Research proof index

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

The variational gradient of the lagrangian of electromagnetic field. -/ noncomputable def gradLagrangian {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) (J : LorentzCurrentDensity d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, lagrangian 𝓕 ⟨q'⟩ J x) /-! ### C.3. The variational gradient in terms of the gradient of the kinetic term -/ lemma...

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...