Electric Field eq propogator cross magnetic Field Matrix
Electromagnetism.ElectromagneticPotential.IsPlaneWave.electricField_eq_propogator_cross_magneticFieldMatrix
Project documentation
The corresponding magnetic field function from ℝ to Fin d × Fin d → ℝ of a plane wave. -/ noncomputable def magneticFunction {d : ℕ} {𝓕 : FreeSpace} {A : ElectromagneticPotential d} {s : Direction d} (hA : IsPlaneWave 𝓕 A s) : ℝ → Fin d × Fin d → ℝ := Classical.choose hA.2 lemma magneticFieldMatrix_eq_magneticFunction {d : ℕ} {𝓕 : FreeSpace} {A : E...
Exact Lean statement
lemma electricField_eq_propogator_cross_magneticFieldMatrix {d : ℕ}
{𝓕 : FreeSpace} {A : ElectromagneticPotential d}
{s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ ∞ A)
(h : IsExtrema 𝓕 A 0) (i : Fin d) :
∃ C, ∀ t x, A.electricField 𝓕.c t x i =
𝓕.c * ∑ j, A.magneticFieldMatrix 𝓕.c t x (i, j) * s.unit j + CFormal artifact
Lean source
lemma electricField_eq_propogator_cross_magneticFieldMatrix {d : ℕ} {𝓕 : FreeSpace} {A : ElectromagneticPotential d} {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ ∞ A) (h : IsExtrema 𝓕 A 0) (i : Fin d) : ∃ C, ∀ t x, A.electricField 𝓕.c t x i = 𝓕.c * ∑ j, A.magneticFieldMatrix 𝓕.c t x (i, j) * s.unit j + C := by have hA2 : ContDiff ℝ 2 A := hA.of_le ENat.LEInfty.out apply Space.equal_up_to_const_of_deriv_eq · exact electricField_apply_differentiable hA2 · exact (Differentiable.fun_sum fun j _ => (magneticFieldMatrix_differentiable A hA2 (i, j)).mul_const _).const_mul _ · intro t x rw [P.time_deriv_electricField_eq_magneticFieldMatrix hA _ t x i] congr · intro t x i rw [P.space_deriv_electricField_eq_magneticFieldMatrix hA] congr- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Vacuum/IsPlaneWave.lean:508-524
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