| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.RegularPatch |
|
|
| namespace OAI |
|
|
| section |
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|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
|
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| lemma TwoFieldRankRegular.comp {u : Coord3 → Fin 2 → ℝ} {O V : Set Coord3} |
| (hO : IsOpen O) (hV : IsOpen V) |
| (hu : ContDiffOn ℝ (↑(⊤:ℕ∞)) u O) (h : TwoFieldRankRegular u O) |
| {P : Coord3 → Coord3} (hP : ContDiffOn ℝ (↑(⊤:ℕ∞)) P V) |
| (hm : MapsTo P V O) (hd : ∀ x∈V,Function.Surjective (fderiv ℝ P x)) : |
| TwoFieldRankRegular (fun x => u (P x)) V := by |
| have hPd (x) (hx : x∈V) := |
| (hP.differentiableOn (by simp) x hx).differentiableAt (hV.mem_nhds hx) |
| rcases h with h|h|⟨κ,hκ⟩ |
| · left |
| intro x hx |
| have hud := (hu.differentiableOn (by simp) (P x) (hm hx)).differentiableAt |
| (hO.mem_nhds (hm hx)) |
| change LinearIndependent ℝ (gradientColumns (fderiv ℝ (u ∘ P) x)).col |
| rw [(hud.hasFDerivAt.comp x (hPd x hx).hasFDerivAt).fderiv] |
| exact gradientColumns_rank _ ((gradientColumns_surjective _ (h (P x) (hm hx))).comp (hd x hx)) |
| · right; left |
| obtain ⟨v,l,κ,hv,hl,hvd,he⟩ := h |
| refine ⟨fun x => v (P x),l,κ,hv.comp hP hm,hl,?_,fun x hx j => he (P x) (hm hx) j⟩ |
| intro x hx hz |
| have hvd' := (hv.differentiableOn (by simp) (P x) (hm hx)).differentiableAt |
| (hO.mem_nhds (hm hx)) |
| change fderiv ℝ (v ∘ P) x=0 at hz |
| rw [(hvd'.hasFDerivAt.comp x (hPd x hx).hasFDerivAt).fderiv] at hz |
| apply hvd (P x) (hm hx) |
| ext y |
| obtain ⟨t,ht⟩ := hd x hx y |
| have ht' := congrArg (fun D : Coord3 →L[ℝ] ℝ => D t) hz |
| simpa only [ContinuousLinearMap.comp_apply,ht,_root_.zero_apply] using ht' |
| · exact Or.inr (Or.inr ⟨κ,fun x hx => hκ (P x) (hm hx)⟩) |
|
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| lemma RegularPatch.comp {u : Coord3 → Fin 2 → ℝ} {A B : Coord3 → Symmetric3} |
| {O V : Set Coord3} (h : RegularPatch u A O) (hV : IsOpen V) |
| {P : Coord3 → Coord3} (hP : ContDiffOn ℝ (↑(⊤:ℕ∞)) P V) |
| (hm : MapsTo P V O) (hd : ∀ x∈V,Function.Surjective (fderiv ℝ P x)) |
| (hB : ContDiffOn ℝ (↑(⊤:ℕ∞)) (fun x => (B x).val) V) : |
| RegularPatch (fun x => u (P x)) B V := |
| ⟨hV,h.2.1.comp hP hm,hB,h.2.2.2.comp h.1 hV h.2.1 hP hm hd⟩ |
|
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| end ScalarConductivity |
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| end |
| end |
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| end OAI |
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|
