Theorems · Theorem · global analysis

VectorField.lieBracketWithin_smul_right

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {V W : E → E} {s : Set E} {x : E} {f : E → 𝕜},
  DifferentiableWithinAt 𝕜 f s x →
    DifferentiableWithinAt 𝕜 W s x →
      UniqueDiffWithinAt 𝕜 s x →
        VectorField.lieBracketWithin 𝕜 V (fun y => f y • W y) s x =
          (fderivWithin 𝕜 f s x) (V x) • W x + f x • VectorField.lieBracketWithin 𝕜 V W s x

Product rule for Lie Brackets: given two vector fields V W : E → E and a function f : E → 𝕜, we have [V, f • W] = (df V) • W + f • [V, W]

Defined in

Mathlib.Analysis.Calculus.VectorField

Cited by

3 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace