Theorems · Theorem · global analysis

VectorField.leibniz_identity_lieBracketWithin

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E],
  minSmoothness 𝕜 2 ≤ n →
    ∀ {U V W : E → E} {s : Set E} {x : E},
      UniqueDiffOn 𝕜 s →
        x ∈ closure (interior s) →
          x ∈ s →
            ContDiffWithinAt 𝕜 n U s x →
              ContDiffWithinAt 𝕜 n V s x →
                ContDiffWithinAt 𝕜 n W s x →
                  VectorField.lieBracketWithin 𝕜 U (VectorField.lieBracketWithin 𝕜 V W s) s x =
                    VectorField.lieBracketWithin 𝕜 (VectorField.lieBracketWithin 𝕜 U V s) W s x +
                      VectorField.lieBracketWithin 𝕜 V (VectorField.lieBracketWithin 𝕜 U W s) s x

The Lie bracket of vector fields in vector spaces satisfies the Leibniz identity [U, [V, W]] = [[U, V], W] + [V, [U, W]].

Defined in

Mathlib.Analysis.Calculus.VectorField

Cited by

2 results in Mathlib

Foundations

Depth 211 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace