Theorems · Theorem · global analysis

VectorField.mpullback_mfderivWithin_apply_smul

∀ {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}
  [IsManifold I 2 M] {f : M → 𝕜},
  MDiffAt[s] f x →
    have V' := VectorField.mpullbackWithin (modelWithCornersSelf 𝕜 E) I (↑(extChartAt I x).symm) V (Set.range ↑I);
    have W' := VectorField.mpullbackWithin (modelWithCornersSelf 𝕜 E) I (↑(extChartAt I x).symm) W (Set.range ↑I);
    VectorField.mpullback I (modelWithCornersSelf 𝕜 E) (↑(extChartAt I x))
        (fun x₀ =>
          (fderivWithin 𝕜 (f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I) x₀) (V' x₀) •
            W' x₀)
        x =
      (mfderiv[s] f x) (V x) • W x

Pulling back through extChartAt the scalar multiplication of a vector field by the derivative of a scalar function equals the scalar multiplication by the manifold derivative.

Defined in

Mathlib.Geometry.Manifold.VectorField.LieBracket

Cited by

1 results in Mathlib

Foundations

Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldTopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceChartedSpaceIsManifold