smul_zero

formal source

  39  simp [eval, add]
  40  ring
  41
  42theorem smul_zero (φ : ScalarField) :
  43  ∀ x, eval (smul 0 φ) x = 0 := by
  44  intro x
  45  simp [eval, smul]
  46
  47/-- Directional derivative of scalar field in direction μ. -/
  48noncomputable def directional_deriv (φ : ScalarField) (μ : Fin 4) (x : Fin 4 → ℝ) : ℝ :=
  49  let h := (0.001 : ℝ)
  50  let x_plus := fun ν => if ν = μ then x ν + h else x ν
  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 :=