laplacianWithin_eq_iteratedFDerivWithin_orthonormalBasis

English

Real-case Laplacian equals iterated derivative within real field.

Русский

Реальный кейс лапласиана равен повторной производной второго порядка.

LaTeX

$$Δ_real f = iteratedDeriv 2 f$$

Lean4

/-- Standard formula, computing the Laplacian from any orthonormal basis.
-/
theorem laplacianWithin_eq_iteratedFDerivWithin_orthonormalBasis {ι : Type*} [Fintype ι] {e : E} (hs : UniqueDiffOn ℝ s)
    (he : e ∈ s) (v : OrthonormalBasis ι ℝ E) : (Δ[s]f) e = ∑ i, iteratedFDerivWithin ℝ 2 f s e ![v i, v i] := by
  simp [InnerProductSpace.laplacianWithin, canonicalCovariantTensor_eq_sum E v,
    tensorIteratedFDerivWithinTwo_eq_iteratedFDerivWithin f hs he]

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