| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CompactDepthReductionTwo |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory Matrix |
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| lemma TwoFieldRankRegular.linearCoordinates {u : Coord3 → Fin 2 → ℝ} {O : Set Coord3} |
| (hO : IsOpen O) (hu : ContDiffOn ℝ (↑(⊤:ℕ∞)) u O) |
| (h : TwoFieldRankRegular u O) (P : Coord3 ≃L[ℝ] Coord3) |
| (Q : (Fin 2 → ℝ) ≃L[ℝ] (Fin 2 → ℝ)) : |
| TwoFieldRankRegular (fun x => Q (u (P x))) (P ⁻¹' O) := by |
| rcases h with h|h|⟨κ,hκ⟩ |
| · left |
| intro x hx |
| have hd := (hu.differentiableOn (by simp) (P x) hx).differentiableAt (hO.mem_nhds hx) |
| have he := Q.hasFDerivAt.comp x (hd.hasFDerivAt.comp x P.hasFDerivAt) |
| change LinearIndependent ℝ (gradientColumns (fderiv ℝ (Q ∘ u ∘ P) x)).col |
| rw [he.fderiv] |
| apply gradientColumns_rank |
| exact Q.surjective.comp ((gradientColumns_surjective _ (h (P x) hx)).comp P.surjective) |
| · right; left |
| obtain ⟨v,l,κ,hv,hl,hd,he⟩ := h |
| refine ⟨fun x => v (P x),Q l,Q κ,?_,?_,?_,?_⟩ |
| · exact hv.comp P.contDiff.contDiffOn (fun _ hx => hx) |
| · obtain ⟨j,hj⟩ := hl |
| by_contra! hh |
| have hz : Q l=0 := funext hh |
| have hl0 : l=0 := Q.injective (hz.trans (map_zero Q).symm) |
| exact hj (congrFun hl0 j) |
| · intro x hx hz |
| have hxv := (hv.differentiableOn (by simp) (P x) hx).differentiableAt (hO.mem_nhds hx) |
| have hh := (hxv.hasFDerivAt.comp x P.hasFDerivAt).fderiv |
| change fderiv ℝ (v ∘ P) x=0 at hz |
| rw [hh] at hz |
| apply hd (P x) hx |
| ext y |
| have h := congrArg (fun D : Coord3 →L[ℝ] ℝ => D (P.symm y)) hz |
| change (fderiv ℝ v (P x)) (P (P.symm y))=0 at h |
| simpa only [P.apply_symm_apply,_root_.zero_apply] using h |
| · intro x hx j |
| have hh : u (P x)=v (P x) • l+κ := by |
| ext i |
| simpa only [Pi.add_apply,Pi.smul_apply,smul_eq_mul,mul_comm] using he (P x) hx i |
| change Q (u (P x)) j=Q l j*v (P x)+Q κ j |
| rw [hh,map_add,map_smul] |
| simp only [Pi.add_apply,Pi.smul_apply,smul_eq_mul,mul_comm] |
| · exact Or.inr (Or.inr ⟨Q κ,fun x hx => congrArg Q (hκ (P x) hx)⟩) |
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| end ScalarConductivity |
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| end |
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| end OAI |
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