  45    (f : ScalarField siteCount) (j : Axis) (x : Fin siteCount) : ℝ :=
  46  (f (Λ.plus j x) - f x) / h
  47
  48def backwardDiff {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  49    (f : ScalarField siteCount) (j : Axis) (x : Fin siteCount) : ℝ :=
  50  (f x - f (Λ.minus j x)) / h
  51
  52def scalarLaplacian {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  53    (f : ScalarField siteCount) (x : Fin siteCount) : ℝ :=
  54  ∑ j : Axis, forwardDiff Λ h (backwardDiff Λ h f j) j x
  55
  56def vectorLaplacian {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  57    (u : VectorField siteCount) (x : Fin siteCount) (i : Axis) : ℝ :=
  58  scalarLaplacian Λ h (fun y => u y i) x
  59
  60def divergence {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  61    (u : VectorField siteCount) (x : Fin siteCount) : ℝ :=
  62  ∑ j : Axis, forwardDiff Λ h (fun y => u y j) j x
  63
  64def advection {siteCount : ℕ} (Λ : LatticeTopology siteCount) (h : ℝ)
  65    (u : VectorField siteCount) (x : Fin siteCount) (i : Axis) : ℝ :=
  66  ∑ j : Axis, u x j * forwardDiff Λ h (fun y => u y i) j x
  67
  68def conservativeTransportField {siteCount : ℕ}
  69    (flux : ScalarField siteCount) (perm : Equiv.Perm (Fin siteCount)) :
  70    ScalarField siteCount :=
  71  fun i => flux i - flux (perm i)
  72
  73theorem total_conservativeTransportField_zero {siteCount : ℕ}
  74    (flux : ScalarField siteCount) (perm : Equiv.Perm (Fin siteCount)) :
  75    total (conservativeTransportField flux perm) = 0 := by
  76  unfold total conservativeTransportField
  77  rw [Finset.sum_sub_distrib]
  78  have hperm : (∑ i : Fin siteCount, flux (perm i)) = ∑ i : Fin siteCount, flux i := by

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.