Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Three Dim Point Particle electric Field

Electromagnetism.DistElectromagneticPotential.threeDimPointParticle_electricField

physics quantum field theory relativity three dimension

Project documentation

The electromagnetic potential of a point particle stationary at r₀ of 3d space. -/ noncomputable def threeDimPointParticle (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 3) : DistElectromagneticPotential 3 := (SpaceTime.distTimeSlice 𝓕.c).symm <| Space.constantTime <| distOfFunction (fun x => (((q * 𝓕.μ₀ * 𝓕.c)/ (4 * π)) * ‖x - r₀‖⁻¹) • Lorentz.Vector.basis (S...

Exact Lean statement

lemma threeDimPointParticle_electricField (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 3) :
    (threeDimPointParticle 𝓕 q r₀).electricField 𝓕.c =
    (q/ (4 * π * 𝓕.ε₀)) • constantTime (distOfFunction (fun x : Space 3 =>
      ‖x - r₀‖ ^ (- 3 : ℤ) • basis.repr (x - r₀))
      ((IsDistBounded.zpow_smul_repr_self (- 3 : ℤ) (by omega)).comp_sub_right r₀))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma threeDimPointParticle_electricField (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 3) :    (threeDimPointParticle 𝓕 q r₀).electricField 𝓕.c =    (q/ (4 * π * 𝓕.ε₀)) • constantTime (distOfFunction (fun x : Space 3 =>      ‖x - r₀‖ ^ (- 3 : ℤ) • basis.repr (x - r₀))      ((IsDistBounded.zpow_smul_repr_self (- 3 : ℤ) (by omega)).comp_sub_right r₀)) := by  simp only [electricField, LinearMap.coe_mk, AddHom.coe_mk, threeDimPointParticle_scalarPotential,    smul_eq_mul, threeDimPointParticle_vectorPotential, map_zero, sub_zero, Int.reduceNeg, zpow_neg]  rw [constantTime_distSpaceGrad, distOfFunction_mul_fun _    (IsDistBounded.inv_shift r₀)]  simp only [map_smul]  trans -(q / (4 * π * 𝓕.ε₀)) • (constantTime <| distGrad <| distTranslate (basis.repr r₀) <|    (distOfFunction (fun x => ‖x‖⁻¹) (IsDistBounded.inv)))  · simp [distTranslate_ofFunction]  rw [Space.distTranslate_distGrad]  have := Space.distGrad_distOfFunction_norm_zpow (d := 3) (-1) (by grind)  simp_all [distOfFunction_neg, distTranslate_ofFunction]

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/Electromagnetism/PointParticle/ThreeDimension.lean:223-238

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