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 kinetic energy from an electromagnetic potential. -/ noncomputable def kineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → ℝ := fun x => - 1/(4 * 𝓕.μ₀) * {η' d | μ μ' ⊗ η' d | ν ν' ⊗ A.toFieldStrength x | μ ν ⊗ A.toFieldStrength x | μ' ν'}ᵀ.toField /-! ### A.1. Lorentz invariance of the kinetic term We show that the kine...

The kinetic energy from an electromagnetic potential. -/ noncomputable def kineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → ℝ := fun x => - 1/(4 * 𝓕.μ₀) * {η' d | μ μ' ⊗ η' d | ν ν' ⊗ A.toFieldStrength x | μ ν ⊗ A.toFieldStrength x | μ' ν'}ᵀ.toField /-! ### A.1. Lorentz invariance of the kinetic term We show that the kine...