Research proof index

The magnetic field from the electromagnetic potential. -/ noncomputable def magneticField (c : SpeedOfLight := 1) (A : ElectromagneticPotential) : MagneticField := fun t x => (∇ ⨯ (A.vectorPotential c t)) x lemma magneticField_eq {c : SpeedOfLight} (A : ElectromagneticPotential) : A.magneticField c = fun t x => (∇ ⨯ (A.vectorPotential c t)) x := rfl /-! #...

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 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 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.

The magnetic field of the electromagnetic potential created from the electric field E and the magnetic field B is B, as long as Gauss's law is satisfied.

ofGradient intertwines the Lorentz action on potentials with composition by Λ⁻¹ on the gauge function: Λ • ofGradient χ = ofGradient (χ ∘ (Λ⁻¹ • ·)).