Research proof index

An electric potential which gives a given constant E-field and B-field. -/ @[nolint unusedArguments] noncomputable def constantEB {d : ℕ} (c : SpeedOfLight) (E₀ : EuclideanSpace ℝ (Fin d)) (B₀ : Fin d × Fin d → ℝ) (B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)) : ElectromagneticPotential d where val := fun x μ => match μ with | Sum.inl _ => - (1/c) * ⟪E₀,...

An electric potential which gives a given constant E-field and B-field. -/ @[nolint unusedArguments] noncomputable def constantEB {d : ℕ} (c : SpeedOfLight) (E₀ : EuclideanSpace ℝ (Fin d)) (B₀ : Fin d × Fin d → ℝ) (B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)) : ElectromagneticPotential d where val := fun x μ => match μ with | Sum.inl _ => - (1/c) * ⟪E₀,...

An electric potential which gives a given constant E-field and B-field. -/ @[nolint unusedArguments] noncomputable def constantEB {d : ℕ} (c : SpeedOfLight) (E₀ : EuclideanSpace ℝ (Fin d)) (B₀ : Fin d × Fin d → ℝ) (B₀_antisymm : ∀ i j, B₀ (i, j) = - B₀ (j, i)) : ElectromagneticPotential d where val := fun x μ => match μ with | Sum.inl _ => - (1/c) * ⟪E₀,...

The matrix corresponding to the magnetic field in general dimensions. In 3 space-dimensions this reduces to a vector. -/ noncomputable def magneticFieldMatrix (c : SpeedOfLight := 1) (A : ElectromagneticPotential d) : Time → Space d → (Fin d × Fin d) → ℝ := timeSlice c <| fun x ij => A.fieldStrengthMatrix x (Sum.inr ij.1, Sum.inr ij.2) lemma magneticFie...

The electromagnetic potential from an electric and a magnetic field. This defines the electromagnetic potential in the Poincare gauge. -/ noncomputable def ofElectromagneticField (c : SpeedOfLight) (E : Time → Space 3 → EuclideanSpace ℝ (Fin 3)) (B : Time → Space 3 → EuclideanSpace ℝ (Fin 3)) : ElectromagneticPotential 3 := let A := fun t (x : Space) => -...

The electric field of the electromagnetic potential created from the electric field E and the magnetic field B is E, as long as Gauss's law for magnetism and Faraday's law are satisfied. -/ lemma ofElectromagneticField_electricField {c : SpeedOfLight} (E : Time → Space 3 → EuclideanSpace ℝ (Fin 3)) (B : Time → Space 3 → EuclideanSpace ℝ (Fin 3)) (E_...