To Field Strength antisymmetric
Electromagnetism.ElectromagneticPotential.toFieldStrength_antisymmetric
Project documentation
Index evaluation of the field strength tensor agrees with the corresponding component of the field strength matrix. -/ lemma toFieldStrength_eval_eq_fieldStrengthMatrix {d} (A : ElectromagneticPotential d) (x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) : toField {A.toFieldStrength x | [μ] [ν]}ᵀ = A.fieldStrengthMatrix x (μ, ν) := by rw [toFieldStrength_eval_eq_b...
Exact Lean statement
lemma toFieldStrength_antisymmetric {d} (A : ElectromagneticPotential d) (x : SpaceTime d) :
{A.toFieldStrength x | μ ν = - (A.toFieldStrength x | ν μ)}ᵀFormal artifact
Lean source
lemma toFieldStrength_antisymmetric {d} (A : ElectromagneticPotential d) (x : SpaceTime d) : {A.toFieldStrength x | μ ν = - (A.toFieldStrength x | ν μ)}ᵀ := by apply (Tensor.basis _).repr.injective ext b rw [toTensor_toFieldStrength_basis_repr, permT_basis_repr_symm_apply, map_neg] simp only [Nat.reduceAdd, Fin.isValue, Nat.reduceSucc, Finsupp.coe_neg, Pi.neg_apply] rw [toTensor_toFieldStrength_basis_repr, ← Finset.sum_neg_distrib] refine Finset.sum_congr rfl fun κ _ => ?_ simp only [Fin.isValue, neg_sub] rfl- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Kinematics/FieldStrength.lean:497-506
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