Theorems · Theorem · global analysis

mfderivWithin_range_extChartAt_symm

∀ {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},
  mfderiv[Set.range ↑I] ↑(extChartAt I x).symm (↑(extChartAt I x) x) =
    ContinuousLinearMap.id 𝕜 (TangentSpace (modelWithCornersSelf 𝕜 E) (↑(extChartAt I x) x))

The manifold derivative within range I of (extChartAt I x).symm at the chart point is the identity.

Defined in

Mathlib.Geometry.Manifold.MFDeriv.Atlas

Cited by

1 results in Mathlib

Foundations

Depth 219 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifold