laplacian_smul — Mathlib

English

The Laplacian commutes with scalar multiplication: Δ(v·f) = v·Δf for all v ∈ ℝ and suitable f.

Русский

Лаплациан коммутирует со скалярным умножением: Δ(v·f) = v·Δf.

LaTeX

$$$ Δ(v \cdot f) = v \cdot (Δ f). $$$

Lean4

/-- The Laplacian commutes with scalar multiplication. -/
theorem laplacian_smul (v : ℝ) (hf : ContDiffAt ℝ 2 f x) : Δ (v • f) x = v • (Δ f) x := by
  simp [laplacian_eq_iteratedFDeriv_stdOrthonormalBasis, iteratedFDeriv_const_smul_apply hf, Finset.smul_sum]

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