Theorems · Theorem · global analysis

ContMDiffAt.mfderiv

∀ {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'] {F : Type u_8}
  [inst_11 : NormedAddCommGroup F] [inst_12 : NormedSpace 𝕜 F] {G : Type u_9} [inst_13 : TopologicalSpace G]
  {J : ModelWithCorners 𝕜 F G} {N : Type u_10} [inst_14 : TopologicalSpace N] [inst_15 : ChartedSpace G N]
  [Js : IsManifold J 1 N] [Is : IsManifold I 1 M] [I's : IsManifold I' 1 M'] {x₀ : N} (f : N → M → M') (g : N → M),
  ContMDiffAt (J.prod I) I' n (Function.uncurry f) (x₀, g x₀) →
    ContMDiffAt J I m g x₀ →
      m + 1 ≤ n →
        ContMDiffAt J (modelWithCornersSelf 𝕜 (E →L[𝕜] E')) m
          (inTangentCoordinates I I' g (fun x => f x (g x)) (fun x => mfderiv% (f x) (g x)) x₀) x₀

The function that sends x to the y-derivative of f (x, y) at g (x) 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, g(x₀)) for n ≥ m + 1 and g is C^m at x₀. We have to insert a coordinate change from x₀ to x to make the derivative sensible. This result is used to show that maps into the 1-jet bundle and cotangent bundle are C^n. ContMDiffAt.mfderiv_const is a special case of this.

Defined in

Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Cited by

2 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpaceIsManifoldIsManifoldIsManifold