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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 | import Mathlib
import OAI.Analysis.Conductivity.Sources.AngularGraph
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology UnitAddTorus Matrix
open scoped ENNReal
theorem sourceAngularJet_attachedCovector {f : (Fin 3 → ℝ) → ℝ}
(hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) {a : ℝ} (ha : a≠0) (b : ℝ) (i j : Fin 4)
{x : Fin 3 → ℝ} (hx : x∈sourceCollarOpenBox) :
(fun k : Fin 3 => (sourceAngularJet f a⁻¹ b k.succ
(a*(x 0-b),torusAngles (sourceFaceAngles i j x))).re)=
attachedFlatTestCovector f a i j x := by
have h1 : |x 1|<1 := by exact hx.2.1
have h2 : |x 2|<1 := by exact hx.2.2
have ht : a⁻¹*(a*(x 0-b))+b=x 0 := by field_simp; ring
rw [sourceAngularJet_face hf a⁻¹ b (a*(x 0-b)) i j ht h1 h2]
have h := attachedFlatTestCovector_chain f ha i j (sourceCollarOpenBox_subset hx)
((hf.differentiable (by simp)).differentiableAt)
have h0 := congrFun h 0
have h1' := congrFun h 1
have h2' := congrFun h 2
simp only [sourceFaceAngleMatrix,endAxialMatrix,mulVec_diagonal] at h0 h1' h2'
change physicalTestCovector (f ∘ sourceCollarPiece i j) x 0=
1*(a*attachedFlatTestCovector f a i j x 0) at h0
change physicalTestCovector (f ∘ sourceCollarPiece i j) x 1=
(2*Real.pi*faceRayDensity 1 i (x 1))*(1*attachedFlatTestCovector f a i j x 1) at h1'
change physicalTestCovector (f ∘ sourceCollarPiece i j) x 2=
(2*Real.pi*faceRayDensity sourceRadialWidth j (x 2))*(1*attachedFlatTestCovector f a i j x 2) at h2'
ext k
fin_cases k
· change a⁻¹*physicalTestCovector (f ∘ sourceCollarPiece i j) x 0=_
rw [h0]; field_simp [ha]; rfl
· change physicalTestCovector (f ∘ sourceCollarPiece i j) x 1/(2*Real.pi*faceRayDensity 1 i (x 1))=_
rw [h1']
field_simp [(faceRayDensity_pos' (w:=1) (by norm_num) i (x 1)).ne',Real.pi_ne_zero]; rfl
· change physicalTestCovector (f ∘ sourceCollarPiece i j) x 2/(2*Real.pi*faceRayDensity sourceRadialWidth j (x 2))=_
rw [h2']
field_simp [(faceRayDensity_pos' (w:=sourceRadialWidth) (by norm_num [sourceRadialWidth,sourceHole]) j (x 2)).ne',Real.pi_ne_zero]; rfl
end ScalarConductivity
end
end
end OAI
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