mathlib documentation

geometry.manifold.times_cont_mdiff_map

Smooth bundled map

In this file we define the type times_cont_mdiff_map of n times continuously differentiable bundled maps.

structure times_cont_mdiff_map {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_6) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_7) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ) :
Type (max u_6 u_7)

Bundled n times continuously differentiable maps.

def smooth_map {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_6) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_7) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] :
Type (max u_6 u_7)

Bundled smooth maps.

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@[instance]
def times_cont_mdiff_map.has_coe_to_fun {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top } :

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@[instance]
def times_cont_mdiff_map.continuous_map.has_coe {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top } :
has_coe C^nI, M; I', M' C(M, M')

Equations
theorem times_cont_mdiff_map.times_cont_mdiff {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top } (f : C^nI, M; I', M') :

theorem times_cont_mdiff_map.smooth {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (f : C^I, M; I', M') :
smooth I I' f

theorem times_cont_mdiff_map.coe_inj {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top } ⦃f g : C^nI, M; I', M' (h : f = g) :
f = g

@[ext]
theorem times_cont_mdiff_map.ext {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top } {f g : C^nI, M; I', M'} (h : ∀ (x : M), f x = g x) :
f = g

def times_cont_mdiff_map.id {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {H : Type u_4} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {n : with_top } :
C^nI, M; I, M

The identity as a smooth map.

Equations
def times_cont_mdiff_map.comp {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {E'' : Type u_8} [normed_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] [smooth_manifold_with_corners I'' M''] {n : with_top } (f : C^nI', M'; I'', M'') (g : C^nI, M; I', M') :
C^nI, M; I'', M''

The composition of smooth maps, as a smooth map.

Equations
@[simp]
theorem times_cont_mdiff_map.comp_apply {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {E'' : Type u_8} [normed_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] [smooth_manifold_with_corners I'' M''] {n : with_top } (f : C^nI', M'; I'', M'') (g : C^nI, M; I', M') (x : M) :
(f.comp g) x = f (g x)

@[instance]
def times_cont_mdiff_map.inhabited {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top } [inhabited M'] :

Equations
def times_cont_mdiff_map.const {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top } (y : M') :
C^nI, M; I', M'

Constant map as a smooth map

Equations
@[instance]
def continuous_linear_map.has_coe_to_times_cont_mdiff_map {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] (n : with_top ) :
has_coe (E →L[𝕜] E') C^n𝓘(𝕜, E), E; 𝓘(𝕜, E'), E'

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