Research proof index

The magnetic field matrix of an electromagnetic potential which is a distribution. -/ noncomputable def magneticFieldMatrix {d} (c : SpeedOfLight) : DistElectromagneticPotential d →ₗ[ℝ] (Time × Space d) →d[ℝ] (EuclideanSpace ℝ (Fin d) ⊗[ℝ] EuclideanSpace ℝ (Fin d)) where toFun A := ⟨TensorProduct.map (Lorentz.Vector.spatialCLM d).toLinearMap (Lorentz.Vect...

The magnetic field matrix of an electromagnetic potential which is a distribution. -/ noncomputable def magneticFieldMatrix {d} (c : SpeedOfLight) : DistElectromagneticPotential d →ₗ[ℝ] (Time × Space d) →d[ℝ] (EuclideanSpace ℝ (Fin d) ⊗[ℝ] EuclideanSpace ℝ (Fin d)) where toFun A := ⟨TensorProduct.map (Lorentz.Vector.spatialCLM d).toLinearMap (Lorentz.Vect...

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 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. -/ lemma ofElectromagneticField_magneticField {c : SpeedOfLight} (E : ElectricField) (B : MagneticField) (B_contDiff : ∀ t, ContDiff ℝ 1 (B t)) (B_grad : ∀ t, ∇ ⬝ (B t) = 0) : (ofElectromagneticFie...

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...