Theorems · Theorem · global analysis

VectorField.fderivWithin_apply_lieBracket

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {V W : E → E}
  {s : Set E} {x : E} {f : E → F} {n : WithTop ℕ∞},
  ContDiffWithinAt 𝕜 n f s x →
    minSmoothness 𝕜 2 ≤ n →
      UniqueDiffOn 𝕜 s →
        x ∈ closure (interior s) →
          x ∈ s →
            DifferentiableWithinAt 𝕜 W s x →
              DifferentiableWithinAt 𝕜 V s x →
                (fderivWithin 𝕜 f s x) (VectorField.lieBracketWithin 𝕜 V W s x) =
                  (fderivWithin 𝕜 (fun x => (fderivWithin 𝕜 f s x) (W x)) s x) (V x) -
                    (fderivWithin 𝕜 (fun x => (fderivWithin 𝕜 f s x) (V x)) s x) (W x)

The differentiation operator along [W, V] is the commutator of the differentiation operators along W and V.

Defined in

Mathlib.Analysis.Calculus.VectorField

Cited by

0 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace