deriv_add

formal source

  51  (φ.ψ x_plus - φ.ψ x) / h
  52
  53/-- Directional derivative is linear in the field. -/
  54theorem deriv_add (φ₁ φ₂ : ScalarField) (μ : Fin 4) (x : Fin 4 → ℝ) :
  55  directional_deriv (add φ₁ φ₂) μ x =
  56    directional_deriv φ₁ μ x + directional_deriv φ₂ μ x := by
  57  simp [directional_deriv, add]
  58  ring
  59
  60theorem deriv_smul (c : ℝ) (φ : ScalarField) (μ : Fin 4) (x : Fin 4 → ℝ) :
  61  directional_deriv (smul c φ) μ x = c * directional_deriv φ μ x := by
  62  simp only [directional_deriv, smul]
  63  ring
  64
  65/-- Derivative of constant field is zero. -/
  66theorem deriv_constant (c : ℝ) (μ : Fin 4) (x : Fin 4 → ℝ) :
  67  directional_deriv (constant c) μ x = 0 := by
  68  simp only [directional_deriv, constant]
  69  norm_num
  70
  71/-- Gradient: collection of all directional derivatives ∂_μ ψ. -/
  72noncomputable def gradient (φ : ScalarField) (x : Fin 4 → ℝ) : Fin 4 → ℝ :=
  73  fun μ => directional_deriv φ μ x
  74
  75/-- Squared gradient g^{μν} (∂_μ ψ)(∂_ν ψ) with inverse metric. -/
  76noncomputable def gradient_squared (φ : ScalarField) (g : MetricTensor) (x : Fin 4 → ℝ) : ℝ :=
  77  Finset.sum (Finset.univ : Finset (Fin 4)) (fun μ =>
  78    Finset.sum (Finset.univ : Finset (Fin 4)) (fun ν =>
  79      (inverse_metric g) x (fun i => if (i : ℕ) = 0 then μ else ν) (fun _ => 0) *
  80      (gradient φ x μ) * (gradient φ x ν)))
  81
  82/-- Gradient squared for Minkowski metric. -/
  83theorem gradient_squared_minkowski (φ : ScalarField) (x : Fin 4 → ℝ) :
  84  gradient_squared φ minkowski_tensor x =