constant_eval

formal source

  17/-- Constant scalar field. -/
  18def constant (c : ℝ) : ScalarField := { ψ := fun _ => c }
  19
  20theorem constant_eval (c : ℝ) (x : Fin 4 → ℝ) :
  21  eval (constant c) x = c := rfl
  22
  23/-- Zero scalar field. -/
  24def zero : ScalarField := constant 0
  25
  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