mathlib documentation

geometry.manifold.mfderiv

The derivative of functions between smooth manifolds #

Let M and M' be two smooth manifolds with corners over a field π•œ (with respective models with corners I on (E, H) and I' on (E', H')), and let f : M β†’ M'. We define the derivative of the function at a point, within a set or along the whole space, mimicking the API for (FrΓ©chet) derivatives. It is denoted by mfderiv I I' f x, where "m" stands for "manifold" and "f" for "FrΓ©chet" (as in the usual derivative fderiv π•œ f x).

Main definitions #

Let f be a map between smooth manifolds. The following definitions follow the fderiv API.

We also establish results on the differential of the identity, constant functions, charts, extended charts. For functions between vector spaces, we show that the usual notions and the manifold notions coincide.

Implementation notes #

The tangent bundle is constructed using the machinery of topological fiber bundles, for which one can define bundled morphisms and construct canonically maps from the total space of one bundle to the total space of another one. One could use this mechanism to construct directly the derivative of a smooth map. However, we want to define the derivative of any map (and let it be zero if the map is not differentiable) to avoid proof arguments everywhere. This means we have to go back to the details of the definition of the total space of a fiber bundle constructed from core, to cook up a suitable definition of the derivative. It is the following: at each point, we have a preferred chart (used to identify the fiber above the point with the model vector space in fiber bundles). Then one should read the function using these preferred charts at x and f x, and take the derivative of f in these charts.

Due to the fact that we are working in a model with corners, with an additional embedding I of the model space H in the model vector space E, the charts taking values in E are not the original charts of the manifold, but those ones composed with I, called extended charts. We define written_in_ext_chart I I' x f for the function f written in the preferred extended charts. Then the manifold derivative of f, at x, is just the usual derivative of written_in_ext_chart I I' x f, at the point (ext_chart_at I x) x.

There is a subtelty with respect to continuity: if the function is not continuous, then the image of a small open set around x will not be contained in the source of the preferred chart around f x, which means that when reading f in the chart one is losing some information. To avoid this, we include continuity in the definition of differentiablity (which is reasonable since with any definition, differentiability implies continuity).

Warning: the derivative (even within a subset) is a linear map on the whole tangent space. Suppose that one is given a smooth submanifold N, and a function which is smooth on N (i.e., its restriction to the subtype N is smooth). Then, in the whole manifold M, the property mdifferentiable_on I I' f N holds. However, mfderiv_within I I' f N is not uniquely defined (what values would one choose for vectors that are transverse to N?), which can create issues down the road. The problem here is that knowing the value of f along N does not determine the differential of f in all directions. This is in contrast to the case where N would be an open subset, or a submanifold with boundary of maximal dimension, where this issue does not appear. The predicate unique_mdiff_on I N indicates that the derivative along N is unique if it exists, and is an assumption in most statements requiring a form of uniqueness.

On a vector space, the manifold derivative and the usual derivative are equal. This means in particular that they live on the same space, i.e., the tangent space is defeq to the original vector space. To get this property is a motivation for our definition of the tangent space as a single copy of the vector space, instead of more usual definitions such as the space of derivations, or the space of equivalence classes of smooth curves in the manifold.

Tags #

Derivative, manifold

Derivative of maps between manifolds #

The derivative of a smooth map f between smooth manifold M and M' at x is a bounded linear map from the tangent space to M at x, to the tangent space to M' at f x. Since we defined the tangent space using one specific chart, the formula for the derivative is written in terms of this specific chart.

We use the names mdifferentiable and mfderiv, where the prefix letter m means "manifold".

def unique_mdiff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] (s : set M) (x : M) :
Prop

Predicate ensuring that, at a point and within a set, a function can have at most one derivative. This is expressed using the preferred chart at the considered point.

Equations
def unique_mdiff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] (s : set M) :
Prop

Predicate ensuring that, at all points of a set, a function can have at most one derivative.

Equations
@[simp]
def written_in_ext_chart_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (x : M) (f : M β†’ M') :
E β†’ E'

Conjugating a function to write it in the preferred charts around x. The manifold derivative of f will just be the derivative of this conjugated function.

Equations
def mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M β†’ M') (s : set M) (x : M) :
Prop

mdifferentiable_within_at I I' f s x indicates that the function f between manifolds has a derivative at the point x within the set s. This is a generalization of differentiable_within_at to manifolds.

We require continuity in the definition, as otherwise points close to x in s could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence written_in_ext_chart_at I I' x f could be differentiable, while this would not mean anything relevant.

Equations
def mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M β†’ M') (x : M) :
Prop

mdifferentiable_at I I' f x indicates that the function f between manifolds has a derivative at the point x. This is a generalization of differentiable_at to manifolds.

We require continuity in the definition, as otherwise points close to x could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence written_in_ext_chart_at I I' x f could be differentiable, while this would not mean anything relevant.

Equations
def mdifferentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M β†’ M') (s : set M) :
Prop

mdifferentiable_on I I' f s indicates that the function f between manifolds has a derivative within s at all points of s. This is a generalization of differentiable_on to manifolds.

Equations
def mdifferentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M β†’ M') :
Prop

mdifferentiable I I' f indicates that the function f between manifolds has a derivative everywhere. This is a generalization of differentiable to manifolds.

Equations
def local_homeomorph.mdifferentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : local_homeomorph M M') :
Prop

Prop registering if a local homeomorphism is a local diffeomorphism on its source

Equations
def has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] (f : M β†’ M') (s : set M) (x : M) (f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)) :
Prop

has_mfderiv_within_at I I' f s x f' indicates that the function f between manifolds has, at the point x and within the set s, the derivative f'. Here, f' is a continuous linear map from the tangent space at x to the tangent space at f x.

This is a generalization of has_fderiv_within_at to manifolds (as indicated by the prefix m). The order of arguments is changed as the type of the derivative f' depends on the choice of x.

We require continuity in the definition, as otherwise points close to x in s could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence written_in_ext_chart_at I I' x f could be differentiable, while this would not mean anything relevant.

Equations
def has_mfderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] (f : M β†’ M') (x : M) (f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)) :
Prop

has_mfderiv_at I I' f x f' indicates that the function f between manifolds has, at the point x, the derivative f'. Here, f' is a continuous linear map from the tangent space at x to the tangent space at f x.

We require continuity in the definition, as otherwise points close to x s could be sent by f outside of the chart domain around f x. Then the chart could do anything to the image points, and in particular by coincidence written_in_ext_chart_at I I' x f could be differentiable, while this would not mean anything relevant.

Equations
def mfderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] (f : M β†’ M') (s : set M) (x : M) :

Let f be a function between two smooth manifolds. Then mfderiv_within I I' f s x is the derivative of f at x within s, as a continuous linear map from the tangent space at x to the tangent space at f x.

Equations
def mfderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] (f : M β†’ M') (x : M) :

Let f be a function between two smooth manifolds. Then mfderiv I I' f x is the derivative of f at x, as a continuous linear map from the tangent space at x to the tangent space at f x.

Equations
def tangent_map_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] (f : M β†’ M') (s : set M) :

The derivative within a set, as a map between the tangent bundles

Equations
def tangent_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I M] [smooth_manifold_with_corners I' M'] (f : M β†’ M') :

The derivative, as a map between the tangent bundles

Equations

Unique differentiability sets in manifolds #

theorem unique_mdiff_within_at_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {x : M} :
theorem unique_mdiff_within_at_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {s : set M} {x : M} :
theorem unique_mdiff_within_at.mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {x : M} {s t : set M} (h : unique_mdiff_within_at I s x) (st : s βŠ† t) :
theorem unique_mdiff_within_at.inter' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {x : M} {s t : set M} (hs : unique_mdiff_within_at I s x) (ht : t ∈ 𝓝[s] x) :
theorem unique_mdiff_within_at.inter {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {x : M} {s t : set M} (hs : unique_mdiff_within_at I s x) (ht : t ∈ 𝓝 x) :
theorem is_open.unique_mdiff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {x : M} {s : set M} (xs : x ∈ s) (hs : is_open s) :
theorem unique_mdiff_on.inter {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {s t : set M} (hs : unique_mdiff_on I s) (ht : is_open t) :
theorem is_open.unique_mdiff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {s : set M} (hs : is_open s) :
theorem unique_mdiff_on_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] :
theorem unique_mdiff_within_at.eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' f₁' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (U : unique_mdiff_within_at I s x) (h : has_mfderiv_within_at I I' f s x f') (h₁ : has_mfderiv_within_at I I' f s x f₁') :
f' = f₁'

unique_mdiff_within_at achieves its goal: it implies the uniqueness of the derivative.

theorem unique_mdiff_on.eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' f₁' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (U : unique_mdiff_on I s) (hx : x ∈ s) (h : has_mfderiv_within_at I I' f s x f') (h₁ : has_mfderiv_within_at I I' f s x f₁') :
f' = f₁'

General lemmas on derivatives of functions between manifolds #

We mimick the API for functions between vector spaces

theorem mdifferentiable_within_at_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s : set M} {x : M} :
theorem mfderiv_within_zero_of_not_mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : Β¬mdifferentiable_within_at I I' f s x) :
mfderiv_within I I' f s x = 0
theorem mfderiv_zero_of_not_mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : Β¬mdifferentiable_at I I' f x) :
mfderiv I I' f x = 0
theorem has_mfderiv_within_at.mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_within_at I I' f t x f') (hst : s βŠ† t) :
has_mfderiv_within_at I I' f s x f'
theorem has_mfderiv_at.has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_at I I' f x f') :
has_mfderiv_within_at I I' f s x f'
theorem has_mfderiv_within_at.mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_within_at I I' f s x f') :
theorem has_mfderiv_at.mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_at I I' f x f') :
@[simp]
theorem has_mfderiv_within_at_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} :
theorem has_mfderiv_at_unique {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {fβ‚€' f₁' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (hβ‚€ : has_mfderiv_at I I' f x fβ‚€') (h₁ : has_mfderiv_at I I' f x f₁') :
fβ‚€' = f₁'
theorem has_mfderiv_within_at_inter' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : t ∈ 𝓝[s] x) :
theorem has_mfderiv_within_at_inter {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : t ∈ 𝓝 x) :
theorem has_mfderiv_within_at.union {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (hs : has_mfderiv_within_at I I' f s x f') (ht : has_mfderiv_within_at I I' f t x f') :
has_mfderiv_within_at I I' f (s βˆͺ t) x f'
theorem has_mfderiv_within_at.nhds_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_within_at I I' f s x f') (ht : s ∈ 𝓝[t] x) :
has_mfderiv_within_at I I' f t x f'
theorem has_mfderiv_within_at.has_mfderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_within_at I I' f s x f') (hs : s ∈ 𝓝 x) :
has_mfderiv_at I I' f x f'
theorem mdifferentiable_within_at.has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_within_at I I' f s x) :
has_mfderiv_within_at I I' f s x (mfderiv_within I I' f s x)
theorem mdifferentiable_within_at.mfderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_within_at I I' f s x) :
theorem mdifferentiable_at.has_mfderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_at I I' f x) :
has_mfderiv_at I I' f x (mfderiv I I' f x)
theorem mdifferentiable_at.mfderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_at I I' f x) :
theorem has_mfderiv_at.mfderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_at I I' f x f') :
mfderiv I I' f x = f'
theorem has_mfderiv_within_at.mfderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_within_at I I' f s x f') (hxs : unique_mdiff_within_at I s x) :
mfderiv_within I I' f s x = f'
theorem mdifferentiable.mfderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_at I I' f x) (hxs : unique_mdiff_within_at I s x) :
mfderiv_within I I' f s x = mfderiv I I' f x
theorem mfderiv_within_subset {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (st : s βŠ† t) (hs : unique_mdiff_within_at I s x) (h : mdifferentiable_within_at I I' f t x) :
mfderiv_within I I' f s x = mfderiv_within I I' f t x
theorem mdifferentiable_within_at.mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} (hst : s βŠ† t) (h : mdifferentiable_within_at I I' f t x) :
theorem mdifferentiable_within_at_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} :
theorem mdifferentiable_within_at_inter {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} (ht : t ∈ 𝓝 x) :
theorem mdifferentiable_within_at_inter' {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} (ht : t ∈ 𝓝[s] x) :
theorem mdifferentiable_at.mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} (h : mdifferentiable_at I I' f x) :
theorem mdifferentiable_within_at.mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} (h : mdifferentiable_within_at I I' f s x) (hs : s ∈ 𝓝 x) :
theorem mdifferentiable_on.mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s t : set M} (h : mdifferentiable_on I I' f t) (st : s βŠ† t) :
theorem mdifferentiable_on_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} :
theorem mdifferentiable.mdifferentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s : set M} (h : mdifferentiable I I' f) :
theorem mdifferentiable_on_of_locally_mdifferentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s : set M} (h : βˆ€ (x : M), x ∈ s β†’ (βˆƒ (u : set M), is_open u ∧ x ∈ u ∧ mdifferentiable_on I I' f (s ∩ u))) :
@[simp]
theorem mfderiv_within_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] :
theorem mfderiv_within_inter {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (ht : t ∈ 𝓝 x) (hs : unique_mdiff_within_at I s x) :
mfderiv_within I I' f (s ∩ t) x = mfderiv_within I I' f s x

Deriving continuity from differentiability on manifolds #

theorem has_mfderiv_within_at.continuous_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} (h : mdifferentiable_within_at I I' f s x) :
theorem has_mfderiv_at.continuous_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_at I I' f x f') :
theorem mdifferentiable_within_at.continuous_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} {s : set M} (h : mdifferentiable_within_at I I' f s x) :
theorem mdifferentiable_at.continuous_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {x : M} (h : mdifferentiable_at I I' f x) :
theorem mdifferentiable_on.continuous_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s : set M} (h : mdifferentiable_on I I' f s) :
theorem mdifferentiable.continuous {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} (h : mdifferentiable I I' f) :
theorem tangent_map_within_subset {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {p : tangent_bundle I M} (st : s βŠ† t) (hs : unique_mdiff_within_at I s p.fst) (h : mdifferentiable_within_at I I' f t p.fst) :
theorem tangent_map_within_univ {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] :
theorem tangent_map_within_eq_tangent_map {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {p : tangent_bundle I M} (hs : unique_mdiff_within_at I s p.fst) (h : mdifferentiable_at I I' f p.fst) :
tangent_map_within I I' f s p = tangent_map I I' f p
@[simp]
theorem tangent_map_within_tangent_bundle_proj {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {p : tangent_bundle I M} :
@[simp]
theorem tangent_map_within_proj {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {p : tangent_bundle I M} :
(tangent_map_within I I' f s p).fst = f p.fst
@[simp]
theorem tangent_map_tangent_bundle_proj {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {p : tangent_bundle I M} :
@[simp]
theorem tangent_map_proj {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M β†’ M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {p : tangent_bundle I M} :
(tangent_map I I' f p).fst = f p.fst

Congruence lemmas for derivatives on manifolds #

theorem has_mfderiv_within_at.congr_of_eventually_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_within_at I I' f s x f') (h₁ : f₁ =αΆ [𝓝[s] x] f) (hx : f₁ x = f x) :
has_mfderiv_within_at I I' f₁ s x f'
theorem has_mfderiv_within_at.congr_mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_within_at I I' f s x f') (ht : βˆ€ (x : M), x ∈ t β†’ f₁ x = f x) (hx : f₁ x = f x) (h₁ : t βŠ† s) :
has_mfderiv_within_at I I' f₁ t x f'
theorem has_mfderiv_at.congr_of_eventually_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} (h : has_mfderiv_at I I' f x f') (h₁ : f₁ =αΆ [𝓝 x] f) :
has_mfderiv_at I I' f₁ x f'
theorem mdifferentiable_within_at.congr_of_eventually_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_within_at I I' f s x) (h₁ : f₁ =αΆ [𝓝[s] x] f) (hx : f₁ x = f x) :
mdifferentiable_within_at I I' f₁ s x
theorem filter.eventually_eq.mdifferentiable_within_at_iff {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h₁ : f₁ =αΆ [𝓝[s] x] f) (hx : f₁ x = f x) :
theorem mdifferentiable_within_at.congr_mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_within_at I I' f s x) (ht : βˆ€ (x : M), x ∈ t β†’ f₁ x = f x) (hx : f₁ x = f x) (h₁ : t βŠ† s) :
mdifferentiable_within_at I I' f₁ t x
theorem mdifferentiable_within_at.congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_within_at I I' f s x) (ht : βˆ€ (x : M), x ∈ s β†’ f₁ x = f x) (hx : f₁ x = f x) :
mdifferentiable_within_at I I' f₁ s x
theorem mdifferentiable_on.congr_mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_on I I' f s) (h' : βˆ€ (x : M), x ∈ t β†’ f₁ x = f x) (h₁ : t βŠ† s) :
mdifferentiable_on I I' f₁ t
theorem mdifferentiable_at.congr_of_eventually_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_at I I' f x) (hL : f₁ =αΆ [𝓝 x] f) :
mdifferentiable_at I I' f₁ x
theorem mdifferentiable_within_at.mfderiv_within_congr_mono {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s t : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : mdifferentiable_within_at I I' f s x) (hs : βˆ€ (x : M), x ∈ t β†’ f₁ x = f x) (hx : f₁ x = f x) (hxt : unique_mdiff_within_at I t x) (h₁ : t βŠ† s) :
mfderiv_within I I' f₁ t x = mfderiv_within I I' f s x
theorem filter.eventually_eq.mfderiv_within_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (hs : unique_mdiff_within_at I s x) (hL : f₁ =αΆ [𝓝[s] x] f) (hx : f₁ x = f x) :
mfderiv_within I I' f₁ s x = mfderiv_within I I' f s x
theorem mfderiv_within_congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (hs : unique_mdiff_within_at I s x) (hL : βˆ€ (x : M), x ∈ s β†’ f₁ x = f x) (hx : f₁ x = f x) :
mfderiv_within I I' f₁ s x = mfderiv_within I I' f s x
theorem tangent_map_within_congr {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {s : set M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (h : βˆ€ (x : M), x ∈ s β†’ f x = f₁ x) (p : tangent_bundle I M) (hp : p.fst ∈ s) (hs : unique_mdiff_within_at I s p.fst) :
tangent_map_within I I' f s p = tangent_map_within I I' f₁ s p
theorem filter.eventually_eq.mfderiv_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M β†’ M'} {x : M} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] (hL : f₁ =αΆ [𝓝 x] f) :
mfderiv I I' f₁ x = mfderiv I I' f x

Composition lemmas #

theorem written_in_ext_chart_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} {x : M} {s : set M} {g : M' β†’ M''} (h : continuous_within_at f s x) :
theorem has_mfderiv_within_at.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} (x : M) {s : set M} {g : M' β†’ M''} {u : set M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} {g' : tangent_space I' (f x) β†’L[π•œ] tangent_space I'' (g (f x))} (hg : has_mfderiv_within_at I' I'' g u (f x) g') (hf : has_mfderiv_within_at I I' f s x f') (hst : s βŠ† f ⁻¹' u) :
has_mfderiv_within_at I I'' (g ∘ f) s x (g'.comp f')
theorem has_mfderiv_at.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} (x : M) {g : M' β†’ M''} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} {g' : tangent_space I' (f x) β†’L[π•œ] tangent_space I'' (g (f x))} (hg : has_mfderiv_at I' I'' g (f x) g') (hf : has_mfderiv_at I I' f x f') :
has_mfderiv_at I I'' (g ∘ f) x (g'.comp f')

The chain rule.

theorem has_mfderiv_at.comp_has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} (x : M) {s : set M} {g : M' β†’ M''} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] {f' : tangent_space I x β†’L[π•œ] tangent_space I' (f x)} {g' : tangent_space I' (f x) β†’L[π•œ] tangent_space I'' (g (f x))} (hg : has_mfderiv_at I' I'' g (f x) g') (hf : has_mfderiv_within_at I I' f s x f') :
has_mfderiv_within_at I I'' (g ∘ f) s x (g'.comp f')
theorem mdifferentiable_within_at.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} (x : M) {s : set M} {g : M' β†’ M''} {u : set M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (hg : mdifferentiable_within_at I' I'' g u (f x)) (hf : mdifferentiable_within_at I I' f s x) (h : s βŠ† f ⁻¹' u) :
theorem mdifferentiable_at.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} (x : M) {g : M' β†’ M''} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (hg : mdifferentiable_at I' I'' g (f x)) (hf : mdifferentiable_at I I' f x) :
theorem mfderiv_within_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} (x : M) {s : set M} {g : M' β†’ M''} {u : set M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (hg : mdifferentiable_within_at I' I'' g u (f x)) (hf : mdifferentiable_within_at I I' f s x) (h : s βŠ† f ⁻¹' u) (hxs : unique_mdiff_within_at I s x) :
mfderiv_within I I'' (g ∘ f) s x = (mfderiv_within I' I'' g u (f x)).comp (mfderiv_within I I' f s x)
theorem mfderiv_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} (x : M) {g : M' β†’ M''} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (hg : mdifferentiable_at I' I'' g (f x)) (hf : mdifferentiable_at I I' f x) :
mfderiv I I'' (g ∘ f) x = (mfderiv I' I'' g (f x)).comp (mfderiv I I' f x)
theorem mdifferentiable_on.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} {s : set M} {g : M' β†’ M''} {u : set M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (hg : mdifferentiable_on I' I'' g u) (hf : mdifferentiable_on I I' f s) (st : s βŠ† f ⁻¹' u) :
theorem mdifferentiable.comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} {g : M' β†’ M''} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (hg : mdifferentiable I' I'' g) (hf : mdifferentiable I I' f) :
theorem tangent_map_within_comp_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} {s : set M} {g : M' β†’ M''} {u : set M'} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (p : tangent_bundle I M) (hg : mdifferentiable_within_at I' I'' g u (f p.fst)) (hf : mdifferentiable_within_at I I' f s p.fst) (h : s βŠ† f ⁻¹' u) (hps : unique_mdiff_within_at I s p.fst) :
tangent_map_within I I'' (g ∘ f) s p = tangent_map_within I' I'' g u (tangent_map_within I I' f s p)
theorem tangent_map_comp_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} {g : M' β†’ M''} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (p : tangent_bundle I M) (hg : mdifferentiable_at I' I'' g (f p.fst)) (hf : mdifferentiable_at I I' f p.fst) :
tangent_map I I'' (g ∘ f) p = tangent_map I' I'' g (tangent_map I I' f p)
theorem tangent_map_comp {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] {I : model_with_corners π•œ E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners π•œ E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_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''] {f : M β†’ M'} {g : M' β†’ M''} [Is : smooth_manifold_with_corners I M] [I's : smooth_manifold_with_corners I' M'] [I''s : smooth_manifold_with_corners I'' M''] (hg : mdifferentiable I' I'' g) (hf : mdifferentiable I I' f) :
tangent_map I I'' (g ∘ f) = tangent_map I' I'' g ∘ tangent_map I I' f

Relations between vector space derivative and manifold derivative #

The manifold derivative mfderiv, when considered on the model vector space with its trivial manifold structure, coincides with the usual Frechet derivative fderiv. In this section, we prove this and related statements.

theorem unique_mdiff_within_at_iff_unique_diff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {s : set E} {x : E} :
theorem unique_diff_within_at.unique_mdiff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {s : set E} {x : E} :
unique_diff_within_at π•œ s x β†’ unique_mdiff_within_at π“˜(π•œ, E) s x

Alias of unique_mdiff_within_at_iff_unique_diff_within_at.

theorem unique_mdiff_within_at.unique_diff_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {s : set E} {x : E} :
unique_mdiff_within_at π“˜(π•œ, E) s x β†’ unique_diff_within_at π•œ s x

Alias of unique_mdiff_within_at_iff_unique_diff_within_at.

theorem unique_mdiff_on_iff_unique_diff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {s : set E} :
theorem unique_diff_on.unique_mdiff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {s : set E} :
unique_diff_on π•œ s β†’ unique_mdiff_on π“˜(π•œ, E) s

Alias of unique_mdiff_on_iff_unique_diff_on.

theorem unique_mdiff_on.unique_diff_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {s : set E} :
unique_mdiff_on π“˜(π•œ, E) s β†’ unique_diff_on π•œ s

Alias of unique_mdiff_on_iff_unique_diff_on.

@[simp]
theorem written_in_ext_chart_model_space {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} :
theorem has_mfderiv_within_at_iff_has_fderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} {x : E} {f' : tangent_space π“˜(π•œ, E) x β†’L[π•œ] tangent_space π“˜(π•œ, E') (f x)} :
theorem has_mfderiv_within_at.has_fderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} {x : E} {f' : tangent_space π“˜(π•œ, E) x β†’L[π•œ] tangent_space π“˜(π•œ, E') (f x)} :
has_mfderiv_within_at π“˜(π•œ, E) π“˜(π•œ, E') f s x f' β†’ has_fderiv_within_at f f' s x

Alias of has_mfderiv_within_at_iff_has_fderiv_within_at.

theorem has_fderiv_within_at.has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} {x : E} {f' : tangent_space π“˜(π•œ, E) x β†’L[π•œ] tangent_space π“˜(π•œ, E') (f x)} :
has_fderiv_within_at f f' s x β†’ has_mfderiv_within_at π“˜(π•œ, E) π“˜(π•œ, E') f s x f'

Alias of has_mfderiv_within_at_iff_has_fderiv_within_at.

theorem has_mfderiv_at_iff_has_fderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} {f' : tangent_space π“˜(π•œ, E) x β†’L[π•œ] tangent_space π“˜(π•œ, E') (f x)} :
has_mfderiv_at π“˜(π•œ, E) π“˜(π•œ, E') f x f' ↔ has_fderiv_at f f' x
theorem has_mfderiv_at.has_fderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} {f' : tangent_space π“˜(π•œ, E) x β†’L[π•œ] tangent_space π“˜(π•œ, E') (f x)} :
has_mfderiv_at π“˜(π•œ, E) π“˜(π•œ, E') f x f' β†’ has_fderiv_at f f' x

Alias of has_mfderiv_at_iff_has_fderiv_at.

theorem has_fderiv_at.has_mfderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} {f' : tangent_space π“˜(π•œ, E) x β†’L[π•œ] tangent_space π“˜(π•œ, E') (f x)} :
has_fderiv_at f f' x β†’ has_mfderiv_at π“˜(π•œ, E) π“˜(π•œ, E') f x f'

Alias of has_mfderiv_at_iff_has_fderiv_at.

theorem mdifferentiable_within_at_iff_differentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} {x : E} :

For maps between vector spaces, mdifferentiable_within_at and fdifferentiable_within_at coincide

theorem differentiable_within_at.mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} {x : E} :
differentiable_within_at π•œ f s x β†’ mdifferentiable_within_at π“˜(π•œ, E) π“˜(π•œ, E') f s x

Alias of mdifferentiable_within_at_iff_differentiable_within_at.

theorem mdifferentiable_within_at.differentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} {x : E} :
mdifferentiable_within_at π“˜(π•œ, E) π“˜(π•œ, E') f s x β†’ differentiable_within_at π•œ f s x

Alias of mdifferentiable_within_at_iff_differentiable_within_at.

theorem mdifferentiable_at_iff_differentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} :
mdifferentiable_at π“˜(π•œ, E) π“˜(π•œ, E') f x ↔ differentiable_at π•œ f x

For maps between vector spaces, mdifferentiable_at and differentiable_at coincide

theorem mdifferentiable_at.differentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} :
mdifferentiable_at π“˜(π•œ, E) π“˜(π•œ, E') f x β†’ differentiable_at π•œ f x

Alias of mdifferentiable_at_iff_differentiable_at.

theorem differentiable_at.mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} :
differentiable_at π•œ f x β†’ mdifferentiable_at π“˜(π•œ, E) π“˜(π•œ, E') f x

Alias of mdifferentiable_at_iff_differentiable_at.

theorem mdifferentiable_on_iff_differentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} :
mdifferentiable_on π“˜(π•œ, E) π“˜(π•œ, E') f s ↔ differentiable_on π•œ f s

For maps between vector spaces, mdifferentiable_on and differentiable_on coincide

theorem mdifferentiable_on.differentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} :
mdifferentiable_on π“˜(π•œ, E) π“˜(π•œ, E') f s β†’ differentiable_on π•œ f s

Alias of mdifferentiable_on_iff_differentiable_on.

theorem differentiable_on.mdifferentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} :
differentiable_on π•œ f s β†’ mdifferentiable_on π“˜(π•œ, E) π“˜(π•œ, E') f s

Alias of mdifferentiable_on_iff_differentiable_on.

theorem mdifferentiable_iff_differentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} :
mdifferentiable π“˜(π•œ, E) π“˜(π•œ, E') f ↔ differentiable π•œ f

For maps between vector spaces, mdifferentiable and differentiable coincide

theorem mdifferentiable.differentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} :
mdifferentiable π“˜(π•œ, E) π“˜(π•œ, E') f β†’ differentiable π•œ f

Alias of mdifferentiable_iff_differentiable.

theorem differentiable.mdifferentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} :
differentiable π•œ f β†’ mdifferentiable π“˜(π•œ, E) π“˜(π•œ, E') f

Alias of mdifferentiable_iff_differentiable.

@[simp]
theorem mfderiv_within_eq_fderiv_within {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {s : set E} {x : E} :
mfderiv_within π“˜(π•œ, E) π“˜(π•œ, E') f s x = fderiv_within π•œ f s x

For maps between vector spaces, mfderiv_within and fderiv_within coincide

@[simp]
theorem mfderiv_eq_fderiv {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_3} [normed_group E'] [normed_space π•œ E'] {f : E β†’ E'} {x : E} :
mfderiv π“˜(π•œ, E) π“˜(π•œ, E') f x = fderiv π•œ f x

For maps between vector spaces, mfderiv and fderiv coincide

Differentiability of specific functions #

theorem continuous_linear_map.has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') {s : set E} {x : E} :
theorem continuous_linear_map.has_mfderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') {x : E} :
has_mfderiv_at π“˜(π•œ, E) π“˜(π•œ, E') ⇑f x f
theorem continuous_linear_map.mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') {s : set E} {x : E} :
theorem continuous_linear_map.mdifferentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') {s : set E} :
theorem continuous_linear_map.mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') {x : E} :
theorem continuous_linear_map.mdifferentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') :
theorem continuous_linear_map.mfderiv_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') {x : E} :
mfderiv π“˜(π•œ, E) π“˜(π•œ, E') ⇑f x = f
theorem continuous_linear_map.mfderiv_within_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E β†’L[π•œ] E') {s : set E} {x : E} (hs : unique_mdiff_within_at π“˜(π•œ, E) s x) :
mfderiv_within π“˜(π•œ, E) π“˜(π•œ, E') ⇑f s x = f
theorem continuous_linear_equiv.has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') {s : set E} {x : E} :
theorem continuous_linear_equiv.has_mfderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') {x : E} :
theorem continuous_linear_equiv.mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') {s : set E} {x : E} :
theorem continuous_linear_equiv.mdifferentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') {s : set E} :
theorem continuous_linear_equiv.mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') {x : E} :
theorem continuous_linear_equiv.mdifferentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') :
theorem continuous_linear_equiv.mfderiv_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') {x : E} :
mfderiv π“˜(π•œ, E) π“˜(π•œ, E') ⇑f x = ↑f
theorem continuous_linear_equiv.mfderiv_within_eq {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] (f : E ≃L[π•œ] E') {s : set E} {x : E} (hs : unique_mdiff_within_at π“˜(π•œ, E) s x) :

Identity #

theorem has_mfderiv_at_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (x : M) :
theorem has_mfderiv_within_at_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (s : set M) (x : M) :
theorem mdifferentiable_at_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {x : M} :
theorem mdifferentiable_within_at_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {s : set M} {x : M} :
theorem mdifferentiable_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] :
theorem mdifferentiable_on_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {s : set M} :
@[simp]
theorem mfderiv_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {x : M} :
theorem mfderiv_within_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {s : set M} {x : M} (hxs : unique_mdiff_within_at I s x) :
@[simp]
theorem tangent_map_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] :
theorem tangent_map_within_id {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {s : set M} {p : tangent_bundle I M} (hs : unique_mdiff_within_at I s (tangent_bundle.proj I M p)) :

Constants #

theorem has_mfderiv_at_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (c : M') (x : M) :
has_mfderiv_at I I' (Ξ» (y : M), c) x 0
theorem has_mfderiv_within_at_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (c : M') (s : set M) (x : M) :
has_mfderiv_within_at I I' (Ξ» (y : M), c) s x 0
theorem mdifferentiable_at_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {x : M} {c : M'} :
mdifferentiable_at I I' (Ξ» (y : M), c) x
theorem mdifferentiable_within_at_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {s : set M} {x : M} {c : M'} :
mdifferentiable_within_at I I' (Ξ» (y : M), c) s x
theorem mdifferentiable_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {c : M'} :
mdifferentiable I I' (Ξ» (y : M), c)
theorem mdifferentiable_on_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {s : set M} {c : M'} :
mdifferentiable_on I I' (Ξ» (y : M), c) s
@[simp]
theorem mfderiv_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {x : M} {c : M'} :
mfderiv I I' (Ξ» (y : M), c) x = 0
theorem mfderiv_within_const {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {E' : Type u_5} [normed_group E'] [normed_space π•œ E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners π•œ E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {s : set M} {x : M} {c : M'} (hxs : unique_mdiff_within_at I s x) :
mfderiv_within I I' (Ξ» (y : M), c) s x = 0

Model with corners #

theorem model_with_corners.has_mfderiv_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {x : H} :
theorem model_with_corners.has_mfderiv_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {s : set H} {x : H} :
theorem model_with_corners.mdifferentiable_within_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {s : set H} {x : H} :
theorem model_with_corners.mdifferentiable_at {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {x : H} :
theorem model_with_corners.mdifferentiable_on {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {s : set H} :
theorem model_with_corners.mdifferentiable {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) :
theorem model_with_corners.has_mfderiv_within_at_symm {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {x : E} (hx : x ∈ set.range ⇑I) :
theorem model_with_corners.mdifferentiable_on_symm {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) :
theorem mdifferentiable_at_atlas {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E] [normed_space π•œ E] {H : Type u_3} [topological_space H] (I : model_with_corners π•œ E H) {M : Type u_4} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {e : local_homeomorph M H} (h : e ∈ atlas H M) {x : M} (hx : x ∈ e.to_local_equiv.source) :
theorem mdifferentiable_on_atlas {π•œ : Type u_1} [nondiscrete_normed_field π•œ] {E : Type u_2} [normed_group E]</