Electric Field space deriv eq time deriv
Electromagnetism.ElectromagneticPotential.IsPlaneWave.electricField_space_deriv_eq_time_deriv
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_space_deriv_eq_time_deriv {d : ℕ} {𝓕 : FreeSpace}
{A : ElectromagneticPotential d}
{s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time)
(x : Space d) (i : Fin d) (k : Fin d) :
∂[k] (A.electricField 𝓕.c t · i) x = - (s.unit k / 𝓕.c.val) •
∂ₜ (A.electricField 𝓕.c · x i) tFormal artifact
Lean source
lemma electricField_space_deriv_eq_time_deriv {d : ℕ} {𝓕 : FreeSpace} {A : ElectromagneticPotential d} {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time) (x : Space d) (i : Fin d) (k : Fin d) : ∂[k] (A.electricField 𝓕.c t · i) x = - (s.unit k / 𝓕.c.val) • ∂ₜ (A.electricField 𝓕.c · x i) t := by rw [Space.deriv_euclid, IsPlaneWave.electricField_space_deriv P hA t x k, Time.deriv_euclid, IsPlaneWave.electricField_time_deriv P hA t x] simp only [fderiv_eq_smul_deriv, one_smul, PiLp.smul_apply, smul_eq_mul, neg_mul, mul_neg, neg_neg] field_simp · exact electricField_differentiable_time hA x · exact electricField_differentiable_space hA t- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Vacuum/IsPlaneWave.lean:281-293
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