Documentation

Mathlib.Geometry.Manifold.VectorBundle.SmoothSection

C^n sections #

In this file we define the type ContMDiffSection of n times continuously differentiable sections of a vector bundle over a manifold M and prove that it's a module.

structure ContMDiffSection {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (F : Type u_5) [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n : WithTop ℕ∞) (V : MType u_6) [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] :
Type (max u_4 u_6)

Bundled n times continuously differentiable sections of a vector bundle. Denoted as Cₛ^n⟮I; F, V⟯ within the Manifold namespace.

@[deprecated ContMDiffSection (since := "2024-11-21")]
def SmoothSection {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (F : Type u_5) [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n : WithTop ℕ∞) (V : MType u_6) [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] :
Type (max u_4 u_6)

Alias of ContMDiffSection.


Bundled n times continuously differentiable sections of a vector bundle. Denoted as Cₛ^n⟮I; F, V⟯ within the Manifold namespace.

Equations

Bundled n times continuously differentiable sections of a vector bundle. Denoted as Cₛ^n⟮I; F, V⟯ within the Manifold namespace.

Equations
  • One or more equations did not get rendered due to their size.
instance ContMDiffSection.instDFunLike {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] :
Equations
@[simp]
theorem ContMDiffSection.coeFn_mk {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : (x : M) → V x) (hs : ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun (x : M) => { proj := x, snd := s x }) :
{ toFun := s, contMDiff_toFun := hs } = s
theorem ContMDiffSection.contMDiff {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : ContMDiffSection I F n V) :
ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun (x : M) => Bundle.TotalSpace.mk' F x (s x)
@[deprecated ContMDiffSection.contMDiff (since := "2024-11-21")]
theorem ContMDiffSection.smooth {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : ContMDiffSection I F n V) :
ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun (x : M) => Bundle.TotalSpace.mk' F x (s x)

Alias of ContMDiffSection.contMDiff.

theorem ContMDiffSection.coe_inj {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] s t : ContMDiffSection I F n V (h : s = t) :
s = t
theorem ContMDiffSection.coe_injective {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] :
theorem ContMDiffSection.ext {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] {s t : ContMDiffSection I F n V} (h : ∀ (x : M), s x = t x) :
s = t
theorem ContMDiffSection.ext_iff {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] {s t : ContMDiffSection I F n V} :
s = t ∀ (x : M), s x = t x
instance ContMDiffSection.instAdd {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
@[simp]
theorem ContMDiffSection.coe_add {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] (s t : ContMDiffSection I F n V) :
⇑(s + t) = s + t
instance ContMDiffSection.instSub {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
@[simp]
theorem ContMDiffSection.coe_sub {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] (s t : ContMDiffSection I F n V) :
⇑(s - t) = s - t
instance ContMDiffSection.instZero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
instance ContMDiffSection.inhabited {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
@[simp]
theorem ContMDiffSection.coe_zero {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
0 = 0
instance ContMDiffSection.instNeg {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
@[simp]
theorem ContMDiffSection.coe_neg {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] (s : ContMDiffSection I F n V) :
⇑(-s) = -s
instance ContMDiffSection.instNSMul {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
@[simp]
theorem ContMDiffSection.coe_nsmul {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] (s : ContMDiffSection I F n V) (k : ) :
⇑(k s) = k s
instance ContMDiffSection.instZSMul {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
@[simp]
theorem ContMDiffSection.coe_zsmul {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] (s : ContMDiffSection I F n V) (z : ) :
⇑(z s) = z s
instance ContMDiffSection.instAddCommGroup {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Equations
instance ContMDiffSection.instSMul {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
SMul 𝕜 (ContMDiffSection I F n V)
Equations
@[simp]
theorem ContMDiffSection.coe_smul {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] (r : 𝕜) (s : ContMDiffSection I F n V) :
⇑(r s) = r s
def ContMDiffSection.coeAddHom {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (F : Type u_5) [NormedAddCommGroup F] [NormedSpace 𝕜 F] (n : WithTop ℕ∞) (V : MType u_6) [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
ContMDiffSection I F n V →+ (x : M) → V x

The additive morphism from C^n sections to dependent maps.

Equations
@[simp]
theorem ContMDiffSection.coeAddHom_apply {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] (s : ContMDiffSection I F n V) :
(coeAddHom I F n V) s = s
instance ContMDiffSection.instModule {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] :
Module 𝕜 (ContMDiffSection I F n V)
Equations
theorem ContMDiffSection.mdifferentiable' {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : ContMDiffSection I F n V) (hn : 1 n) :
MDifferentiable I (I.prod (modelWithCornersSelf 𝕜 F)) fun (x : M) => Bundle.TotalSpace.mk' F x (s x)
theorem ContMDiffSection.mdifferentiable {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : ContMDiffSection I F (↑) V) :
MDifferentiable I (I.prod (modelWithCornersSelf 𝕜 F)) fun (x : M) => Bundle.TotalSpace.mk' F x (s x)
theorem ContMDiffSection.mdifferentiableAt {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : MType u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] (s : ContMDiffSection I F (↑) V) {x : M} :
MDifferentiableAt I (I.prod (modelWithCornersSelf 𝕜 F)) (fun (x : M) => Bundle.TotalSpace.mk' F x (s x)) x