Theorems · Theorem · global analysis

mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4}
  [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] [IsManifold I 1 M] {x : M} {y : E},
  y ∈ (extChartAt I x).target →
    mfderiv[Set.range ↑I] ↑(extChartAt I x).symm y ∘SL mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) =
      ContinuousLinearMap.id 𝕜 (TangentSpace I (↑(extChartAt I x).symm y))

The composition of the derivative of the inverse of extChartAt with the derivative of extChartAt gives the identity. Version where the basepoint belongs to (extChartAt I x).target.

Defined in

Mathlib.Geometry.Manifold.MFDeriv.Atlas

Cited by

3 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifold