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.

@[reducible]
def smooth_map {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] (I : model_with_corners š•œ E H) (I' : model_with_corners š•œ 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 {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} :
has_coe_to_fun (cont_mdiff_map I I' M M' n) (Ī» (_x : cont_mdiff_map I I' M M' n), M ā†’ M')
Equations
@[protected, instance]
def cont_mdiff_map.continuous_map.has_coe {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} :
has_coe (cont_mdiff_map I I' M M' n) C(M, M')
Equations
@[simp]
theorem cont_mdiff_map.coe_fn_mk {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} (f : M ā†’ M') (hf : cont_mdiff I I' n f) :
@[protected]
theorem cont_mdiff_map.cont_mdiff {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} (f : cont_mdiff_map I I' M M' n) :
@[protected]
theorem cont_mdiff_map.smooth {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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' āŠ¤) :
@[protected]
theorem cont_mdiff_map.mdifferentiable' {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} (f : cont_mdiff_map I I' M M' n) (hn : 1 ā‰¤ n) :
@[protected]
theorem cont_mdiff_map.mdifferentiable {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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' āŠ¤) :
@[protected]
theorem cont_mdiff_map.mdifferentiable_at {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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' āŠ¤) {x : M} :
theorem cont_mdiff_map.coe_inj {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} ā¦ƒf g : cont_mdiff_map I I' M M' nā¦„ (h : ā‡‘f = ā‡‘g) :
f = g
@[ext]
theorem cont_mdiff_map.ext {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} {f g : cont_mdiff_map I I' M M' n} (h : āˆ€ (x : M), ā‡‘f x = ā‡‘g x) :
f = g
def cont_mdiff_map.id {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {H : Type u_4} [topological_space H] {I : model_with_corners š•œ E H} {M : Type u_6} [topological_space M] [charted_space H M] {n : ā„•āˆž} :
cont_mdiff_map I I M M n

The identity as a smooth map.

Equations
def cont_mdiff_map.comp {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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_add_comm_group E''] [normed_space š•œ E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners š•œ E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {n : ā„•āˆž} (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 {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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_add_comm_group E''] [normed_space š•œ E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners š•œ E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {n : ā„•āˆž} (f : cont_mdiff_map I' I'' M' M'' n) (g : cont_mdiff_map I I' M M' n) (x : M) :
ā‡‘(f.comp g) x = ā‡‘f (ā‡‘g x)
@[protected, instance]
def cont_mdiff_map.inhabited {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} [inhabited M'] :
Equations
def cont_mdiff_map.const {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners š•œ E H} {I' : model_with_corners š•œ 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 : ā„•āˆž} (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 {š•œ : Type u_1} [nontrivially_normed_field š•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space š•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space š•œ E'] (n : ā„•āˆž) :
has_coe (E ā†’L[š•œ] E') (cont_mdiff_map (model_with_corners_self š•œ E) (model_with_corners_self š•œ E') E E' n)
Equations