`BookProof.ChapterRadialLaplacian.laplacian_radial` [FiniteDimensional ℝ E] {g : ℝ → ℝ} {x : E} (hx : x ≠ 0) (hg : ContDiffAt ℝ 2 g ‖x‖) : (Δ fun y : E => g ‖y‖) x = deriv...

Proved

BookProof.ChapterRadialLaplacian.laplacian_radial

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterRadialLaplacian.

BookProof.ChapterRadialLaplacian.laplacian_radial [FiniteDimensional ℝ E] {g : ℝ → ℝ} {x : E} (hx : x ≠ 0) (hg : ContDiffAt ℝ 2 g ‖x‖) : (Δ fun y : E => g ‖y‖) x = deriv (deriv g) ‖x‖ + (((Module.finrank ℝ E : ℝ) - 1) / ‖x‖) * deriv g ‖x‖

Formalization note: Lean 4 identifier BookProof.ChapterRadialLaplacian.laplacian_radial.

Preamble

-- Generated from ChapterRadialLaplacian.lean — theorem BookProof.ChapterRadialLaplacian.laplacian_radial
import Mathlib
import Definitions.Def_ChapterRadialLaplacian
open BookProof.ChapterRadialLaplacian



open Filter Laplacian InnerProductSpace
open scoped InnerProductSpace RealInnerProductSpace

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]

Formal statement

theorem BookProof.ChapterRadialLaplacian.laplacian_radial [FiniteDimensional ℝ E] {g : ℝ → ℝ} {x : E} (hx : x ≠ 0)
    (hg : ContDiffAt ℝ 2 g ‖x‖) :
    (Δ fun y : E => g ‖y‖) x
      = deriv (deriv g) ‖x‖
        + (((Module.finrank ℝ E : ℝ) - 1) / ‖x‖) * deriv g ‖x‖ := by sorry

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