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 {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] (I : model_with_corners π•œ E H) (I' : model_with_corners π•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 {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} :
has_coe_to_fun (cont_mdiff_map I I' M M' n) (Ξ» (_x : cont_mdiff_map I I' M M' n), M β†’ M')
Equations
@[protected, instance]
def cont_mdiff_map.continuous_map.has_coe {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} :
has_coe (cont_mdiff_map I I' M M' n) C(M, M')
Equations
@[simp]
theorem cont_mdiff_map.coe_fn_mk {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} (f : M β†’ M') (hf : cont_mdiff I I' n f) :
@[protected]
theorem cont_mdiff_map.cont_mdiff {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} (f : cont_mdiff_map I I' M M' n) :
@[protected]
theorem cont_mdiff_map.smooth {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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' ⊀) :
@[protected]
theorem cont_mdiff_map.mdifferentiable' {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} (f : cont_mdiff_map I I' M M' n) (hn : 1 ≀ n) :
@[protected]
theorem cont_mdiff_map.mdifferentiable {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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' ⊀) :
@[protected]
theorem cont_mdiff_map.mdifferentiable_at {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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' ⊀) {x : M} :
theorem cont_mdiff_map.coe_inj {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} ⦃f g : cont_mdiff_map I I' M M' n⦄ (h : ⇑f = ⇑g) :
f = g
@[ext]
theorem cont_mdiff_map.ext {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} {f g : cont_mdiff_map I I' M M' n} (h : βˆ€ (x : M), ⇑f x = ⇑g x) :
f = g
def cont_mdiff_map.id {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {H : Type u_4} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_6} [topological_space M] [charted_space H M] {n : β„•βˆž} :
cont_mdiff_map I I M M n

The identity as a smooth map.

Equations
def cont_mdiff_map.comp {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 π•œ E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners π•œ E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {n : β„•βˆž} (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 {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 π•œ E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners π•œ E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {n : β„•βˆž} (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 {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} [inhabited M'] :
Equations
def cont_mdiff_map.const {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners π•œ E H} {I' : model_with_corners π•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 : β„•βˆž} (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 {π•œ : Type u_1} [nontrivially_normed_field π•œ] {E : Type u_2} [normed_add_comm_group E] [normed_space π•œ E] {E' : Type u_3} [normed_add_comm_group E'] [normed_space π•œ E'] (n : β„•βˆž) :
has_coe (E β†’L[π•œ] E') (cont_mdiff_map (model_with_corners_self π•œ E) (model_with_corners_self π•œ E') E E' n)
Equations