Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Three Dim Point Particle div electric Field

Electromagnetism.DistElectromagneticPotential.threeDimPointParticle_div_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_div_electricField {𝓕} (q : ℝ) (r₀ : Space 3) :
    distSpaceDiv ((threeDimPointParticle 𝓕 q r₀).electricField 𝓕.c) =
    (1/𝓕.ε₀) • constantTime (q • diracDelta ℝ r₀)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma threeDimPointParticle_div_electricField {𝓕} (q : ℝ) (r₀ : Space 3) :    distSpaceDiv ((threeDimPointParticle 𝓕 q r₀).electricField 𝓕.c) =    (1/𝓕.ε₀) • constantTime (q • diracDelta ℝ r₀) := by  rw [threeDimPointParticle_electricField]  simp only [Int.reduceNeg, zpow_neg, map_smul, smul_smul]  have h1 := Space.distDiv_inv_pow_eq_dim (d := 3)  simp at h1  trans (q / (4 * π * 𝓕.ε₀)) •    distSpaceDiv (constantTime <|    distTranslate (basis.repr r₀) <|    (distOfFunction (fun x => ‖x‖ ^ (-3 : ℤ) • basis.repr x)      (IsDistBounded.zpow_smul_repr_self (- 3 : ℤ) (by omega))))  · ext η    simp [distTranslate_ofFunction]  simp only [Int.reduceNeg, zpow_neg, one_div]  rw [constantTime_distSpaceDiv, distDiv_distTranslate, h1]  simp only [map_smul, smul_smul]  ext η  simp [constantTime_apply, diracDelta_apply, distTranslate_apply]  left  ring_nf  field_simp

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/Electromagnetism/PointParticle/ThreeDimension.lean:278-299

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