Theorems · Theorem · global analysis

VectorField.mlieBracket_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] {x : M} {V W : (x : M) → TangentSpace I x}
  [inst_6 : IsManifold I 2 M] [CompleteSpace E] {f : M → 𝕜},
  MDiffAt f x →
    (MDiffAt fun x => ⟨x, W x⟩) x →
      VectorField.mlieBracket I V (f • W) x = (d% f x) (V x) • W x + f x • VectorField.mlieBracket I V W 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].

Defined in

Mathlib.Geometry.Manifold.VectorField.LieBracket

Cited by

0 results in Mathlib

Foundations

Depth 226 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceIsManifoldCompleteSpace