external force on a test charge and the continuum force and power densities(i) gives the four-force $K^{μ} = q F^{μ ν} U_{ν}$ . Write $w$ for its laboratory coordinate velocity, $∣ w ∣ < c$ , and $γ : = (1 - ∣ w ∣^{2} / c^{2})^{- 1 / 2} > 0$ . In the laboratory frame $U^{0} = γ c$ and $U_{0} = - γ c$ , while $K^{i} = γ f_{i}$ with $f = d p / d t$ the laboratory three-force (Four-force identities and the work-energy relation in an inertial frame, Statement and (c)). When $B = 0$ , The electromagnetic two-form, its observer decomposition and the four-current(a) gives $F^{i 0} = - E_{i} / c$ and $F^{i j} = 0$ , hence $γ f_{i} = q (- E_{i} / c) (- γ c) = q γ E_{i}$ . Since $γ > 0$ , $f = q E$ . This identifies the force used by the work definition whenever the external particle model applies; the prescribed path itself is not assumed to be a particle worldline.

Proof

technique · direct

1.1givenF1F2

By [F1] the potential $φ \in C^{2} (Ω)$ satisfies $\nabla φ = - E$ ; the field $q E$ is $C^{1}$ on $Ω$ for the constant $q$ , and the work $W (γ) = \int_{γ} q E \cdot d r$ is defined by the vector line integral of [F2], which is finite because the integrand is continuous on each smooth piece of the compact parameter interval.

2.1step 1.1F1F2F3

Clause (a): by the linearity of the line integral in the field and the constancy of $q$ , $W (γ) = q \int_{γ} E \cdot d r$ ([F2]); substituting $E = - \nabla φ$ and applying the gradient theorem [F3] gives $W (γ) = - q \int_{γ} \nabla φ \cdot d r = - q (φ (γ (t_{1})) - φ (γ (t_{0}))) = q (φ (γ (t_{0})) - φ (γ (t_{1}))),$ which is the displayed formula, including the degenerate case $t_{0} = t_{1}$ where the line integral is $0$ and the right side vanishes too.

3.1step 2.1

Clause (b): if $γ_{1}, γ_{2}$ are piecewise $C^{1}$ paths in $Ω$ with the same initial point $x$ and the same terminal point $y$ , then by step 2.1 applied to each path $W (γ_{1}) = q (φ (x) - φ (y)) = W (γ_{2})$ ; for a closed path, where $x = y$ , both formulas give $W (γ) = 0$ .

4.1step 3.1

Clause (c): with $U : = q φ$ , the identity of step 2.1 reads $W (γ) = U (γ (t_{0})) - U (γ (t_{1})) = - (U (γ (t_{1})) - U (γ (t_{0})))$ , the potential energy being the constant multiple $q φ$ of the potential.

5.1step 4.1F4

Clause (d): let $φ^{'} \in C^{2} (Ω)$ satisfy $\nabla φ^{'} = - E = \nabla φ$ . Then $\nabla (φ^{'} - φ) = 0$ on $Ω$ , and by [F4] the difference is constant on every piecewise- $C^{1}$ path component of $Ω$ ; since $Ω$ is star-shaped, every point is joined to the centre by a $C^{1}$ segment, so all of $Ω$ is one path component ([F4]) and $φ^{'} = φ + c$ for a constant $c \in R$ . Substituting into the formula of step 2.1 replaces $φ$ by $φ + c$ and leaves the right side unchanged, so the work is independent of the additive normalisation of the potential.

6.1step 3.1step 4.1step 5.1F5∎

Clauses (a)-(d) are steps 2.1, 3.1, 4.1 and 5.1. The force whose work was computed is the electric term $q E$ of the adopted external coupling [F5] along the prescribed path; the computation used only the potential representation, the gradient theorem and the constancy of the potential up to an additive constant, and asserted no equation of motion, no radiation, no work done on the sources and no claim that the particle follows $γ$ , which is exactly the scope stated.

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

Finite-source assembly energy equals the electric field energy

Physical scope

Electrostatic energy of a compactly supported $C^{1}$ charge distribution in vacuum within the pure electrostatic regime: the field energy $(ε_{0} / 2) \int ∣ E ∣^{2}$ , the assembly integral $\frac{1}{2} \int ρ φ$ and the symmetric double-kernel energy $(2 ε_{0})^{- 1} \iint Φ (x - y) ρ (x) ρ (y)$ are finite and equal. The assembly is an ideal quasi-static model with sources held by external support; no radiation, time-dependent work, material model, stability claim or identification of field energy with particle rest mass is asserted.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) as carried by the Lebesgue, polar and surface interfaces cited below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $ρ \in C_{c}^{1} (R^{3}; R)$ with support in ${\overline{B}}_{R}$ ( $C^{k}$ Euclidean maps and diffeomorphisms, Open ball, closed ball and sphere in a metric space), let $φ = ε_{0}^{- 1} N ρ$ and $E = - \nabla φ$ be the smooth-branch Coulomb solution of The localized Newtonian potential and the ideal point-charge Coulomb field(a) with the constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values and the pure electrostatic regime of The static and pure electrostatic regimes and their fixed-time reading, and let $Φ$ be the $n = 3$ kernel, $Φ (z) = 1 / (4 π ∣ z ∣)$ (Fundamental solution for the positive operator minus Laplacian, The closed form for the volume of the unit $n$ -ball, Agreement with the existing polar sphere measure). Then:

(a) [Finiteness] the three quantities $\frac{ε_{0}}{2} \int_{R^{3}} ∣ E ∣^{2} d λ_{3}, \frac{1}{2} \int_{R^{3}} ρ φ d λ_{3}, + \frac{1}{2 ε_{0}} \iint_{R^{3} \times R^{3}} Φ (x - y) ρ (x) ρ (y) d (λ_{3} \times λ_{3}) (x, y)$ are finite real numbers;

(b) [Identity] they are equal: $\frac{ε_{0}}{2} \int_{R^{3}} ∣ E ∣^{2} d λ_{3} = \frac{1}{2} \int_{R^{3}} ρ φ d λ_{3} = \frac{1}{2 ε_{0}} \iint_{R^{3} \times R^{3}} Φ (x - y) ρ (x) ρ (y) d (λ_{3} \times λ_{3}) (x, y) .$ The last expression is the external assembly work of the ideal quasistatic bookkeeping family $ρ_{s} = s ρ$ , $φ_{s} = s φ$ for $0 \leq s \leq 1$ . An added test charge brought from infinity, where the potential is zero, has electric-force work $- d q φ_{s}$ by Work of the electrostatic force equals charge times the potential drop and is path independent; the balancing external work is $+ d q φ_{s}$ . Therefore $W_{e x t} = \int_{0}^{1} s d s \int ρ φ = \frac{1}{2} \int ρ φ$ . Both signs of $ρ$ are allowed. The symmetric double kernel gives the equivalent ordered-pair interpretation of the factor $1 / 2$ .

(c) [Scope] the identity is a statement about the ideal quasi-static assembly of a compactly supported $C^{1}$ charge distribution in vacuum, all sources being held in place by external support (The static and pure electrostatic regimes and their fixed-time reading). It asserts no time-dependent work, no radiation, no material or stability model, and it does not identify the field energy with the rest mass of any particle; the point-source limit is treated separately below and is divergent.

Facts & Assumptions

Given: The Axiom of Countable Choice, a real $C^{1}$ function $ρ$ compactly supported in ${\overline{B}}_{R}$ with $R > 0$ as in the smooth Coulomb branch, the potential $φ = ε_{0}^{- 1} N ρ$ and field $E = - \nabla φ$ of the smooth Coulomb branch, and the kernel $Φ (z) = 1 / (4 π ∣ z ∣)$ off the origin.

[F1]

The Coulomb solution gives $φ \in C^{2} (R^{3})$ , $E = - \nabla φ \in C^{1}$ , $- Δ φ = ρ / ε_{0}$ , the far-field bound $∣ E (x) ∣ \leq ∥ ρ ∥_{1} / (π ε_{0} ∣ x ∣^{2})$ for $∣ x ∣ > 2 R$ , and, for $∣ x ∣ > 2 R$ , the potential bound $∣ φ (x) ∣ \leq ∣ Q ∣ / (4 π ε_{0} ∣ x ∣) + R ∥ ρ ∥_{1} / (π ε_{0} ∣ x ∣^{2})$ , where $Q : = \int ρ d λ_{3}$ ; the kernel is $Φ (z) = 1 / (4 π ∣ z ∣)$ with $σ (S^{2}) = 4 π$ (The localized Newtonian potential and the ideal point-charge Coulomb field, Fundamental solution for the positive operator minus Laplacian, Newtonian potential of compactly supported data, The far-field gradient of a compactly supported Newtonian potential, Far-field asymptotics of compact-source Newtonian potentials, The closed form for the volume of the unit $n$ -ball, Agreement with the existing polar sphere measure, The polar surface set function on the unit sphere).

[F2]

Green's first identity holds on a bounded $C^{1}$ domain for real $u = v \in C^{2}$ up to the closure, with the outward normal; the Euclidean balls are bounded $C^{1}$ domains with outward normal $x / ∣ x ∣$ on their bounding spheres, and $∣ S_{R^{'}} ∣ = 4 π R^{' 2}$ (First Green identity, Euclidean balls, annuli and spheres as bounded smooth domains and hypersurfaces, Agreement with the existing polar sphere measure, The closed form for the volume of the unit $n$ -ball).

[F3]

The Laplacian, divergence and gradient are the classical operators of the cited definitions, with the partial derivative convention of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ and the algebra rules for derivatives; the regime, field and unit conventions are those of the primitives (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Divergence and curl of a $C^{1}$ vector field, $C^{k}$ Euclidean maps and diffeomorphisms, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The static and pure electrostatic regimes and their fixed-time reading, 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, Framework-local SI units, the vacuum constants and the exact/measured status of their values, Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

The polar formula, translation invariance of Lebesgue measure, the product-measure identification for Euclidean spaces, Fubini and Tonelli, monotone and dominated convergence, and the order and scalar rules for the nonnegative integral are available; continuous functions on nonempty compact sets are bounded and attain extrema, and are Borel (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Integral invariance under measure-preserving maps, Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Fubini's theorem for L^1 functions on a sigma-finite product, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Monotone convergence for the integral, Dominated convergence, Monotonicity and nonnegative homogeneity of the nonnegative integral, Integral over a measurable subset, Integrable real and complex functions, and their integrals, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Continuous functions on Euclidean spaces are Borel measurable, Open ball, closed ball and sphere in a metric space, The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , Riemann–Lebesgue comparison for distribution test integrands).

[F5]

The work of the electric force along a prescribed path is $q$ times the potential drop and is additive in the added charge; the potential is linear in the source (Work of the electrostatic force equals charge times the potential drop and is path independent, Newtonian potential of compactly supported data).

Proof

technique · direct

1.1givenF1F4

Finiteness of the field energy: on the bounded ball ${\overline{B}}_{2 R}$ the continuous function $∣ E ∣^{2}$ is bounded by [F4] and $λ_{3} ({\overline{B}}_{2 R}) < \infty$ , so $\int_{∣ x ∣ \leq 2 R} ∣ E ∣^{2} < \infty$ ; on the far field the bound of [F1] gives $\int_{∣ x ∣ > 2 R} ∣ E ∣^{2} \leq \frac{∥ ρ ∥_{1}^{2}}{π^{2} ε_{0}^{2}} \int_{∣ x ∣ > 2 R} ∣ x ∣^{- 4} d λ_{3}$ , and the polar formula computes $\int_{2 R}^{\infty} r^{- 4} \cdot 4 π r^{2} d r = 4 π / (2 R) < \infty$ ([F1], [F4]).

2.1step 1.1F1F4

Finiteness of the other two quantities: $ρ$ is bounded and supported in ${\overline{B}}_{R}$ and $φ$ is continuous, so $∣ ρ φ ∣$ is bounded with compact support and $\int ∣ ρ φ ∣ < \infty$ . Put $M : = \sup ∣ ρ ∣$ . Splitting at the positive length $R$ , translation invariance and the polar formula give, uniformly in $x$ , $\int_{R^{3}} Φ (x - y) ∣ ρ (y) ∣ d λ_{3} (y) \leq M \int_{∣ z ∣ < R} Φ (z) d λ_{3} (z) + \frac{∥ ρ ∥_{1}}{4 π R} = \frac{M R^{2}}{2} + \frac{∥ ρ ∥_{1}}{4 π R},$ since $\int_{∣ z ∣ < R} Φ (z) d λ_{3} = \int_{0}^{R} r d r = R^{2} / 2$ . Tonelli therefore bounds the absolute double integral by $∥ ρ ∥_{1} (M R^{2} / 2 + ∥ ρ ∥_{1} / (4 π R)) < \infty$ . This proves the remaining finiteness claims of (a), with both terms in the kernel estimate carrying charge per length.

3.1step 2.1F1F2F3

Green's identity on the ball: fix $R^{'} > R$ . By [F2] the ball $B_{R^{'}}$ is a bounded $C^{1}$ domain with outward normal $ν (x) = x / ∣ x ∣$ on $S_{R^{'}}$ , and $φ \in C^{2} ({\overline{B}}_{R^{'}})$ ; Green's first identity with $u = v = φ$ gives $\int_{B_{R^{'}}} (φ Δ φ + ∣ D φ ∣^{2}) d λ_{3} = \int_{S_{R^{'}}} φ \partial_{ν} φ d S .$ Substituting $Δ φ = - ρ / ε_{0}$ and $∣ D φ ∣^{2} = ∣ E ∣^{2}$ from [F1], [F3] and rearranging gives $\int_{B_{R^{'}}} ∣ E ∣^{2} d λ_{3} = \frac{1}{ε_{0}} \int_{B_{R^{'}}} ρ φ d λ_{3} + \int_{S_{R^{'}}} φ \partial_{ν} φ d S .$

4.1step 3.1F1F2F4

Boundary estimate: take $R^{'} > 2 R$ . On $S_{R^{'}}$ the decay bound of [F1] gives $∣ φ (x) ∣ \leq C / R^{'}$ with $C : = (∣ Q ∣ + 2 ∥ ρ ∥_{1}) / (4 π ε_{0})$ , because $R / R^{'} < 1 / 2$ , the far-field bound gives $∣ \nabla φ (x) ∣ \leq ∥ ρ ∥_{1} / (π ε_{0} R^{' 2})$ , and $\partial_{ν} φ = \nabla φ \cdot x / ∣ x ∣$ has the same bound; with $∣ S_{R^{'}} ∣ = 4 π R^{' 2}$ from [F2] this yields $∣ \int_{S_{R^{'}}} φ \partial_{ν} φ d S ∣ \leq \frac{C}{R^{'}} \cdot \frac{∥ ρ ∥_{1}}{π ε_{0} R^{' 2}} \cdot 4 π R^{' 2} = \frac{4 C ∥ ρ ∥_{1}}{ε_{0} R^{'}},$ which tends to $0$ as $R^{'} \to \infty$ ([F1], [F2], [F4]).

5.1step 3.1step 4.1F4

First equality: the sets $B_{R^{'}}$ increase to $R^{3}$ along the increasing sequence $R_{n}^{'} = (n + 2) R$ , $n \geq 1$ , and $∣ E ∣^{2} \geq 0$ , so monotone convergence gives $\int_{B_{R^{'}}} ∣ E ∣^{2} \to \int_{R^{3}} ∣ E ∣^{2}$ ; since $ρ$ is supported in ${\overline{B}}_{R} \subset B_{R^{'}}$ , the integrals $\int_{B_{R^{'}}} ρ φ$ are eventually constant equal to $\int_{R^{3}} ρ φ$ , which is finite by step 2.1; the boundary term vanishes by step 4.1. Passing to the limit in the identity of step 3.1 therefore gives $\int ∣ E ∣^{2} = \frac{1}{ε_{0}} \int ρ φ$ , that is $\frac{ε_{0}}{2} \int ∣ E ∣^{2} = \frac{1}{2} \int ρ φ$ ([F4]).

6.1step 5.1F1F4

Second equality: by [F1] the potential is $φ = ε_{0}^{- 1} N ρ$ , and the product $ρ (x) Φ (x - y) ρ (y)$ is absolutely integrable over $R^{3} \times R^{3}$ by step 2.1, so Fubini applies: $\int_{R^{3}} ρ φ d λ_{3} = \frac{1}{ε_{0}} \int_{R^{3}} ρ (x) (\int_{R^{3}} Φ (x - y) ρ (y) d λ_{3} (y)) d λ_{3} (x) = \frac{1}{ε_{0}} \iint Φ (x - y) ρ (x) ρ (y) d (λ_{3} \times λ_{3}) .$ Since $Φ (x - y) = Φ (y - x)$ and $ρ (x) ρ (y) = ρ (y) ρ (x)$ , the double integral is unchanged by exchanging $x$ and $y$ , which is the ordered-pair reading of the factor; multiplying the identity by $\frac{1}{2}$ gives $\frac{1}{2} \int ρ φ = \frac{1}{2 ε_{0}} \iint Φ (x - y) ρ (x) ρ (y) d (λ_{3} \times λ_{3})$ , the second equality of (b) ([F1], [F4]).

7.1step 6.1F1F4F5

Assembly reading: for $0 \leq s \leq 1$ put $ρ_{s} : = s ρ$ and $φ_{s} : = ε_{0}^{- 1} N ρ_{s} = s φ$ , so that $E_{s} = s E$ and all the above estimates hold with $ρ_{s}$ in place of $ρ$ ([F1], [F5]). An increment of charge $d ρ_{s} = ρ d s$ added at $x$ to a configuration whose potential is $φ_{s}$ has electric-force work $- d ρ_{s} φ_{s} (x)$ when the charge is brought from infinity, by the work formula of [F5] together with the decay $φ_{s} (x) \to 0$ at infinity; the balancing external work is $+ d ρ_{s} φ_{s} (x)$ , so the total external assembly work is $W_{e x t} = \int_{0}^{1} (\int_{R^{3}} ρ (x) φ_{s} (x) d λ_{3} (x)) d s = \int_{0}^{1} s d s \int_{R^{3}} ρ φ d λ_{3} = \frac{1}{2} \int_{R^{3}} ρ φ d λ_{3},$ the interchange being justified by the absolute finiteness of step 2.1 applied to $ρ_{s}$ ; both signs of $ρ$ are allowed because all integrals involved are absolutely convergent ([F4], [F5]).

8.1step 1.1step 5.1step 6.1step 7.1F1∎

Clauses (a), (b) and (c): finiteness is steps 1.1-2.1, the two equalities are steps 5.1 and 6.1, the assembly reading is step 7.1, and the scope clause records that the construction is an ideal quasi-static bookkeeping with external support, uses the pure electrostatic regime, and neither models radiation or material dynamics nor identifies the field energy with a particle rest mass; the divergent point-source limit is treated separately below and is not asserted here.

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

Divergence of the point-charge field energy and the limits of mass inference

Physical scope

The truncated electric field energy of the idealised point charge in vacuum: the shell energy is $q^{2} / (8 π ε_{0}) (1 / δ - 1 / R)$ , finite for $0 < δ < R$ but divergent as the inner cutoff $δ ↓ 0$ for $q \neq 0$ . The item records this divergence as a property of the point idealisation and explicitly excludes inferences about particle rest mass, self-force, empirical measurement or inconsistency of the framework; finite-energy statements remain those of the compact-source branches.

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ ) as carried by the Lebesgue and polar interfaces (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $q \in R$ , $x_{0} \in R^{3}$ , and let $E_{q}$ be the ideal point-charge field of The localized Newtonian potential and the ideal point-charge Coulomb field(d) (The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values). For $0 < δ < R$ put $U_{q} (δ, R) : = \frac{ε_{0}}{2} \int_{{δ \leq ∣ x - x_{0} ∣ \leq R}} ∣ E_{q} (x) ∣^{2} d λ_{3} (x)$ (the truncated field energy of the shell between the two spheres, Open ball, closed ball and sphere in a metric space). Then:

(a) [Exact truncated energy] for all $0 < δ < R$ , $U_{q} (δ, R) = \frac{q^{2}}{8 π ε_{0}} (\frac{1}{δ} - \frac{1}{R});$

(b) [Divergence] $\lim_{R \to \infty} U_{q} (δ, R) = q^{2} / (8 π ε_{0} δ)$ for every $δ > 0$ , and for $q \neq 0$ $\lim_{δ ↓ 0} U_{q} (δ, R) = + \infty for every R > 0;$ so the field energy of the ideal point charge is locally infinite at its own location, and any finite-energy treatment of a point-like charge must modify the source/field near the singularity or introduce an explicit cutoff;

(c) [Limits of the mass inference] the identity is a divergence of the field energy of the idealised point source in the framework's SI conventions. It does not assert that a physical particle has infinite rest mass, does not identify field energy with rest mass, contains no measurement, and involves neither a self-force nor a contradiction claim for classical electrodynamics: the framework supplies no mass-energy identification for a charged particle, and the smooth compact-source field/assembly energy of Finite-source assembly energy equals the electric field energy remains finite. The compact $L^{1}$ branch of The localized Newtonian potential and the ideal point-charge Coulomb field retains its locally integrable potential and distributional Poisson equation; it does not guarantee finite field energy for arbitrary $L^{1}$ densities. The point source itself is the idealisation of Distributional Maxwell equations with point and stationary-sheet sources(b) (The static and pure electrostatic regimes and their fixed-time reading(e)); its energy divergence is a mathematical property of that idealisation, not an empirical result.

Facts & Assumptions

Given: The Axiom of Countable Choice, $q \in R$ , $x_{0} \in R^{3}$ , the point-charge field $E_{q} (z) = q z / (4 π ε_{0} ∣ z ∣^{3})$ off $z = 0$ of the Coulomb solution, and $0 < δ < R$ .

[F1]

The ideal point-charge field satisfies $∣ E_{q} (z) ∣ = ∣ q ∣ / (4 π ε_{0} ∣ z ∣^{2})$ for $z \neq 0$ ; its value at the pole is arbitrary and immaterial; the compact-source field/assembly energy of a $C^{1}$ compactly supported density is finite and equal to $\frac{1}{2} \int ρ φ$ , and the compact $L^{1}$ branch keeps a locally integrable potential with distributional Poisson equation but asserts no finite field energy (The localized Newtonian potential and the ideal point-charge Coulomb field, Finite-source assembly energy equals the electric field energy, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values).

[F2]

The polar formula computes radial integrals, the sphere area is $σ (S^{2}) = 4 π$ from $∣ S^{n - 1} ∣ = n ∣ B_{1} ∣$ and $∣ B_{1} ∣ = 4 π / 3$ , and the fundamental theorem evaluates $\int_{δ}^{R} r^{- 2} d r$ as the increment of the primitive $- 1 / r$ on compact subintervals, the Riemann and Lebesgue values agreeing for the continuous integrand; a singleton or the pole sphere set ${x_{0}}$ is null and does not affect the integral (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, The polar surface set function on the unit sphere, The closed form for the volume of the unit $n$ -ball, Agreement with the existing polar sphere measure, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Continuity and derivatives of positive-base real powers, Riemann–Lebesgue comparison for distribution test integrands, A nonnegative integral over a null set vanishes).

[F3]

The nonnegative integral is monotone and homogeneous, the integral of a radial density is the polar integral, and monotone convergence passes increasing limits; the point source is the idealisation of the weak point-source definition and of the static-to-distributional reading (Monotonicity and nonnegative homogeneity of the nonnegative integral, Integral over a measurable subset, Integrable real and complex functions, and their integrals, Monotone convergence for the integral, Continuous functions on Euclidean spaces are Borel measurable, Distributional Maxwell equations with point and stationary-sheet sources, The static and pure electrostatic regimes and their fixed-time reading, Open ball, closed ball and sphere in a metric space, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ ).

Proof

technique · direct

1.1givenF1F2F3

For $z \neq 0$ the field satisfies $∣ E_{q} (z) ∣^{2} = q^{2} / (16 π^{2} ε_{0}^{2} ∣ z ∣^{4})$ by [F1]; the shell ${δ \leq ∣ x - x_{0} ∣ \leq R}$ excludes $x_{0}$ because $δ > 0$ , and the integrand is continuous on the shell, hence Borel and locally bounded there; the value assigned to $E_{q}$ at $x_{0}$ does not affect the integral since ${x_{0}}$ is null ([F1], [F2], [F3]).

2.1step 1.1F2F3

Polar integration about $x_{0}$ : substituting $x = x_{0} + z$ and using the polar formula in $z$ with the radial density $q^{2} / (16 π^{2} ε_{0}^{2} r^{4})$ gives $\int_{{δ \leq ∣ x - x_{0} ∣ \leq R}} ∣ E_{q} (x) ∣^{2} d λ_{3} (x) = \frac{q^{2}}{16 π^{2} ε_{0}^{2}} \int_{δ}^{R} r^{- 4} \cdot 4 π r^{2} d r = \frac{q^{2}}{4 π ε_{0}^{2}} \int_{δ}^{R} r^{- 2} d r,$ where $σ (S^{2}) = 4 π$ and the radial integrals are taken over the compact interval $[δ, R]$ ([F2], [F3]).

3.1step 2.1F2F3

The fundamental theorem with the primitive $r \mapsto - r^{- 1}$ of $r \mapsto r^{- 2}$ gives $\int_{δ}^{R} r^{- 2} d r = 1 / δ - 1 / R$ ([F2]); substituting into step 2.1 and multiplying by $ε_{0} / 2$ yields $U_{q} (δ, R) = \frac{ε_{0}}{2} \cdot \frac{q^{2}}{4 π ε_{0}^{2}} (\frac{1}{δ} - \frac{1}{R}) = \frac{q^{2}}{8 π ε_{0}} (\frac{1}{δ} - \frac{1}{R}),$ which is clause (a); the quantity is finite for $0 < δ < R$ and of the sign of $q^{2}$ ([F3]).

4.1step 3.1F2F3

Clause (b): letting $R \to \infty$ in the exact formula gives $\lim_{R \to \infty} U_{q} (δ, R) = q^{2} / (8 π ε_{0} δ)$ for every $δ > 0$ , and letting $δ ↓ 0$ gives $\lim_{δ ↓ 0} U_{q} (δ, R) = + \infty$ whenever $q \neq 0$ , while for $q = 0$ both expressions vanish identically; the limits are read off the continuous elementary functions $R \mapsto 1 / R$ and $δ \mapsto 1 / δ$ on $(0, \infty)$ ([F2], [F3]).

5.1step 4.1F1F3∎

Clause (c): the computation used only the idealised point field of [F1] and the radial integration of [F2]; it yields a divergence of the truncated field energy of that idealisation. No identification of field energy with rest mass, no self-force, no measurement and no inconsistency claim is made; the smooth compact-source energy of Finite-source assembly energy equals the electric field energy remains finite by [F1], the compact $L^{1}$ branch keeps only its locally integrable potential and distributional Poisson equation, and the point source is the idealisation of the weak point-source definition together with the static-to-distributional reading [F3].

5 · Examples, experiments and counterexamples

None yet.

Electrostatics in vacuum

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The far-field gradient of a compactly supported Newtonian potential

Statement

Facts & Assumptions

Proof

The distributional gradient of a locally integrable function smooth off one point

Statement

Facts & Assumptions

Proof

Euclidean balls, annuli and spheres as bounded smooth domains and hypersurfaces

Statement

Facts & Assumptions

Proof

The static and pure electrostatic regimes and their fixed-time reading

Definition

Facts & Assumptions

Verification

Scalar potential and the Poisson equation in a star-shaped electrostatic region

Statement

Facts & Assumptions

Proof

The localized Newtonian potential and the ideal point-charge Coulomb field

Statement

Facts & Assumptions

Proof

Work of the electrostatic force equals charge times the potential drop and is path independent

Statement

Facts & Assumptions

Proof

Finite-source assembly energy equals the electric field energy

Statement

Facts & Assumptions

Proof

Divergence of the point-charge field energy and the limits of mass inference

Statement

Facts & Assumptions

Proof

5 · Examples, experiments and counterexamples

Sources