`Equations`

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module: IndisputableMonolith.MaxwellDEC
domain: MaxwellDEC
line: 57 · github
papers citing: none yet

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IndisputableMonolith.MaxwellDEC on GitHub at line 57.

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  54    Faraday and Gauss-M are stated cleanly. Constitutive relations (const_D, const_B)
  55    and source equations (ampere_qs, gauss_e) require dimension n=3 for form-degree
  56    alignment, so they are parameterized by the dimension hypothesis `hn`. -/
  57structure Equations (α : Type) [HasCoboundary α] [inst : HasHodge α] (M : Medium α) where
  58  E : DForm α 1
  59  H : DForm α 1
  60  B : DForm α 2
  61  D : DForm α 2
  62  src : Sources α
  63  /-- Faraday QS: curl E = 0 (no time-varying B in quasistatic limit) -/
  64  faraday_qs : HasCoboundary.d E = fun _ => 0
  65  /-- Ampere QS: d H = star J (requires n-1=2 for form-degree alignment) -/
  66  ampere_qs  : ∀ (hn : inst.n - 1 = 2),
  67    HasCoboundary.d H = cast (congrArg (DForm α) hn) (HasHodge.star src.J)
  68  /-- Gauss E: d D = star rho (requires n-0=3 for form-degree alignment) -/
  69  gauss_e    : ∀ (hn : inst.n - 0 = 3),
  70    HasCoboundary.d D = cast (congrArg (DForm α) hn) (HasHodge.star src.ρ)
  71  /-- Gauss M: div B = 0 (no magnetic monopoles) -/
  72  gauss_m    : HasCoboundary.d B = fun _ => 0
  73  /-- Constitutive: D = ε⋆E (requires n-1=2 for form-degree alignment) -/
  74  const_D    : ∀ (hn : inst.n - 1 = 2), D = hn ▸ (fun s => M.eps * HasHodge.star E s)
  75  /-- Constitutive: B = μ⋆H (requires n-1=2 for form-degree alignment) -/
  76  const_B    : ∀ (hn : inst.n - 1 = 2), B = hn ▸ (fun s => M.mu * HasHodge.star H s)
  77
  78/-- Pointwise Hodge energy density for 2-forms: ω · (⋆ω) on each 2-simplex.
  79    Requires n = 4 (spacetime dimension). -/
  80def energy2 (ω : DForm α 2) [inst : HasHodge α] (h : inst.n - 2 = 2) : DForm α 2 :=
  81  fun s => ω s * (h ▸ (HasHodge.star (k:=2) ω)) s
  82
  83/-- Admissibility: strictly positive material parameters. -/
  84def Admissible [HasHodge α] (M : Medium α) : Prop := 0 < M.eps ∧ 0 < M.mu
  85
  86/-- Positivity of the Hodge energy density for admissible media, provided the
  87    instance supplies `star2_psd`. This is signature-agnostic and delegates the

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