mathlib documentation

geometry.manifold.cont_mdiff_map

Smooth bundled map #

In this file we define the type cont_mdiff_map of n times continuously differentiable bundled 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] (M' : Type u_7) [topological_space M'] [charted_space H' M'] :
Type (max u_6 u_7)

Bundled smooth maps.

Equations
@[protected, instance]
def 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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } :
has_coe_to_fun (cont_mdiff_map I I' M M' n) (λ (_x : cont_mdiff_map I I' M M' n), M → M')
Equations
@[protected, instance]
def 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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } :
has_coe (cont_mdiff_map I I' M M' n) C(M, M')
Equations
@[simp]
theorem cont_mdiff_map.coe_fn_mk {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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } (f : M → M') (hf : cont_mdiff I I' n f) :
@[protected]
theorem cont_mdiff_map.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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } (f : cont_mdiff_map I I' M M' n) :
cont_mdiff I I' n f
@[protected]
theorem 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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : cont_mdiff_map I I' M M' ) :
smooth I I' f
@[protected]
theorem cont_mdiff_map.mdifferentiable' {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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } (f : cont_mdiff_map I I' M M' n) (hn : 1 n) :
@[protected]
theorem cont_mdiff_map.mdifferentiable {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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : cont_mdiff_map I I' M M' ) :
@[protected]
theorem cont_mdiff_map.mdifferentiable_at {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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : cont_mdiff_map I I' M M' ) {x : M} :
theorem 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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } ⦃f g : cont_mdiff_map I I' M M' n⦄ (h : f = g) :
f = g
@[ext]
theorem 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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } {f g : cont_mdiff_map I I' M M' n} (h : ∀ (x : M), f x = g x) :
f = g
def 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] {n : with_top } :
cont_mdiff_map I I M M n

The identity as a smooth map.

Equations
def 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] {M' : Type u_7} [topological_space M'] [charted_space H' 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''] {n : with_top } (f : cont_mdiff_map I' I'' M' M'' n) (g : cont_mdiff_map I I' M M' n) :
cont_mdiff_map I I'' M M'' n

The composition of smooth maps, as a smooth map.

Equations
@[simp]
theorem 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] {M' : Type u_7} [topological_space M'] [charted_space H' 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''] {n : with_top } (f : cont_mdiff_map I' I'' M' M'' n) (g : cont_mdiff_map I I' M M' n) (x : M) :
(f.comp g) x = f (g x)
@[protected, instance]
def 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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } [inhabited M'] :
Equations
def 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] {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : with_top } (y : M') :
cont_mdiff_map I I' M M' n

Constant map as a smooth map

Equations
@[protected, instance]
def continuous_linear_map.has_coe_to_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 ) :
Equations