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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CompactDepthReductionTwo
namespace OAI
noncomputable section
namespace ScalarConductivity
open Real Set Filter Topology MeasureTheory Matrix
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)⟩)
end ScalarConductivity
end
end OAI
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