| 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 |
|
|
