Research proof index

The electromagnetic potential for a Harmonic wave travelling in the x-direction with wave number k. -/ noncomputable def harmonicWaveX (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) : ElectromagneticPotential d.succ where val := fun x μ => match μ with | Sum.inl 0 => 0 | Sum.inr 0 => 0 | Sum.inr ⟨Nat.succ i, h⟩ => -E₀ ⟨i, Nat.succ_lt_succ_i...

The electromagnetic potential for a Harmonic wave travelling in the x-direction with wave number k. -/ noncomputable def harmonicWaveX (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) : ElectromagneticPotential d.succ where val := fun x μ => match μ with | Sum.inl 0 => 0 | Sum.inr 0 => 0 | Sum.inr ⟨Nat.succ i, h⟩ => -E₀ ⟨i, Nat.succ_lt_succ_i...

The electromagnetic potential for a Harmonic wave travelling in the x-direction with wave number k. -/ noncomputable def harmonicWaveX (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) : ElectromagneticPotential d.succ where val := fun x μ => match μ with | Sum.inl 0 => 0 | Sum.inr 0 => 0 | Sum.inr ⟨Nat.succ i, h⟩ => -E₀ ⟨i, Nat.succ_lt_succ_i...

The electromagnetic potential for a Harmonic wave travelling in the x-direction with wave number k. -/ noncomputable def harmonicWaveX (𝓕 : FreeSpace) (k : ℝ) (E₀ : Fin d → ℝ) (φ : Fin d → ℝ) : ElectromagneticPotential d.succ where val := fun x μ => match μ with | Sum.inl 0 => 0 | Sum.inr 0 => 0 | Sum.inr ⟨Nat.succ i, h⟩ => -E₀ ⟨i, Nat.succ_lt_succ_i...

The condition on an electromagnetic potential to be an extrema of the lagrangian. -/ def IsExtrema {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : Prop := gradLagrangian 𝓕 A J = 0 lemma isExtrema_iff_gradLagrangian {𝓕 : FreeSpace} (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : IsExtrema 𝓕 A J ↔ A....

The condition on an electromagnetic potential to be an extrema of the lagrangian. -/ def IsExtrema {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : Prop := gradLagrangian 𝓕 A J = 0 lemma isExtrema_iff_gradLagrangian {𝓕 : FreeSpace} (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : IsExtrema 𝓕 A J ↔ A....