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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 | import OAI.Geometry.IsometricImmersion.Calculus.VerticalSliceJets
import OAI.Geometry.IsometricImmersion.Darboux.ActualDriftEquation
import Mathlib.Analysis.Calculus.Deriv.Pi
namespace OAI
noncomputable section
open Set Function
open scoped ContDiff Matrix
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry
theorem fderiv_axis_eq_deriv_update {n : ℕ} {f : (Fin n → ℝ) → ℝ} {w : Fin n → ℝ}
(hf : DifferentiableAt ℝ f w) (i : Fin n) :
fderiv ℝ f w (Pi.single i 1) = deriv (fun t => f (Function.update w i t)) (w i) := by
have hd : HasFDerivAt f (fderiv ℝ f w) (Function.update w i (w i)) := by
simpa using hf.hasFDerivAt
exact (hd.comp_hasDerivAt (w i) (hasDerivAt_update w i (w i))).deriv.symm
def gradientStateIndex (i : Fin 2) : Fin 6 := if i = 0 then 2 else 3
theorem sixVariableP_update_gradient (g : MetricField) (w : DarbouxState) (i : Fin 2) (t : ℝ) :
sixVariableP g (Function.update w (gradientStateIndex i) t) =
solvedDarboux g (statePoint w) (w 4) (w 5) (Function.update (stateGradient w) i t) := by
have hp : statePoint (Function.update w (gradientStateIndex i) t) = statePoint w := by
ext j
fin_cases i <;> fin_cases j <;> simp [gradientStateIndex, statePoint]
have hv : stateGradient (Function.update w (gradientStateIndex i) t) =
Function.update (stateGradient w) i t := by
ext j
fin_cases i <;> fin_cases j <;> simp [gradientStateIndex, stateGradient]
unfold sixVariableP
rw [hp, hv]
fin_cases i <;> simp [gradientStateIndex]
theorem sixVariableP_update_mixed (g : MetricField) (w : DarbouxState) (t : ℝ) :
sixVariableP g (Function.update w 4 t) =
solvedDarboux g (statePoint w) t (w 5) (stateGradient w) := by
have hp : statePoint (Function.update w 4 t) = statePoint w := by
ext j
fin_cases j <;> simp [statePoint]
have hv : stateGradient (Function.update w 4 t) = stateGradient w := by
ext j
fin_cases j <;> simp [stateGradient]
unfold sixVariableP
rw [hp, hv]
simp
theorem sixVariableP_update_yy (g : MetricField) (w : DarbouxState) (t : ℝ) :
sixVariableP g (Function.update w 5 t) =
solvedDarboux g (statePoint w) (w 4) t (stateGradient w) := by
have hp : statePoint (Function.update w 5 t) = statePoint w := by
ext j
fin_cases j <;> simp [statePoint]
have hv : stateGradient (Function.update w 5 t) = stateGradient w := by
ext j
fin_cases j <;> simp [stateGradient]
unfold sixVariableP
rw [hp, hv]
simp
theorem solvedDarboux_first_variables_differentiableAt (g : MetricField) (p u : Coord) (b c : ℝ)
(hyy : jetYY g p c u ≠ 0) : DifferentiableAt ℝ (solvedDarboux g p b c) u := by
have hd := (jetConnection_differentiableAt g p u 0 0).add
((jetNumerator_differentiableAt g p u b).mul ((jetYY_differentiableAt g p u c).inv hyy))
exact hd
theorem sixVariableP_gradient_axis
{g : MetricField} {U : Set Coord} {w : DarbouxState}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hw : w ∈ darbouxStateDomain g U) (i : Fin 2) :
fderiv ℝ (sixVariableP g) w (Pi.single (gradientStateIndex i) 1) =
coordPartial i (solvedDarboux g (statePoint w) (w 4) (w 5)) (stateGradient w) := by
have hP : DifferentiableAt ℝ (sixVariableP g) w :=
((sixVariableP_contDiffOn hg hU).contDiffAt
((darbouxStateDomain_isOpen hg hU).mem_nhds hw)).differentiableAt (by simp)
rw [fderiv_axis_eq_deriv_update hP]
have heq : (fun t => sixVariableP g (Function.update w (gradientStateIndex i) t)) =
(fun t => solvedDarboux g (statePoint w) (w 4) (w 5) (Function.update (stateGradient w) i t)) :=
funext (sixVariableP_update_gradient g w i)
rw [heq]
have hi : w (gradientStateIndex i) = stateGradient w i := by
fin_cases i <;> simp [gradientStateIndex, stateGradient]
rw [hi]
exact (fderiv_axis_eq_deriv_update
(solvedDarboux_first_variables_differentiableAt g (statePoint w) (stateGradient w) (w 4) (w 5) hw.2) i).symm
theorem sixVariableP_gradient_axis_at_height
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U)
(hyy : covHessian g z p 1 1 ≠ 0) (i : Fin 2) :
fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single (gradientStateIndex i) 1) =
heightPFirst g z i p := by
rw [sixVariableP_gradient_axis hg hU (solutionJet_mem_domain g z hp hyy) i,
statePoint_solutionJet, stateGradient_solutionJet]
rfl
theorem sixVariableP_mixed_axis_at_height
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U)
(hyy : covHessian g z p 1 1 ≠ 0) :
fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 4 1) = 2 * hessianQuotient g z p := by
have hP := (sixVariableP_contDiffAt_solutionJet hg hU hp hyy).differentiableAt (by simp)
rw [fderiv_axis_eq_deriv_update hP]
have heq : (fun t => sixVariableP g (Function.update (solutionJet z p) 4 t)) =
(fun t => solvedDarboux g p t (coordPartial 1 (coordPartial 1 z) p) (fun i => coordPartial i z p)) := by
funext t
rw [sixVariableP_update_mixed, statePoint_solutionJet, stateGradient_solutionJet]
rfl
rw [heq]
change deriv (fun t => solvedDarboux g p t (coordPartial 1 (coordPartial 1 z) p)
(fun i => coordPartial i z p)) (coordPartial 0 (coordPartial 1 z) p) = _
rw [solvedDarboux_deriv_b g p _ _ _ hyy, jetMixed_at_height, jetYY_at_height]
rfl
theorem sixVariableP_yy_axis_at_height
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U)
(hyy : covHessian g z p 1 1 ≠ 0) :
fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 5 1) =
-((hessianQuotient g z p)^2 + gaussianCurvature g p * darbouxG g z p) := by
have hP := (sixVariableP_contDiffAt_solutionJet hg hU hp hyy).differentiableAt (by simp)
rw [fderiv_axis_eq_deriv_update hP]
have heq : (fun t => sixVariableP g (Function.update (solutionJet z p) 5 t)) =
(fun t => solvedDarboux g p (coordPartial 0 (coordPartial 1 z) p) t (fun i => coordPartial i z p)) := by
funext t
rw [sixVariableP_update_yy, statePoint_solutionJet, stateGradient_solutionJet]
rfl
rw [heq]
change deriv (fun t => solvedDarboux g p (coordPartial 0 (coordPartial 1 z) p) t
(fun i => coordPartial i z p)) (coordPartial 1 (coordPartial 1 z) p) = _
rw [solvedDarboux_deriv_c g p _ _ _ hyy, jetMixed_at_height, jetYY_at_height,
jetEnergy_at_height hg hp z]
unfold hessianQuotient darbouxG
ring
theorem sixVariableP_vertical_direction
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hp : p ∈ U)
(hyy : covHessian g z p 1 1 ≠ 0) (v : DarbouxState) (hv0 : v 0 = 0) (hv1 : v 1 = 0) :
fderiv ℝ (sixVariableP g) (solutionJet z p) v =
heightPFirst g z 0 p * v 2 + heightPFirst g z 1 p * v 3 +
2 * hessianQuotient g z p * v 4 -
((hessianQuotient g z p)^2 + gaussianCurvature g p * darbouxG g z p) * v 5 := by
have he : v = v 2 • (Pi.single 2 (1 : ℝ) : DarbouxState) +
v 3 • Pi.single 3 1 + v 4 • Pi.single 4 1 + v 5 • Pi.single 5 1 := by
ext i
fin_cases i <;> simp [hv0, hv1]
have h0 : fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 2 1) = heightPFirst g z 0 p := by
simpa only [gradientStateIndex, ↓reduceIte] using sixVariableP_gradient_axis_at_height hg hU hp hyy 0
have h1 : fderiv ℝ (sixVariableP g) (solutionJet z p) (Pi.single 3 1) = heightPFirst g z 1 p := by
simpa only [gradientStateIndex, show (1 : Fin 2) ≠ 0 by decide, ↓reduceIte] using
sixVariableP_gradient_axis_at_height hg hU hp hyy 1
have hl := congrArg (fderiv ℝ (sixVariableP g) (solutionJet z p)) he
simp only [map_add, map_smul, h0, h1, sixVariableP_mixed_axis_at_height hg hU hp hyy,
sixVariableP_yy_axis_at_height hg hU hp hyy, smul_eq_mul] at hl
rw [hl]
ring
end SmoothLocal.HighEquation
end
end OAI
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