Theorems · Theorem · global analysis

ContMDiffWithinAt.mfderivWithin_const

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {m n : WithTop ℕ∞} {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] {E' : Type u_5} [inst_6 : NormedAddCommGroup E']
  [inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'}
  {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {s : Set M} [Is : IsManifold I 1 M]
  [I's : IsManifold I' 1 M'] {x₀ : M} {f : M → M'},
  ContMDiffWithinAt I I' n f s x₀ →
    m + 1 ≤ n →
      x₀ ∈ s →
        UniqueMDiff[s] →
          ContMDiffWithinAt I (modelWithCornersSelf 𝕜 (E →L[𝕜] E')) m (inTangentCoordinates I I' id f (mfderiv[s] f) x₀)
            s x₀

The derivative D_yf(y) is C^m at x₀, where the derivative is taken as a continuous linear map. We have to assume that f is C^n at x₀ for some n ≥ m + 1. We have to insert a coordinate change from x₀ to x to make the derivative sensible. This is a special case of ContMDiffWithinAt.mfderivWithin where f does not contain any parameters and g = id.

Defined in

Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Cited by

3 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifold