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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 | import OAI.Geometry.IsometricImmersion.Calculus.CoordinateDerivatives
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
namespace OAI
noncomputable section
open scoped ContDiff Topology
namespace SmoothLocal.Geometry
variable {f h : Coord → ℝ} {p : Coord} {U : Set Coord}
theorem coordPartial_div
(hf : DifferentiableAt ℝ f p) (hh : DifferentiableAt ℝ h p)
(hne : h p ≠ 0) (i : Fin 2) :
coordPartial i (fun x => f x / h x) p =
(coordPartial i f p * h p - f p * coordPartial i h p) / (h p) ^ 2 := by
have hinv : HasFDerivAt (fun x : Coord => (h x)⁻¹)
(-(h p ^ 2)⁻¹ • fderiv ℝ h p) p := by
simpa only [Function.comp_def] using
(hasDerivAt_inv hne).comp_hasFDerivAt p hh.hasFDerivAt
have hquot : HasFDerivAt (fun x : Coord => f x / h x)
(f p • (-(h p ^ 2)⁻¹ • fderiv ℝ h p) +
(h p)⁻¹ • fderiv ℝ f p) p := by
simpa only [div_eq_mul_inv] using hf.hasFDerivAt.fun_mul hinv
unfold coordPartial
rw [hquot.fderiv]
simp only [add_apply, smul_apply, smul_eq_mul]
field_simp [hne]
ring
theorem coordPartial_div_on
(hU : IsOpen U) (hp : p ∈ U)
(hf : DifferentiableOn ℝ f U) (hh : DifferentiableOn ℝ h U)
(hne : h p ≠ 0) (i : Fin 2) :
coordPartial i (fun x => f x / h x) p =
(coordPartial i f p * h p - f p * coordPartial i h p) / (h p) ^ 2 := by
exact coordPartial_div (hf.differentiableAt (hU.mem_nhds hp))
(hh.differentiableAt (hU.mem_nhds hp)) hne i
def coordinateDrift (q w : Coord → ℝ) (p : Coord) : ℝ :=
coordPartial 0 w p - q p * coordPartial 1 w p
theorem coordinateDrift_quotient (q : Coord → ℝ)
(hf : DifferentiableAt ℝ f p) (hh : DifferentiableAt ℝ h p)
(hne : h p ≠ 0) :
coordinateDrift q (fun x => f x / h x) p =
(coordinateDrift q f p * h p - f p * coordinateDrift q h p) / (h p) ^ 2 := by
unfold coordinateDrift
rw [coordPartial_div hf hh hne 0, coordPartial_div hf hh hne 1]
field_simp [hne]
ring
theorem coordinateDrift_self_quotient
(hf : DifferentiableAt ℝ f p) (hh : DifferentiableAt ℝ h p)
(hne : h p ≠ 0) :
coordinateDrift (fun x => f x / h x) (fun x => f x / h x) p =
(coordPartial 0 f p - (f p / h p) * coordPartial 1 f p) / h p -
(f p / h p) *
(coordPartial 0 h p - (f p / h p) * coordPartial 1 h p) / h p := by
rw [coordinateDrift_quotient (fun x => f x / h x) hf hh hne]
unfold coordinateDrift
field_simp [hne]
theorem quotient_contDiffOn
(hf : ContDiffOn ℝ ∞ f U) (hh : ContDiffOn ℝ ∞ h U)
(hne : ∀ p ∈ U, h p ≠ 0) :
ContDiffOn ℝ ∞ (fun p => f p / h p) U := by
exact hf.div hh hne
def explicitDriftError (h E T J : Coord → ℝ) (p : Coord) : ℝ :=
E p / (h p) ^ 2 * (T p - coordPartial 1 h p / h p) + J p / h p
theorem explicitDriftError_contDiffOn
{E T J : Coord → ℝ} (hU : IsOpen U)
(hh : ContDiffOn ℝ ∞ h U) (hE : ContDiffOn ℝ ∞ E U)
(hT : ContDiffOn ℝ ∞ T U) (hJ : ContDiffOn ℝ ∞ J U)
(hne : ∀ p ∈ U, h p ≠ 0) :
ContDiffOn ℝ ∞ (explicitDriftError h E T J) U := by
have hG : ContDiffOn ℝ ∞ (fun p => E p / (h p) ^ 2) U :=
hE.div (hh.pow 2) (fun p hp => pow_ne_zero 2 (hne p hp))
have hhy : ContDiffOn ℝ ∞ (coordPartial 1 h) U :=
partial_contDiffOn hh hU 1
exact (hG.mul (hT.sub (hhy.div hh hne))).add (hJ.div hh hne)
theorem height_energy_quotient_pos (E h : ℝ) (hE : 0 < E) (hh : h ≠ 0) :
0 < E / h ^ 2 :=
div_pos hE (sq_pos_of_ne_zero hh)
end SmoothLocal.Geometry
end
end OAI
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