| 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 |
|
|
