Dist Time Slice dist Deriv inl
SpaceTime.distTimeSlice_distDeriv_inl
Project documentation
The time slice of a distribution on SpaceTime d to form a distribution on Time × Space d. -/ noncomputable def distTimeSlice {M d} [NormedAddCommGroup M] [NormedSpace ℝ M] (c : SpeedOfLight := 1) : ((SpaceTime d) →d[ℝ] M) ≃L[ℝ] ((Time × Space d) →d[ℝ] M) where toFun f := f ∘L compCLMOfContinuousLinearEquiv (F := ℝ) ℝ (SpaceTime.toTimeAndSpace c (d :=...
Exact Lean statement
lemma distTimeSlice_distDeriv_inl {M d} [NormedAddCommGroup M] [NormedSpace ℝ M]
{c : SpeedOfLight}
(f : (SpaceTime d) →d[ℝ] M) :
distTimeSlice c (distDeriv (Sum.inl 0) f) =
(1/c.val) • Space.distTimeDeriv (distTimeSlice c f)Formal artifact
Lean source
lemma distTimeSlice_distDeriv_inl {M d} [NormedAddCommGroup M] [NormedSpace ℝ M] {c : SpeedOfLight} (f : (SpaceTime d) →d[ℝ] M) : distTimeSlice c (distDeriv (Sum.inl 0) f) = (1/c.val) • Space.distTimeDeriv (distTimeSlice c f) := by ext κ rw [distTimeSlice_apply, distDeriv_apply, fderivD_apply] simp only [Fin.isValue, one_div, FunLike.coe_smul, Pi.smul_apply] rw [distTimeDeriv_apply, fderivD_apply, distTimeSlice_apply] simp only [Fin.isValue, smul_neg, neg_inj] rw [← map_smul] congr ext x change fderiv ℝ (κ ∘ toTimeAndSpace c) x (Lorentz.Vector.basis (Sum.inl 0)) = c.val⁻¹ • fderiv ℝ κ (toTimeAndSpace c x) (1, 0) rw [fderiv_comp] simp only [toTimeAndSpace_fderiv, Fin.isValue, ContinuousLinearMap.coe_comp, ContinuousLinearEquiv.coe_coe, Function.comp_apply, smul_eq_mul] rw [toTimeAndSpace_basis_inl'] rw [map_smul] simp only [one_div, smul_eq_mul] · apply Differentiable.differentiableAt exact SchwartzMap.differentiable κ · fun_prop- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/SpaceAndTime/SpaceTime/TimeSlice.lean:161-184
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.