Theorems · Theorem · potential theory

InnerProductSpace.laplacianWithin_eq_iteratedFDerivWithin_stdOrthonormalBasis

∀ {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) {s : Set E} {e : E},
  UniqueDiffOn ℝ s →
    e ∈ s →
      InnerProductSpace.laplacianWithin f s e =
        ∑ i, (iteratedFDerivWithin ℝ 2 f s e) ![(stdOrthonormalBasis ℝ E) i, (stdOrthonormalBasis ℝ E) i]

Standard formula, computing the Laplacian from the standard orthonormal basis of a real inner product space.

Defined in

Mathlib.Analysis.InnerProductSpace.Laplacian

Cited by

7 results in Mathlib

Foundations

Depth 244 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupNormedSpace