Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

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.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources