openAI-math-dataset / lean /OAI /Analysis /Conductivity /Geometry /ParametricPotentialCoordinates.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Geometry.RegularPotentialCoordinates | |
| import OAI.Analysis.Conductivity.Variational.ParametricDiffeomorph | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set Filter Topology | |
| variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] | |
| theorem exists_parametric_potential_coordinates | |
| {u : P×Coord3 → Fin 2 → ℝ} (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) | |
| (p : P) (x : Coord3) (hD : Function.Surjective (fderiv ℝ (fun y => u (p,y)) x)) | |
| {U : Set (P×Coord3)} (hU : IsOpen U) (hpx : (p,x)∈U) : | |
| ∃ Y : OpenPartialHomeomorph (P×Coord3) (P×Coord3), | |
| (p,x)∈Y.source ∧ Y.source⊆U ∧ | |
| (∀ q y,(Y (q,y)).1=q) ∧ | |
| (∀ q y,coordinatePair (Y (q,y)).2=u (q,y)) ∧ | |
| ContDiff ℝ (↑(⊤:ℕ∞)) Y ∧ ContDiffOn ℝ (↑(⊤:ℕ∞)) Y.symm Y.target := by | |
| let D := fderiv ℝ (fun y => u (p,y)) x | |
| obtain ⟨Q,hQ⟩ := ContinuousLinearMap.HasRightInverse.of_surjective_of_finiteDimensional hD | |
| let L := potentialCompletionEquiv D hD | |
| let φ : P×Coord3 → Coord3 := fun q => L (completedPotential (fun y => u (q.1,y)) D Q q.2) | |
| have hφ : ContDiff ℝ (↑(⊤:ℕ∞)) φ := | |
| L.contDiff.comp (((Q.contDiff.comp hu).add contDiff_snd).sub (Q.contDiff.comp (D.contDiff.comp contDiff_snd))) | |
| have hup : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y => u (p,y)) := | |
| hu.comp (contDiff_const.prodMk contDiff_id) | |
| have hφd : fderiv ℝ (fun y => φ (p,y)) x=(L : Coord3 →L[ℝ] Coord3) := by | |
| have hh := L.hasFDerivAt.comp x (completedPotential_derivative | |
| ((hup.differentiable (by simp) x).hasFDerivAt) Q) | |
| simpa only [φ,D,Function.comp_def,ContinuousLinearMap.comp_id] using hh.fderiv | |
| obtain ⟨Y,hp,hsub,he,hY,hYi⟩ := exists_parametric_diffeomorph hφ p x L hφd hU hpx | |
| refine ⟨Y,hp,hsub,fun q y => by rw [he],?_,hY,hYi⟩ | |
| intro q y | |
| rw [he] | |
| exact (potentialCompletionEquiv_pairing D hD _).trans | |
| (completedPotential_pairing (fun y => u (q,y)) D Q hQ y) | |
| omit [FiniteDimensional ℝ P] in | |
| lemma fiberSlice_smooth | |
| (Y : OpenPartialHomeomorph (P×Coord3) (P×Coord3)) | |
| (hY : ∀ p x,(Y (p,x)).1=p) | |
| (hsm : ContDiff ℝ (↑(⊤:ℕ∞)) Y) | |
| (hinv : ContDiffOn ℝ (↑(⊤:ℕ∞)) Y.symm Y.target) (p : P) : | |
| ContDiff ℝ (↑(⊤:ℕ∞)) (fiberSlice Y hY p) ∧ | |
| ContDiffOn ℝ (↑(⊤:ℕ∞)) (fiberSlice Y hY p).symm (fiberSlice Y hY p).target := by | |
| constructor | |
| · exact contDiff_snd.comp (hsm.comp (contDiff_const.prodMk contDiff_id)) | |
| · exact contDiff_snd.contDiffOn.comp | |
| (hinv.comp (contDiff_const.prodMk contDiff_id).contDiffOn (fun _ hx => hx)) | |
| (fun _ _ => mem_univ _) | |
| end ScalarConductivity | |
| end | |
| end OAI | |