Theorems · Theorem · potential theory

InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis

∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E]
  {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] (f : E → F) {ι : Type u_5}
  [inst_5 : Fintype ι] (v : OrthonormalBasis ι ℝ E),
  Laplacian.laplacian f = fun x => ∑ i, (iteratedFDeriv ℝ 2 f x) ![v i, v i]

Standard formula, computing the Laplacian from any orthonormal basis.

Defined in

Mathlib.Analysis.InnerProductSpace.Laplacian

Cited by

3 results in Mathlib

Foundations

Depth 243 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupNormedSpaceFintype