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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 | import OAI.Geometry.IsometricImmersion.Taylor.TopChainExtraction
import OAI.Geometry.IsometricImmersion.Estimates.LargestBlockLinearization
import OAI.Geometry.IsometricImmersion.Immersions.HeightCoefficientRegularity
namespace OAI
noncomputable section
open Set Function Filter
open scoped ContDiff Topology BigOperators Matrix
namespace SmoothLocal.HighEquation
open SmoothLocal.Geometry
theorem second_fderiv_comp_constant_direction
{F : DarbouxState → ℝ} {J : Coord → DarbouxState} {p : Coord}
(hF : ContDiffAt ℝ ∞ F (J p)) (hJ : DifferentiableAt ℝ J p) (v : DarbouxState) :
iteratedFDeriv ℝ 2 F (J p) ![coordPartial 1 J p, v] =
coordPartial 1 (fun r => fderiv ℝ F (J r) v) p := by
have hdF : DifferentiableAt ℝ (fderiv ℝ F) (J p) :=
(hF.fderiv_right (m := 1) (WithTop.coe_le_coe.mpr le_top)).differentiableAt (by norm_num)
have hc : DifferentiableAt ℝ (fun r => fderiv ℝ F (J r)) p := hdF.comp p hJ
rw [iteratedFDeriv_two_apply]
simp only [Matrix.cons_val_zero, Matrix.cons_val_one]
unfold coordPartial
have hcomp : fderiv ℝ (fun r => fderiv ℝ F (J r)) p =
(fderiv ℝ (fderiv ℝ F) (J p)).comp (fderiv ℝ J p) := by
convert fderiv_comp p hdF hJ using 1
rfl
rw [fderiv_clm_apply hc (differentiableAt_const (c := v)), hcomp]
simp
theorem sixVariableP_second_vertical_direction
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hyy : ∀ r ∈ U, covHessian g z r 1 1 ≠ 0) (hp : p ∈ U)
(v : DarbouxState) (hv0 : v 0 = 0) (hv1 : v 1 = 0) :
iteratedFDeriv ℝ 2 (sixVariableP g) (solutionJet z p)
![coordPartial 1 (solutionJet z) p, v] =
coordPartial 1 (heightPFirst g z 0) p * v 2 +
coordPartial 1 (heightPFirst g z 1) p * v 3 +
2 * coordPartial 1 (hessianQuotient g z) p * v 4 -
coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
gaussianCurvature g r * darbouxG g z r) p * v 5 := by
rw [second_fderiv_comp_constant_direction
(sixVariableP_contDiffAt_solutionJet hg hU hp (hyy _ hp))
(((solutionJet_contDiffOn hU hz).contDiffAt (hU.mem_nhds hp)).differentiableAt (by simp))]
have heq : (fun r => fderiv ℝ (sixVariableP g) (solutionJet z r) v) =ᶠ[𝓝 p]
(fun r => heightPFirst g z 0 r * v 2 + heightPFirst g z 1 r * v 3 +
2 * hessianQuotient g z r * v 4 -
((hessianQuotient g z r)^2 + gaussianCurvature g r * darbouxG g z r) * v 5) := by
filter_upwards [hU.mem_nhds hp] with r hr
exact sixVariableP_vertical_direction hg hU hr (hyy _ hr) v hv0 hv1
have h0 := ((heightPFirst_contDiffOn hg hU hz hyy 0).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp)
have h1 := ((heightPFirst_contDiffOn hg hU hz hyy 1).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp)
have hq := ((hessianQuotient_contDiffOn hg hU hz hyy).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp)
have hK := ((gaussianCurvature_contDiffOn hg hU).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp)
have hG := ((darbouxG_contDiffOn hg hU hz hyy).contDiffAt
(hU.mem_nhds hp)).differentiableAt (by simp)
have ha : DifferentiableAt ℝ (fun r => (hessianQuotient g z r)^2 +
gaussianCurvature g r * darbouxG g z r) p := (hq.pow 2).add (hK.mul hG)
have hd := (((h0.hasFDerivAt.mul_const (v 2)).add
(h1.hasFDerivAt.mul_const (v 3))).add
((hq.hasFDerivAt.const_mul 2).mul_const (v 4))).sub (ha.hasFDerivAt.mul_const (v 5))
simp only [Pi.add_def, Pi.sub_def] at hd
unfold coordPartial
rw [heq.fderiv_eq]
change fderiv ℝ (fun r => heightPFirst g z 0 r * v 2 + heightPFirst g z 1 r * v 3 +
2 * hessianQuotient g z r * v 4 -
((hessianQuotient g z r)^2 + gaussianCurvature g r * darbouxG g z r) * v 5) p
(Pi.single 1 1) = _
rw [hd.fderiv]
simp only [add_apply, sub_apply, smul_apply, smul_eq_mul]
ring
def firstJetPairRemainder (g : MetricField) (z : Coord → ℝ) (n : ℕ) (p : Coord) : ℝ :=
coordPartial 1 (heightPFirst g z 0) p * coordPartial 0 (verticalJet z n) p +
coordPartial 1 (heightPFirst g z 1) p * coordPartial 1 (verticalJet z n) p
def secondJetCorrection (g : MetricField) (z u : Coord → ℝ) (p : Coord) : ℝ :=
2 * coordPartial 1 (hessianQuotient g z) p * coordPartial 0 u p -
coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
gaussianCurvature g r * darbouxG g z r) p * coordPartial 1 u p
theorem actualP_pair_extract_secondJet
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {x y : ℝ}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hyy : ∀ p ∈ U, covHessian g z p 1 1 ≠ 0)
(hp : (![x, y] : Coord) ∈ U) (n : ℕ) (hn : 2 ≤ n) :
chainTerm (sixVariableP g) (solutionJetCurve z x) (ChainWord.pair 1 n) y =
secondJetCorrection g z (verticalJet z (n + 1)) ![x, y] +
firstJetPairRemainder g z n ![x, y] := by
have hfirst : iteratedDeriv 1 (solutionJetCurve z x) y =
coordPartial 1 (solutionJet z) ![x, y] :=
verticalJet_slice hU (solutionJet_contDiffOn hU hz) x 1 y hp
rw [chainTerm_pair_eq, hfirst, solutionJetCurve_iteratedDeriv hU hz hp n hn]
change iteratedFDeriv ℝ 2 (sixVariableP g) (solutionJet z ![x, y]) _ = _
rw [sixVariableP_second_vertical_direction hg hU hz hyy hp _ (by rfl) (by rfl)]
change
coordPartial 1 (heightPFirst g z 0) ![x, y] * coordPartial 0 (verticalJet z n) ![x, y] +
coordPartial 1 (heightPFirst g z 1) ![x, y] * coordPartial 1 (verticalJet z n) ![x, y] +
2 * coordPartial 1 (hessianQuotient g z) ![x, y] *
coordPartial 0 (coordPartial 1 (verticalJet z n)) ![x, y] -
coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
gaussianCurvature g r * darbouxG g z r) ![x, y] *
coordPartial 1 (coordPartial 1 (verticalJet z n)) ![x, y] = _
unfold secondJetCorrection firstJetPairRemainder
change _ =
(2 * coordPartial 1 (hessianQuotient g z) ![x, y] *
coordPartial 0 (coordPartial 1 (verticalJet z n)) ![x, y] -
coordPartial 1 (fun r => (hessianQuotient g z r)^2 +
gaussianCurvature g r * darbouxG g z r) ![x, y] *
coordPartial 1 (coordPartial 1 (verticalJet z n)) ![x, y]) + _
ring
end SmoothLocal.HighEquation
end
end OAI
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