Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Grad Kinetic Term add

Electromagnetism.ElectromagneticPotential.gradKineticTerm_add

physics quantum field theory relativity kinetic term

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The variational gradient of the kinetic term of an electromagnetic potential. -/ noncomputable def gradKineticTerm {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d) : SpaceTime d → Lorentz.Vector d := (δ (q':=A), ∫ x, kineticTerm 𝓕 ⟨q'⟩ x) /-! ### B.1. Variational gradient in terms of fderiv We give a first simplification of the variational gradient...

Exact Lean statement

lemma gradKineticTerm_add {d} {𝓕 : FreeSpace} (A1 A2 : ElectromagneticPotential d)
    (hA1 : ContDiff ℝ ∞ A1) (hA2 : ContDiff ℝ ∞ A2) :
    (A1 + A2).gradKineticTerm 𝓕 = A1.gradKineticTerm 𝓕 + A2.gradKineticTerm 𝓕

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Full Lean sourceLean 4

lemma gradKineticTerm_add {d} {𝓕 : FreeSpace} (A1 A2 : ElectromagneticPotential d)    (hA1 : ContDiff ℝ ∞ A1) (hA2 : ContDiff ℝ ∞ A2) :    (A1 + A2).gradKineticTerm 𝓕 = A1.gradKineticTerm 𝓕 + A2.gradKineticTerm 𝓕 := by  funext x  rw [gradKineticTerm_eq_fieldStrength (A1 + A2) x (hA1.add hA2)]  simp only [Pi.add_apply]  rw [gradKineticTerm_eq_fieldStrength A1 x hA1, gradKineticTerm_eq_fieldStrength A2 x hA2,    ← Finset.sum_add_distrib]  apply Finset.sum_congr rfl (fun ν _ => ?_)  rw [← smul_add, ← add_smul, ← Finset.sum_add_distrib]  congr  funext μ  rw [SpaceTime.deriv_eq, SpaceTime.deriv_eq, SpaceTime.deriv_eq]  conv_lhs =>    enter [1, 2, x]    rw [fieldStrengthMatrix_add _ _ _ (hA1.differentiable (by simp))      (hA2.differentiable (by simp))]    simp [Finsupp.coe_add, Pi.add_apply]  rw [fderiv_fun_add    (fieldStrengthMatrix_differentiable (hA1.of_le ENat.LEInfty.out)).differentiableAt    (fieldStrengthMatrix_differentiable (hA2.of_le ENat.LEInfty.out)).differentiableAt]  rfl

Project: Physlib
License: Apache-2.0
Commit: dd43e9e65791
Source: Physlib/Electromagnetism/Dynamics/KineticTerm.lean:602-623

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