Magnetic Field Matrix basis repr eq field Strength
Electromagnetism.DistElectromagneticPotential.magneticFieldMatrix_basis_repr_eq_fieldStrength
Project documentation
The magnetic field matrix of an electromagnetic potential which is a distribution. -/ noncomputable def magneticFieldMatrix {d} (c : SpeedOfLight) : DistElectromagneticPotential d →ₗ[ℝ] (Time × Space d) →d[ℝ] (EuclideanSpace ℝ (Fin d) ⊗[ℝ] EuclideanSpace ℝ (Fin d)) where toFun A := ⟨TensorProduct.map (Lorentz.Vector.spatialCLM d).toLinearMap (Lorentz.Vect...
Exact Lean statement
lemma magneticFieldMatrix_basis_repr_eq_fieldStrength {c : SpeedOfLight}
(A : DistElectromagneticPotential d)
(ε : 𝓢(Time × Space d, ℝ)) (i j : Fin d) :
((PiLp.basisFun 2 ℝ (Fin d)).tensorProduct (PiLp.basisFun 2 ℝ (Fin d))).repr
(A.magneticFieldMatrix c ε) (i, j) =
(Lorentz.Vector.basis.tensorProduct Lorentz.Vector.basis).repr
(distTimeSlice c A.fieldStrength ε) (Sum.inr i, Sum.inr j)Formal artifact
Lean source
lemma magneticFieldMatrix_basis_repr_eq_fieldStrength {c : SpeedOfLight} (A : DistElectromagneticPotential d) (ε : 𝓢(Time × Space d, ℝ)) (i j : Fin d) : ((PiLp.basisFun 2 ℝ (Fin d)).tensorProduct (PiLp.basisFun 2 ℝ (Fin d))).repr (A.magneticFieldMatrix c ε) (i, j) = (Lorentz.Vector.basis.tensorProduct Lorentz.Vector.basis).repr (distTimeSlice c A.fieldStrength ε) (Sum.inr i, Sum.inr j) := by simp only [magneticFieldMatrix_eq_vectorPotential, EuclideanSpace.basisFun_apply, map_sum, map_smul, Finsupp.coe_finsetSum, Finsupp.coe_smul, Finset.sum_apply, Pi.smul_apply, Basis.tensorProduct_repr_tmul_apply, PiLp.basisFun_repr, PiLp.single_apply, smul_eq_mul, mul_ite, mul_one, mul_zero, Finset.sum_ite_irrel, Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, Finset.sum_const_zero, distTimeSlice_apply, fieldStrength_basis_repr_eq_single, inr_i_inr_i, neg_mul, one_mul, sub_neg_eq_add] simp only [vectorPotential, Vector.spatialCLM, LinearMap.coe_mk, AddHom.coe_mk, Space.distSpaceDeriv_apply_CLM, ContinuousLinearMap.coe_comp, ContinuousLinearMap.coe_mk', Function.comp_apply, ← distTimeSlice_apply, distTimeSlice_distDeriv_inr] ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Electromagnetism/Distributional/MagneticField.lean:120-136
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Adiabatic relation log
adiabatic_relation_log
Plain-language statement
Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.
Source project: Physlib
Person-level attribution pending.
Adiabatic relation Ua Ub Va Vb
adiabatic_relation_UaUbVaVb
Plain-language statement
Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.
Source project: Physlib
Person-level attribution pending.
Deriv Within mean Energy Beta eq neg variance
CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance
Plain-language statement
(∂U/∂β) = -Var(E) for finite systems.
Source project: Physlib
Person-level attribution pending.