Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Three Dim Point Particle scalar Potential

Electromagnetism.DistElectromagneticPotential.threeDimPointParticle_scalarPotential

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_scalarPotential (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 3) :
    (threeDimPointParticle 𝓕 q r₀).scalarPotential 𝓕.c =
    Space.constantTime (distOfFunction (fun x => (q/ (4 * π * 𝓕.ε₀))• ‖x - r₀‖⁻¹)
      (((IsDistBounded.inv_shift _).const_mul_fun _)))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma threeDimPointParticle_scalarPotential (𝓕 : FreeSpace) (q : ℝ) (r₀ : Space 3) :    (threeDimPointParticle 𝓕 q r₀).scalarPotential 𝓕.c =    Space.constantTime (distOfFunction (fun x => (q/ (4 * π * 𝓕.ε₀))• ‖x - r₀‖⁻¹)      (((IsDistBounded.inv_shift _).const_mul_fun _))) := by  ext ε  simp only [scalarPotential, Lorentz.Vector.temporalCLM, Fin.isValue, map_smul,    ContinuousLinearMap.comp_smulₛₗ, ringHom_apply, threeDimPointParticle, LinearMap.coe_mk,    AddHom.coe_mk, ContinuousLinearEquiv.apply_symm_apply, FunLike.coe_smul,    ContinuousLinearMap.coe_comp, LinearMap.coe_toContinuousLinearMap', Pi.smul_apply,    Function.comp_apply, constantTime_apply, distOfFunction_vector_eval, Lorentz.Vector.apply_smul,    Lorentz.Vector.basis_apply, ↓reduceIte, mul_one, smul_eq_mul]  rw [distOfFunction_mul_fun _ (IsDistBounded.inv_shift _),    distOfFunction_mul_fun _ (IsDistBounded.inv_shift _)]  simp only [FunLike.coe_smul, Pi.smul_apply, smul_eq_mul]  ring_nf  simp only [𝓕.c_sq, one_div, mul_inv_rev, mul_eq_mul_right_iff, inv_eq_zero, OfNat.ofNat_ne_zero,    or_false]  field_simp

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/Electromagnetism/PointParticle/ThreeDimension.lean:177-194

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