add

formal source

  26theorem zero_eval (x : Fin 4 → ℝ) : eval zero x = 0 := rfl
  27
  28/-- Scalar field addition. -/
  29def add (φ₁ φ₂ : ScalarField) : ScalarField :=
  30  { ψ := fun x => φ₁.ψ x + φ₂.ψ x }
  31
  32/-- Scalar multiplication. -/
  33def smul (c : ℝ) (φ : ScalarField) : ScalarField :=
  34  { ψ := fun x => c * φ.ψ x }
  35
  36theorem add_comm (φ₁ φ₂ : ScalarField) :
  37  ∀ x, eval (add φ₁ φ₂) x = eval (add φ₂ φ₁) x := by
  38  intro x
  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