File size: 3,678 Bytes
59d1c80 | 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Geometry.IsometricImmersion.Flows.FlowChart
import OAI.Geometry.IsometricImmersion.Flows.FlowTransport
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.LinearAlgebra.Determinant
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
theorem capChart_injOn_of_pairChart
{Y : ℝ → ℝ → ℝ} {e : OpenPartialHomeomorph (ℝ × ℝ) (ℝ × ℝ)}
(hsource : e.source = pairRectangle 2 (-2) 2)
(heq : (e : (ℝ × ℝ) → (ℝ × ℝ)) = triangularFlow Y) :
InjOn (capChart Y) capChartDomain := by
intro p hp r hr hsame
have hp' : (p 0,p 1) ∈ e.source := by rw [hsource]; exact hp
have hr' : (r 0,r 1) ∈ e.source := by rw [hsource]; exact hr
have hpair : e (p 0,p 1) = e (r 0,r 1) := by
rw [heq]
apply Prod.ext
· simpa only [triangularFlow,capChart_zero] using congrArg (fun q : Coord => q 0) hsame
· simpa only [triangularFlow,capChart_one,capFlowHeight] using congrArg (fun q : Coord => q 1) hsame
have hinj := e.injOn hp' hr' hpair
ext i
fin_cases i
· exact congrArg Prod.fst hinj
· exact congrArg Prod.snd hinj
theorem capChart_fderiv_matrix {Y : ℝ → ℝ → ℝ}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
{p : Coord} (hp : p ∈ capChartDomain) :
LinearMap.toMatrix' (fderiv ℝ (capChart Y) p).toLinearMap =
!![1,0; coordPartial 0 (capFlowHeight Y) p,coordPartial 1 (capFlowHeight Y) p] := by
ext i j
rw [LinearMap.toMatrix'_apply]
change coordPartial j (capChart Y) p i = _
rw [capChart_partial hYs hp j]
fin_cases i <;> fin_cases j <;> simp
theorem capChart_det_fderiv {Y : ℝ → ℝ → ℝ}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
{p : Coord} (hp : p ∈ capChartDomain) :
(fderiv ℝ (capChart Y) p).det = coordPartial 1 (capFlowHeight Y) p := by
rw [ContinuousLinearMap.det,←LinearMap.det_toMatrix',capChart_fderiv_matrix hYs hp]
simp [Matrix.det_fin_two]
theorem capChart_abs_det_fderiv {Y : ℝ → ℝ → ℝ}
(hYs : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s)
{p : Coord} (hp : p ∈ capChartDomain) :
|(fderiv ℝ (capChart Y) p).det| = coordPartial 1 (capFlowHeight Y) p := by
rw [capChart_det_fderiv hYs hp,abs_of_pos (capFlowHeight_partial_initial_pos hYs hvar hp)]
theorem capChart_jacobian_bounds
{q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {M : ℝ}
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
(hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, Y s t ∈ Icc (-3 : ℝ) 3)
(hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
(hM : 0 ≤ M) (hMq : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
{p : Coord} (hp : p ∈ capChartDomain) :
Real.exp (-2*M) ≤ coordPartial 1 (capFlowHeight Y) p ∧
coordPartial 1 (capFlowHeight Y) p ≤ Real.exp (2*M) := by
have hYs := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
rw [←(capFlowHeight_hasDerivAt_initial hYs hp).deriv]
exact cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM hMq hp.2 hp.1
end SmoothLocal.Flow
end
end OAI
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