To Field Strength eval eq basis repr
Electromagnetism.ElectromagneticPotential.toFieldStrength_eval_eq_basis_repr
Plain-language statement
Evaluating both tensor indices of the field strength gives the coefficient in the standard tensor-product basis.
Exact Lean statement
lemma toFieldStrength_eval_eq_basis_repr {d} (A : ElectromagneticPotential d)
(x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) :
toField {A.toFieldStrength x | [μ] [ν]}ᵀ =
(Lorentz.CoVector.basis.tensorProduct Lorentz.Vector.basis).repr
(A.toFieldStrength x) (μ, ν)Formal artifact
Lean source
lemma toFieldStrength_eval_eq_basis_repr {d} (A : ElectromagneticPotential d) (x : SpaceTime d) (μ ν : Fin 1 ⊕ Fin d) : toField {A.toFieldStrength x | [μ] [ν]}ᵀ = (Lorentz.CoVector.basis.tensorProduct Lorentz.Vector.basis).repr (A.toFieldStrength x) (μ, ν) := by trans (Lorentz.Vector.basis.tensorProduct Lorentz.Vector.basis).repr (A.toFieldStrength x) (μ, ν) · conv_rhs => rw [prod_eq_sum_eval Vector.basis_eq_map_tensor_basis Vector.basis_eq_map_tensor_basis (A.toFieldStrength x)] simp [Basis.tensorProduct_repr_tmul_apply, Finsupp.single_apply] · rfl- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Kinematics/FieldStrength.lean:354-365
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