secondDeriv_smul_local

formal source

  74
  75/-- Localized version of second derivative linearity (scalar multiplication).
  76    This only requires differentiability in a neighborhood of the point x. -/
  77lemma secondDeriv_smul_local (f : (Fin 4 → ℝ) → ℝ) (c : ℝ) (μ ν : Fin 4)
  78    (x : Fin 4 → ℝ)
  79    (h1 : ∀ᶠ s in 𝓝 0, DifferentiableAt ℝ (fun t => f (coordRay (coordRay x ν s) μ t)) 0)
  80    (h2 : DifferentiableAt ℝ (fun s => partialDeriv_v2 f μ (coordRay x ν s)) 0) :
  81  secondDeriv (fun y => c * f y) μ ν x = c * secondDeriv f μ ν x := by
  82  unfold secondDeriv
  83  have h_ev : ∀ᶠ s in 𝓝 0, partialDeriv_v2 (fun z => c * f z) μ (coordRay x ν s) =
  84                          c * partialDeriv_v2 f μ (coordRay x ν s) := by
  85    apply h1.mono
  86    intro s hs
  87    exact partialDeriv_v2_smul f c μ (coordRay x ν s) hs
  88  rw [Filter.EventuallyEq.deriv_eq h_ev]
  89  exact deriv_const_mul c h2
  90
  91/-- Second derivative linearity (scalar multiplication). -/
  92lemma secondDeriv_smul (f : (Fin 4 → ℝ) → ℝ) (c : ℝ) (μ ν : Fin 4)
  93    (x : Fin 4 → ℝ)
  94    (h1 : ∀ y, DifferentiableAt ℝ (fun t => f (coordRay y μ t)) 0)
  95    (h2 : DifferentiableAt ℝ (fun s => partialDeriv_v2 f μ (coordRay x ν s)) 0) :
  96  secondDeriv (fun y => c * f y) μ ν x = c * secondDeriv f μ ν x := by
  97  unfold secondDeriv
  98  have h_partial : ∀ y, partialDeriv_v2 (fun z => c * f z) μ y = c * partialDeriv_v2 f μ y := by
  99    intro y
 100    exact partialDeriv_v2_smul f c μ y (h1 y)
 101  simp only [h_partial]
 102  exact deriv_const_mul c h2
 103
 104/-- Laplacian linearity (scalar multiplication). -/
 105lemma laplacian_smul (f : (Fin 4 → ℝ) → ℝ) (c : ℝ) (x : Fin 4 → ℝ)
 106    (h1 : ∀ μ y, DifferentiableAt ℝ (fun t => f (coordRay y μ t)) 0)
 107    (h2 : ∀ μ ν, DifferentiableAt ℝ (fun s => partialDeriv_v2 f μ (coordRay x ν s)) 0) :