Theorems · Theorem · global analysis

VectorField.mlieBracketWithin_smul_right

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {H : Type u_2} [inst_1 : TopologicalSpace H] {E : Type u_3}
  [inst_2 : NormedAddCommGroup E] [inst_3 : NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {s : Set M} {x : M} {V W : (x : M) → TangentSpace I x}
  [inst_6 : IsManifold I 2 M] [CompleteSpace E] {f : M → 𝕜},
  MDiffAt[s] f x →
    (MDiffAt[s] fun x => ⟨x, W x⟩) x →
      UniqueMDiffAt[s] x →
        VectorField.mlieBracketWithin I V (f • W) s x =
          (d[s] f x) (V x) • W x + f x • VectorField.mlieBracketWithin I V W s x

Product rule for Lie brackets: given two vector fields V and W on M and a function f : M → 𝕜, we have [V, f • W] = (df V) • W + f • [V, W]. Version within a set.

Defined in

Mathlib.Geometry.Manifold.VectorField.LieBracket

Cited by

3 results in Mathlib

Foundations

Depth 225 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceIsManifoldCompleteSpace