The derivative of functions between smooth manifolds #
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Let M and M' be two smooth manifolds with corners over a field 𝕜 (with respective models with
corners I on (E, H) and I' on (E', H')), and let f : M → M'. We define the
derivative of the function at a point, within a set or along the whole space, mimicking the API
for (Fréchet) derivatives. It is denoted by mfderiv I I' f x, where "m" stands for "manifold" and
"f" for "Fréchet" (as in the usual derivative fderiv 𝕜 f x).
Main definitions #
unique_mdiff_on I s: predicate saying that, at each point of the sets, a function can have at most one derivative. This technical condition is important when we definemfderiv_withinbelow, as otherwise there is an arbitrary choice in the derivative, and many properties will fail (for instance the chain rule). This is analogous tounique_diff_on 𝕜 sin a vector space.
Let f be a map between smooth manifolds. The following definitions follow the fderiv API.
mfderiv I I' f x: the derivative offatx, as a continuous linear map from the tangent space atxto the tangent space atf x. If the map is not differentiable, this is0.mfderiv_within I I' f s x: the derivative offatxwithins, as a continuous linear map from the tangent space atxto the tangent space atf x. If the map is not differentiable withins, this is0.mdifferentiable_at I I' f x: Prop expressing whetherfis differentiable atx.mdifferentiable_within_at 𝕜 f s x: Prop expressing whetherfis differentiable withinsatx.has_mfderiv_at I I' f s x f': Prop expressing whetherfhasf'as a derivative atx.has_mfderiv_within_at I I' f s x f': Prop expressing whetherfhasf'as a derivative withinsatx.mdifferentiable_on I I' f s: Prop expressing thatfis differentiable on the sets.mdifferentiable I I' f: Prop expressing thatfis differentiable everywhere.tangent_map I I' f: the derivative off, as a map from the tangent bundle ofMto the tangent bundle ofM'.
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
differentiable_within_at_prop
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
(f : H → H')
(s : set H)
(x : H) :
Prop
Property in the model space of a model with corners of being differentiable within at set at a point, when read in the model vector space. This property will be lifted to manifolds to define differentiable functions between manifolds.
theorem
differentiable_within_at_local_invariant_prop
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H') :
Being differentiable in the model space is a local property, invariant under smooth maps. Therefore, it will lift nicely to manifolds.
def
unique_mdiff_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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
- unique_mdiff_within_at I s x = unique_diff_within_at 𝕜 (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ set.range ⇑I) (⇑(ext_chart_at I x) x)
def
unique_mdiff_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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
- unique_mdiff_on I s = ∀ (x : M), x ∈ s → unique_mdiff_within_at I s x
def
mdifferentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
(f : M → 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
- mdifferentiable_within_at I I' f s x = (continuous_within_at f s x ∧ differentiable_within_at 𝕜 (written_in_ext_chart_at I I' x f) (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ set.range ⇑I) (⇑(ext_chart_at I x) x))
theorem
mdifferentiable_within_at_iff_lift_prop_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
(f : M → M')
(s : set M)
(x : M) :
mdifferentiable_within_at I I' f s x ↔ charted_space.lift_prop_within_at (differentiable_within_at_prop I I') f s x
def
mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
(f : M → 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
- mdifferentiable_at I I' f x = (continuous_at f x ∧ differentiable_within_at 𝕜 (written_in_ext_chart_at I I' x f) (set.range ⇑I) (⇑(ext_chart_at I x) x))
theorem
mdifferentiable_at_iff_lift_prop_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
(f : M → M')
(x : M) :
mdifferentiable_at I I' f x ↔ charted_space.lift_prop_at (differentiable_within_at_prop I I') f x
def
mdifferentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
(f : M → 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
- mdifferentiable_on I I' f s = ∀ (x : M), x ∈ s → mdifferentiable_within_at I I' f s x
def
mdifferentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
(f : M → 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
- mdifferentiable I I' f = ∀ (x : M), mdifferentiable_at I I' f x
def
local_homeomorph.mdifferentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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
- local_homeomorph.mdifferentiable I I' f = (mdifferentiable_on I I' ⇑f f.to_local_equiv.source ∧ mdifferentiable_on I' I ⇑(f.symm) f.to_local_equiv.target)
def
has_mfderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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 → M')
(s : set M)
(x : M)
(f' : tangent_space I x →L[𝕜] 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
- has_mfderiv_within_at I I' f s x f' = (continuous_within_at f s x ∧ has_fderiv_within_at (written_in_ext_chart_at I I' x f) f' (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ set.range ⇑I) (⇑(ext_chart_at I x) x))
def
has_mfderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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 → M')
(x : M)
(f' : tangent_space I x →L[𝕜] 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
- has_mfderiv_at I I' f x f' = (continuous_at f x ∧ has_fderiv_within_at (written_in_ext_chart_at I I' x f) f' (set.range ⇑I) (⇑(ext_chart_at I x) x))
noncomputable
def
mfderiv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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 → M')
(s : set M)
(x : M) :
tangent_space I x →L[𝕜] tangent_space I' (f x)
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
- mfderiv_within I I' f s x = ite (mdifferentiable_within_at I I' f s x) (fderiv_within 𝕜 (written_in_ext_chart_at I I' x f) (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ set.range ⇑I) (⇑(ext_chart_at I x) x)) 0
noncomputable
def
mfderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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 → M')
(x : M) :
tangent_space I x →L[𝕜] tangent_space I' (f x)
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
- mfderiv I I' f x = ite (mdifferentiable_at I I' f x) (fderiv_within 𝕜 (written_in_ext_chart_at I I' x f) (set.range ⇑I) (⇑(ext_chart_at I x) x)) 0
noncomputable
def
tangent_map_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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 → M')
(s : set M) :
tangent_bundle I M → tangent_bundle I' M'
The derivative within a set, as a map between the tangent bundles
Equations
- tangent_map_within I I' f s = λ (p : tangent_bundle I M), {proj := f p.proj, snd := ⇑(mfderiv_within I I' f s p.proj) p.snd}
noncomputable
def
tangent_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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 → M') :
tangent_bundle I M → tangent_bundle I' M'
The derivative, as a map between the tangent bundles
Unique differentiability sets in manifolds #
theorem
unique_mdiff_within_at_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{x : M} :
theorem
unique_mdiff_within_at_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{s : set M}
{x : M} :
unique_mdiff_within_at I s x ↔ unique_diff_within_at 𝕜 (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ (ext_chart_at I x).target) (⇑(ext_chart_at I x) x)
theorem
unique_mdiff_within_at.mono
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 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 ⊆ t) :
unique_mdiff_within_at I t x
theorem
unique_mdiff_within_at.inter'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 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 ∈ nhds_within x s) :
unique_mdiff_within_at I (s ∩ t) x
theorem
unique_mdiff_within_at.inter
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 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 ∈ nhds x) :
unique_mdiff_within_at I (s ∩ t) x
theorem
is_open.unique_mdiff_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 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) :
unique_mdiff_within_at I s x
theorem
unique_mdiff_on.inter
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 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) :
unique_mdiff_on I (s ∩ t)
theorem
is_open.unique_mdiff_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{s : set M}
(hs : is_open s) :
unique_mdiff_on I s
theorem
unique_mdiff_on_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M] :
theorem
unique_mdiff_within_at.eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{s : set M}
{x : M} :
mdifferentiable_within_at I I' f s x ↔ continuous_within_at f s x ∧ differentiable_within_at 𝕜 (written_in_ext_chart_at I I' x f) ((ext_chart_at I x).target ∩ ⇑((ext_chart_at I x).symm) ⁻¹' s) (⇑(ext_chart_at I x) x)
theorem
mdifferentiable_within_at_iff_of_mem_source
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s : set M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{x' : M}
{y : M'}
(hx : x' ∈ (charted_space.chart_at H x).to_local_equiv.source)
(hy : f x' ∈ (charted_space.chart_at H' y).to_local_equiv.source) :
mdifferentiable_within_at I I' f s x' ↔ continuous_within_at f s x' ∧ differentiable_within_at 𝕜 (⇑(ext_chart_at I' y) ∘ f ∘ ⇑((ext_chart_at I x).symm)) (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ set.range ⇑I) (⇑(ext_chart_at I x) x')
One can reformulate differentiability within a set at a point as continuity within this set at this point, and differentiability in any chart containing that point.
theorem
mfderiv_within_zero_of_not_mdifferentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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) :
mfderiv_within I I' f s x = 0
theorem
mfderiv_zero_of_not_mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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_within_at.mono
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_within_at I I' f t x f')
(hst : s ⊆ t) :
has_mfderiv_within_at I I' f s x f'
theorem
has_mfderiv_at.has_mfderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_within_at I I' f s x f') :
mdifferentiable_within_at I I' f s x
theorem
has_mfderiv_at.mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{f' : tangent_space I x →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_at I I' f x f') :
mdifferentiable_at I I' f x
@[simp]
theorem
has_mfderiv_within_at_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{f' : tangent_space I x →L[𝕜] tangent_space I' (f x)} :
has_mfderiv_within_at I I' f set.univ x f' ↔ has_mfderiv_at I I' f x f'
theorem
has_mfderiv_at_unique
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{f₀' f₁' : tangent_space I x →L[𝕜] tangent_space I' (f x)}
(h₀ : has_mfderiv_at I I' f x f₀')
(h₁ : has_mfderiv_at I I' f x f₁') :
f₀' = f₁'
theorem
has_mfderiv_within_at_inter'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : t ∈ nhds_within x s) :
has_mfderiv_within_at I I' f (s ∩ t) x f' ↔ has_mfderiv_within_at I I' f s x f'
theorem
has_mfderiv_within_at_inter
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : t ∈ nhds x) :
has_mfderiv_within_at I I' f (s ∩ t) x f' ↔ has_mfderiv_within_at I I' f s x f'
theorem
has_mfderiv_within_at.union
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] 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 ∪ t) x f'
theorem
has_mfderiv_within_at.nhds_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_within_at I I' f s x f')
(ht : s ∈ nhds_within x t) :
has_mfderiv_within_at I I' f t x f'
theorem
has_mfderiv_within_at.has_mfderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_within_at I I' f s x f')
(hs : s ∈ nhds x) :
has_mfderiv_at I I' f x f'
theorem
mdifferentiable_within_at.has_mfderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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) :
mfderiv_within I I' f s x = fderiv_within 𝕜 (written_in_ext_chart_at I I' x f) (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ set.range ⇑I) (⇑(ext_chart_at I x) x)
theorem
mdifferentiable_at.has_mfderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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) :
mfderiv I I' f x = fderiv_within 𝕜 (written_in_ext_chart_at I I' x f) (set.range ⇑I) (⇑(ext_chart_at I x) x)
theorem
has_mfderiv_at.mfderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{f' : tangent_space I x →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_at I I' f x f') :
theorem
has_mfderiv_within_at.mfderiv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s t : set M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
(st : s ⊆ 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s t : set M}
(hst : s ⊆ t)
(h : mdifferentiable_within_at I I' f t x) :
mdifferentiable_within_at I I' f s x
theorem
mdifferentiable_within_at_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M} :
mdifferentiable_within_at I I' f set.univ x ↔ mdifferentiable_at I I' f x
theorem
mdifferentiable_within_at_inter
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s t : set M}
(ht : t ∈ nhds x) :
mdifferentiable_within_at I I' f (s ∩ t) x ↔ mdifferentiable_within_at I I' f s x
theorem
mdifferentiable_within_at_inter'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s t : set M}
(ht : t ∈ nhds_within x s) :
mdifferentiable_within_at I I' f (s ∩ t) x ↔ mdifferentiable_within_at I I' f s x
theorem
mdifferentiable_at.mdifferentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s : set M}
(h : mdifferentiable_at I I' f x) :
mdifferentiable_within_at I I' f s x
theorem
mdifferentiable_within_at.mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s : set M}
(h : mdifferentiable_within_at I I' f s x)
(hs : s ∈ nhds x) :
mdifferentiable_at I I' f x
theorem
mdifferentiable_on.mono
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{s t : set M}
(h : mdifferentiable_on I I' f t)
(st : s ⊆ t) :
mdifferentiable_on I I' f s
theorem
mdifferentiable_on_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'} :
mdifferentiable_on I I' f set.univ ↔ mdifferentiable I I' f
theorem
mdifferentiable.mdifferentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{s : set M}
(h : mdifferentiable I I' f) :
mdifferentiable_on I I' f s
theorem
mdifferentiable_on_of_locally_mdifferentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{s : set M}
(h : ∀ (x : M), x ∈ s → (∃ (u : set M), is_open u ∧ x ∈ u ∧ mdifferentiable_on I I' f (s ∩ u))) :
mdifferentiable_on I I' f s
@[simp]
theorem
mfderiv_within_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M'] :
mfderiv_within I I' f set.univ = mfderiv I I' f
theorem
mfderiv_within_inter
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s t : set M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
(ht : t ∈ nhds x) :
mfderiv_within I I' f (s ∩ t) x = mfderiv_within I I' f s x
theorem
mdifferentiable_at_iff_of_mem_source
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{x' : M}
{y : M'}
(hx : x' ∈ (charted_space.chart_at H x).to_local_equiv.source)
(hy : f x' ∈ (charted_space.chart_at H' y).to_local_equiv.source) :
mdifferentiable_at I I' f x' ↔ continuous_at f x' ∧ differentiable_within_at 𝕜 (⇑(ext_chart_at I' y) ∘ f ∘ ⇑((ext_chart_at I x).symm)) (set.range ⇑I) (⇑(ext_chart_at I x) x')
Deriving continuity from differentiability on manifolds #
theorem
has_mfderiv_within_at.continuous_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_within_at I I' f s x f') :
continuous_within_at f s x
theorem
has_mfderiv_at.continuous_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{f' : tangent_space I x →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_at I I' f x f') :
continuous_at f x
theorem
mdifferentiable_within_at.continuous_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
{s : set M}
(h : mdifferentiable_within_at I I' f s x) :
continuous_within_at f s x
theorem
mdifferentiable_at.continuous_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
(h : mdifferentiable_at I I' f x) :
continuous_at f x
theorem
mdifferentiable_on.continuous_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{s : set M}
(h : mdifferentiable_on I I' f s) :
continuous_on f s
theorem
mdifferentiable.continuous
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
(h : mdifferentiable I I' f) :
theorem
tangent_map_within_subset
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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 ⊆ t)
(hs : unique_mdiff_within_at I s p.proj)
(h : mdifferentiable_within_at I I' f t p.proj) :
tangent_map_within I I' f s p = tangent_map_within I I' f t p
theorem
tangent_map_within_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M'] :
tangent_map_within I I' f set.univ = tangent_map I I' f
theorem
tangent_map_within_eq_tangent_map
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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.proj)
(h : mdifferentiable_at I I' f p.proj) :
tangent_map_within I I' f s p = tangent_map I I' f p
@[simp]
theorem
tangent_map_within_proj
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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).proj = f p.proj
@[simp]
theorem
tangent_map_proj
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → 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).proj = f p.proj
theorem
mdifferentiable_within_at.prod_mk
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{x : M}
{s : set M}
{f : M → M'}
{g : M → M''}
(hf : mdifferentiable_within_at I I' f s x)
(hg : mdifferentiable_within_at I I'' g s x) :
mdifferentiable_within_at I (I'.prod I'') (λ (x : M), (f x, g x)) s x
theorem
mdifferentiable_at.prod_mk
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{x : M}
{f : M → M'}
{g : M → M''}
(hf : mdifferentiable_at I I' f x)
(hg : mdifferentiable_at I I'' g x) :
mdifferentiable_at I (I'.prod I'') (λ (x : M), (f x, g x)) x
theorem
mdifferentiable_on.prod_mk
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{s : set M}
{f : M → M'}
{g : M → M''}
(hf : mdifferentiable_on I I' f s)
(hg : mdifferentiable_on I I'' g s) :
mdifferentiable_on I (I'.prod I'') (λ (x : M), (f x, g x)) s
theorem
mdifferentiable.prod_mk
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{g : M → M''}
(hf : mdifferentiable I I' f)
(hg : mdifferentiable I I'' g) :
mdifferentiable I (I'.prod I'') (λ (x : M), (f x, g x))
theorem
mdifferentiable_within_at.prod_mk_space
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{x : M}
{s : set M}
{f : M → E'}
{g : M → E''}
(hf : mdifferentiable_within_at I (model_with_corners_self 𝕜 E') f s x)
(hg : mdifferentiable_within_at I (model_with_corners_self 𝕜 E'') g s x) :
mdifferentiable_within_at I (model_with_corners_self 𝕜 (E' × E'')) (λ (x : M), (f x, g x)) s x
theorem
mdifferentiable_at.prod_mk_space
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{x : M}
{f : M → E'}
{g : M → E''}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f x)
(hg : mdifferentiable_at I (model_with_corners_self 𝕜 E'') g x) :
mdifferentiable_at I (model_with_corners_self 𝕜 (E' × E'')) (λ (x : M), (f x, g x)) x
theorem
mdifferentiable_on.prod_mk_space
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{s : set M}
{f : M → E'}
{g : M → E''}
(hf : mdifferentiable_on I (model_with_corners_self 𝕜 E') f s)
(hg : mdifferentiable_on I (model_with_corners_self 𝕜 E'') g s) :
mdifferentiable_on I (model_with_corners_self 𝕜 (E' × E'')) (λ (x : M), (f x, g x)) s
theorem
mdifferentiable.prod_mk_space
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{f : M → E'}
{g : M → E''}
(hf : mdifferentiable I (model_with_corners_self 𝕜 E') f)
(hg : mdifferentiable I (model_with_corners_self 𝕜 E'') g) :
mdifferentiable I (model_with_corners_self 𝕜 (E' × E'')) (λ (x : M), (f x, g x))
Congruence lemmas for derivatives on manifolds #
theorem
has_mfderiv_within_at.congr_of_eventually_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_within_at I I' f s x f')
(h₁ : f₁ =ᶠ[nhds_within x s] f)
(hx : f₁ x = f x) :
has_mfderiv_within_at I I' f₁ s x f'
theorem
has_mfderiv_within_at.congr_mono
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_within_at I I' f s x f')
(ht : ∀ (x : M), x ∈ t → f₁ x = f x)
(hx : f₁ x = f x)
(h₁ : t ⊆ s) :
has_mfderiv_within_at I I' f₁ t x f'
theorem
has_mfderiv_at.congr_of_eventually_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{f' : tangent_space I x →L[𝕜] tangent_space I' (f x)}
(h : has_mfderiv_at I I' f x f')
(h₁ : f₁ =ᶠ[nhds x] f) :
has_mfderiv_at I I' f₁ x f'
theorem
mdifferentiable_within_at.congr_of_eventually_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 =ᶠ[nhds_within x s] f)
(hx : f₁ x = f x) :
mdifferentiable_within_at I I' f₁ s x
theorem
filter.eventually_eq.mdifferentiable_within_at_iff
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → M'}
{x : M}
{s : set M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
(h₁ : f₁ =ᶠ[nhds_within x s] f)
(hx : f₁ x = f x) :
mdifferentiable_within_at I I' f s x ↔ mdifferentiable_within_at I I' f₁ s x
theorem
mdifferentiable_within_at.congr_mono
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 : ∀ (x : M), x ∈ t → f₁ x = f x)
(hx : f₁ x = f x)
(h₁ : t ⊆ s) :
mdifferentiable_within_at I I' f₁ t x
theorem
mdifferentiable_within_at.congr
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 : ∀ (x : M), x ∈ s → f₁ x = f x)
(hx : f₁ x = f x) :
mdifferentiable_within_at I I' f₁ s x
theorem
mdifferentiable_on.congr_mono
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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' : ∀ (x : M), x ∈ t → f₁ x = f x)
(h₁ : t ⊆ s) :
mdifferentiable_on I I' f₁ t
theorem
mdifferentiable_at.congr_of_eventually_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 =ᶠ[nhds x] f) :
mdifferentiable_at I I' f₁ x
theorem
mdifferentiable_within_at.mfderiv_within_congr_mono
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 : ∀ (x : M), x ∈ t → f₁ x = f x)
(hx : f₁ x = f x)
(hxt : unique_mdiff_within_at I t x)
(h₁ : t ⊆ s) :
mfderiv_within I I' f₁ t x = mfderiv_within I I' f s x
theorem
filter.eventually_eq.mfderiv_within_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 =ᶠ[nhds_within x s] f)
(hx : f₁ x = f x) :
mfderiv_within I I' f₁ s x = mfderiv_within I I' f s x
theorem
mfderiv_within_congr
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → 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 : ∀ (x : M), x ∈ s → 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → M'}
{s : set M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
(h : ∀ (x : M), x ∈ s → f x = f₁ x)
(p : tangent_bundle I M)
(hp : p.proj ∈ s)
(hs : unique_mdiff_within_at I s p.proj) :
tangent_map_within I I' f s p = tangent_map_within I I' f₁ s p
theorem
filter.eventually_eq.mfderiv_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f f₁ : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
(hL : f₁ =ᶠ[nhds x] f) :
theorem
mfderiv_congr_point
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{x' : M}
(h : x = x') :
A congruence lemma for mfderiv, (ab)using the fact that tangent_space I' (f x) is
definitionally equal to E'.
theorem
mfderiv_congr
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{f : M → M'}
{x : M}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
{f' : M → M'}
(h : f = f') :
A congruence lemma for mfderiv, (ab)using the fact that tangent_space I' (f x) is
definitionally equal to E'.
Composition lemmas #
theorem
written_in_ext_chart_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{x : M}
{s : set M}
{g : M' → M''}
(h : continuous_within_at f s x) :
{y : E | written_in_ext_chart_at I I'' x (g ∘ f) y = (written_in_ext_chart_at I' I'' (f x) g ∘ written_in_ext_chart_at I I' x f) y} ∈ nhds_within (⇑(ext_chart_at I x) x) (⇑((ext_chart_at I x).symm) ⁻¹' s ∩ set.range ⇑I)
theorem
has_mfderiv_within_at.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
(x : M)
{s : set M}
{g : M' → 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 →L[𝕜] tangent_space I' (f x)}
{g' : tangent_space I' (f x) →L[𝕜] 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 ⊆ f ⁻¹' u) :
has_mfderiv_within_at I I'' (g ∘ f) s x (g'.comp f')
theorem
has_mfderiv_at.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
(x : M)
{g : M' → 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 →L[𝕜] tangent_space I' (f x)}
{g' : tangent_space I' (f x) →L[𝕜] 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
(x : M)
{s : set M}
{g : M' → 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 →L[𝕜] tangent_space I' (f x)}
{g' : tangent_space I' (f x) →L[𝕜] 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
(x : M)
{s : set M}
{g : M' → 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 ⊆ f ⁻¹' u) :
mdifferentiable_within_at I I'' (g ∘ f) s x
theorem
mdifferentiable_at.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
(x : M)
{g : M' → 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) :
mdifferentiable_at I I'' (g ∘ f) x
theorem
mfderiv_within_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
(x : M)
{s : set M}
{g : M' → 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 ⊆ 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
(x : M)
{g : M' → 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_comp_of_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{g : M' → M''}
[Is : smooth_manifold_with_corners I M]
[I's : smooth_manifold_with_corners I' M']
[I''s : smooth_manifold_with_corners I'' M'']
{x : M}
{y : M'}
(hg : mdifferentiable_at I' I'' g y)
(hf : mdifferentiable_at I I' f x)
(hy : f x = y) :
theorem
mdifferentiable_on.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{s : set M}
{g : M' → 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 ⊆ f ⁻¹' u) :
mdifferentiable_on I I'' (g ∘ f) s
theorem
mdifferentiable.comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{g : M' → 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) :
mdifferentiable I I'' (g ∘ f)
theorem
tangent_map_within_comp_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{s : set M}
{g : M' → 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.proj))
(hf : mdifferentiable_within_at I I' f s p.proj)
(h : s ⊆ f ⁻¹' u)
(hps : unique_mdiff_within_at I s p.proj) :
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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{g : M' → 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.proj))
(hf : mdifferentiable_at I I' f p.proj) :
tangent_map I I'' (g ∘ f) p = tangent_map I' I'' g (tangent_map I I' f p)
theorem
tangent_map_comp
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{f : M → M'}
{g : M' → 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
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E} :
unique_mdiff_within_at (model_with_corners_self 𝕜 E) s x ↔ unique_diff_within_at 𝕜 s x
theorem
unique_diff_within_at.unique_mdiff_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E} :
unique_diff_within_at 𝕜 s x → unique_mdiff_within_at (model_with_corners_self 𝕜 E) s x
Alias of the reverse direction of unique_mdiff_within_at_iff_unique_diff_within_at.
theorem
unique_mdiff_within_at.unique_diff_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E}
{x : E} :
unique_mdiff_within_at (model_with_corners_self 𝕜 E) s x → unique_diff_within_at 𝕜 s x
Alias of the forward direction of unique_mdiff_within_at_iff_unique_diff_within_at.
theorem
unique_mdiff_on_iff_unique_diff_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E} :
unique_mdiff_on (model_with_corners_self 𝕜 E) s ↔ unique_diff_on 𝕜 s
theorem
unique_diff_on.unique_mdiff_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E} :
unique_diff_on 𝕜 s → unique_mdiff_on (model_with_corners_self 𝕜 E) s
Alias of the reverse direction of unique_mdiff_on_iff_unique_diff_on.
theorem
unique_mdiff_on.unique_diff_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{s : set E} :
unique_mdiff_on (model_with_corners_self 𝕜 E) s → unique_diff_on 𝕜 s
Alias of the forward direction of unique_mdiff_on_iff_unique_diff_on.
@[simp]
theorem
written_in_ext_chart_model_space
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E} :
written_in_ext_chart_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') x f = f
theorem
has_mfderiv_within_at_iff_has_fderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E}
{x : E}
{f' : tangent_space (model_with_corners_self 𝕜 E) x →L[𝕜] tangent_space (model_with_corners_self 𝕜 E') (f x)} :
has_mfderiv_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s x f' ↔ has_fderiv_within_at f f' s x
theorem
has_mfderiv_within_at.has_fderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E}
{x : E}
{f' : tangent_space (model_with_corners_self 𝕜 E) x →L[𝕜] tangent_space (model_with_corners_self 𝕜 E') (f x)} :
has_mfderiv_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s x f' → has_fderiv_within_at f f' s x
Alias of the forward direction of has_mfderiv_within_at_iff_has_fderiv_within_at.
theorem
has_fderiv_within_at.has_mfderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E}
{x : E}
{f' : tangent_space (model_with_corners_self 𝕜 E) x →L[𝕜] tangent_space (model_with_corners_self 𝕜 E') (f x)} :
has_fderiv_within_at f f' s x → has_mfderiv_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s x f'
Alias of the reverse direction of has_mfderiv_within_at_iff_has_fderiv_within_at.
theorem
has_mfderiv_at_iff_has_fderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E}
{f' : tangent_space (model_with_corners_self 𝕜 E) x →L[𝕜] tangent_space (model_with_corners_self 𝕜 E') (f x)} :
has_mfderiv_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f x f' ↔ has_fderiv_at f f' x
theorem
has_mfderiv_at.has_fderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E}
{f' : tangent_space (model_with_corners_self 𝕜 E) x →L[𝕜] tangent_space (model_with_corners_self 𝕜 E') (f x)} :
has_mfderiv_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f x f' → has_fderiv_at f f' x
Alias of the forward direction of has_mfderiv_at_iff_has_fderiv_at.
theorem
has_fderiv_at.has_mfderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E}
{f' : tangent_space (model_with_corners_self 𝕜 E) x →L[𝕜] tangent_space (model_with_corners_self 𝕜 E') (f x)} :
has_fderiv_at f f' x → has_mfderiv_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f x f'
Alias of the reverse direction of has_mfderiv_at_iff_has_fderiv_at.
theorem
mdifferentiable_within_at_iff_differentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E}
{x : E} :
mdifferentiable_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s x ↔ differentiable_within_at 𝕜 f s x
For maps between vector spaces, mdifferentiable_within_at and fdifferentiable_within_at
coincide
theorem
differentiable_within_at.mdifferentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E}
{x : E} :
differentiable_within_at 𝕜 f s x → mdifferentiable_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s x
Alias of the reverse direction of mdifferentiable_within_at_iff_differentiable_within_at.
theorem
mdifferentiable_within_at.differentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E}
{x : E} :
mdifferentiable_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s x → differentiable_within_at 𝕜 f s x
Alias of the forward direction of mdifferentiable_within_at_iff_differentiable_within_at.
theorem
mdifferentiable_at_iff_differentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E} :
mdifferentiable_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f x ↔ differentiable_at 𝕜 f x
For maps between vector spaces, mdifferentiable_at and differentiable_at coincide
theorem
mdifferentiable_at.differentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E} :
mdifferentiable_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f x → differentiable_at 𝕜 f x
Alias of the forward direction of mdifferentiable_at_iff_differentiable_at.
theorem
differentiable_at.mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E} :
differentiable_at 𝕜 f x → mdifferentiable_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f x
Alias of the reverse direction of mdifferentiable_at_iff_differentiable_at.
theorem
mdifferentiable_on_iff_differentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E} :
mdifferentiable_on (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s ↔ differentiable_on 𝕜 f s
For maps between vector spaces, mdifferentiable_on and differentiable_on coincide
theorem
mdifferentiable_on.differentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E} :
mdifferentiable_on (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s → differentiable_on 𝕜 f s
Alias of the forward direction of mdifferentiable_on_iff_differentiable_on.
theorem
differentiable_on.mdifferentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E} :
differentiable_on 𝕜 f s → mdifferentiable_on (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s
Alias of the reverse direction of mdifferentiable_on_iff_differentiable_on.
theorem
mdifferentiable_iff_differentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'} :
mdifferentiable (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f ↔ differentiable 𝕜 f
For maps between vector spaces, mdifferentiable and differentiable coincide
theorem
mdifferentiable.differentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'} :
mdifferentiable (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f → differentiable 𝕜 f
Alias of the forward direction of mdifferentiable_iff_differentiable.
theorem
differentiable.mdifferentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'} :
differentiable 𝕜 f → mdifferentiable (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f
Alias of the reverse direction of mdifferentiable_iff_differentiable.
@[simp]
theorem
mfderiv_within_eq_fderiv_within
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{s : set E}
{x : E} :
mfderiv_within (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f s x = fderiv_within 𝕜 f s x
For maps between vector spaces, mfderiv_within and fderiv_within coincide
@[simp]
theorem
mfderiv_eq_fderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_3}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : E → E'}
{x : E} :
mfderiv (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') f x = fderiv 𝕜 f x
Differentiability of specific functions #
@[protected]
theorem
continuous_linear_map.has_mfderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E')
{s : set E}
{x : E} :
has_mfderiv_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s x f
@[protected]
theorem
continuous_linear_map.has_mfderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E')
{x : E} :
has_mfderiv_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f x f
@[protected]
theorem
continuous_linear_map.mdifferentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E')
{s : set E}
{x : E} :
mdifferentiable_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s x
@[protected]
theorem
continuous_linear_map.mdifferentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E')
{s : set E} :
mdifferentiable_on (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s
@[protected]
theorem
continuous_linear_map.mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E')
{x : E} :
mdifferentiable_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f x
@[protected]
theorem
continuous_linear_map.mdifferentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E') :
mdifferentiable (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f
theorem
continuous_linear_map.mfderiv_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E')
{x : E} :
mfderiv (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f x = f
theorem
continuous_linear_map.mfderiv_within_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E →L[𝕜] E')
{s : set E}
{x : E}
(hs : unique_mdiff_within_at (model_with_corners_self 𝕜 E) s x) :
mfderiv_within (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s x = f
@[protected]
theorem
continuous_linear_equiv.has_mfderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E')
{s : set E}
{x : E} :
has_mfderiv_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s x ↑f
@[protected]
theorem
continuous_linear_equiv.has_mfderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E')
{x : E} :
has_mfderiv_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f x ↑f
@[protected]
theorem
continuous_linear_equiv.mdifferentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E')
{s : set E}
{x : E} :
mdifferentiable_within_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s x
@[protected]
theorem
continuous_linear_equiv.mdifferentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E')
{s : set E} :
mdifferentiable_on (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s
@[protected]
theorem
continuous_linear_equiv.mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E')
{x : E} :
mdifferentiable_at (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f x
@[protected]
theorem
continuous_linear_equiv.mdifferentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E') :
mdifferentiable (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f
theorem
continuous_linear_equiv.mfderiv_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E')
{x : E} :
mfderiv (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f x = ↑f
theorem
continuous_linear_equiv.mfderiv_within_eq
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : E ≃L[𝕜] E')
{s : set E}
{x : E}
(hs : unique_mdiff_within_at (model_with_corners_self 𝕜 E) s x) :
mfderiv_within (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') ⇑f s x = ↑f
Identity #
theorem
has_mfderiv_at_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
(x : M) :
has_mfderiv_at I I id x (continuous_linear_map.id 𝕜 (tangent_space I x))
theorem
has_mfderiv_within_at_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
(s : set M)
(x : M) :
has_mfderiv_within_at I I id s x (continuous_linear_map.id 𝕜 (tangent_space I x))
theorem
mdifferentiable_at_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{x : M} :
mdifferentiable_at I I id x
theorem
mdifferentiable_within_at_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
{x : M} :
mdifferentiable_within_at I I id s x
theorem
mdifferentiable_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M] :
mdifferentiable I I id
theorem
mdifferentiable_on_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M} :
mdifferentiable_on I I id s
@[simp]
theorem
mfderiv_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{x : M} :
mfderiv I I id x = continuous_linear_map.id 𝕜 (tangent_space I x)
theorem
mfderiv_within_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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) :
mfderiv_within I I id s x = continuous_linear_map.id 𝕜 (tangent_space I x)
@[simp]
theorem
tangent_map_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M] :
tangent_map I I id = id
theorem
tangent_map_within_id
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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 p.proj) :
tangent_map_within I I id s p = p
Constants #
theorem
has_mfderiv_at_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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' (λ (y : M), c) x 0
theorem
has_mfderiv_within_at_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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' (λ (y : M), c) s x 0
theorem
mdifferentiable_at_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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' (λ (y : M), c) x
theorem
mdifferentiable_within_at_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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' (λ (y : M), c) s x
theorem
mdifferentiable_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{c : M'} :
mdifferentiable I I' (λ (y : M), c)
theorem
mdifferentiable_on_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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' (λ (y : M), c) s
@[simp]
theorem
mfderiv_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{x : M}
{c : M'} :
theorem
mfderiv_within_const
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 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' (λ (y : M), c) s x = 0
Operations on the product of two manifolds
theorem
has_mfderiv_at_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
(x : M × M') :
has_mfderiv_at (I.prod I') I prod.fst x (continuous_linear_map.fst 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd))
theorem
has_mfderiv_within_at_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
(s : set (M × M'))
(x : M × M') :
has_mfderiv_within_at (I.prod I') I prod.fst s x (continuous_linear_map.fst 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd))
theorem
mdifferentiable_at_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{x : M × M'} :
mdifferentiable_at (I.prod I') I prod.fst x
theorem
mdifferentiable_within_at_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')}
{x : M × M'} :
mdifferentiable_within_at (I.prod I') I prod.fst s x
theorem
mdifferentiable_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M'] :
mdifferentiable (I.prod I') I prod.fst
theorem
mdifferentiable_on_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')} :
mdifferentiable_on (I.prod I') I prod.fst s
@[simp]
theorem
mfderiv_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{x : M × M'} :
mfderiv (I.prod I') I prod.fst x = continuous_linear_map.fst 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd)
theorem
mfderiv_within_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')}
{x : M × M'}
(hxs : unique_mdiff_within_at (I.prod I') s x) :
mfderiv_within (I.prod I') I prod.fst s x = continuous_linear_map.fst 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd)
@[simp]
theorem
tangent_map_prod_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{p : tangent_bundle (I.prod I') (M × M')} :
theorem
tangent_map_within_prod_fst
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')}
{p : tangent_bundle (I.prod I') (M × M')}
(hs : unique_mdiff_within_at (I.prod I') s p.proj) :
theorem
has_mfderiv_at_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
(x : M × M') :
has_mfderiv_at (I.prod I') I' prod.snd x (continuous_linear_map.snd 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd))
theorem
has_mfderiv_within_at_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
(s : set (M × M'))
(x : M × M') :
has_mfderiv_within_at (I.prod I') I' prod.snd s x (continuous_linear_map.snd 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd))
theorem
mdifferentiable_at_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{x : M × M'} :
mdifferentiable_at (I.prod I') I' prod.snd x
theorem
mdifferentiable_within_at_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')}
{x : M × M'} :
mdifferentiable_within_at (I.prod I') I' prod.snd s x
theorem
mdifferentiable_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M'] :
mdifferentiable (I.prod I') I' prod.snd
theorem
mdifferentiable_on_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')} :
mdifferentiable_on (I.prod I') I' prod.snd s
@[simp]
theorem
mfderiv_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{x : M × M'} :
mfderiv (I.prod I') I' prod.snd x = continuous_linear_map.snd 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd)
theorem
mfderiv_within_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')}
{x : M × M'}
(hxs : unique_mdiff_within_at (I.prod I') s x) :
mfderiv_within (I.prod I') I' prod.snd s x = continuous_linear_map.snd 𝕜 (tangent_space I x.fst) (tangent_space I' x.snd)
@[simp]
theorem
tangent_map_prod_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{p : tangent_bundle (I.prod I') (M × M')} :
theorem
tangent_map_within_prod_snd
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{s : set (M × M')}
{p : tangent_bundle (I.prod I') (M × M')}
(hs : unique_mdiff_within_at (I.prod I') s p.proj) :
theorem
mdifferentiable_at.mfderiv_prod
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
[smooth_manifold_with_corners I'' M'']
{f : M → M'}
{g : M → M''}
{x : M}
(hf : mdifferentiable_at I I' f x)
(hg : mdifferentiable_at I I'' g x) :
theorem
mfderiv_prod_left
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{x₀ : M}
{y₀ : M'} :
mfderiv I (I.prod I') (λ (x : M), (x, y₀)) x₀ = continuous_linear_map.inl 𝕜 (tangent_space I x₀) (tangent_space I' y₀)
theorem
mfderiv_prod_right
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{x₀ : M}
{y₀ : M'} :
mfderiv I' (I.prod I') (λ (y : M'), (x₀, y)) y₀ = continuous_linear_map.inr 𝕜 (tangent_space I x₀) (tangent_space I' y₀)
theorem
mfderiv_prod_eq_add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
(I' : model_with_corners 𝕜 E' H')
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
[smooth_manifold_with_corners I' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
(I'' : model_with_corners 𝕜 E'' H'')
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
[smooth_manifold_with_corners I'' M'']
{f : M × M' → M''}
{p : M × M'}
(hf : mdifferentiable_at (I.prod I') I'' f p) :
The total derivative of a function in two variables is the sum of the partial derivatives.
Note that to state this (without casts) we need to be able to see through the definition of
tangent_space.
Arithmetic #
Note that in the in has_mfderiv_at lemmas there is an abuse of the defeq between E' and
tangent_space 𝓘(𝕜, E') (f z) (similarly for g',F',p',q'). In general this defeq is not
canonical, but in this case (the tangent space of a vector space) it is canonical.
theorem
has_mfderiv_at.add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f g : M → E'}
{f' g' : tangent_space I z →L[𝕜] E'}
(hf : has_mfderiv_at I (model_with_corners_self 𝕜 E') f z f')
(hg : has_mfderiv_at I (model_with_corners_self 𝕜 E') g z g') :
has_mfderiv_at I (model_with_corners_self 𝕜 E') (f + g) z (f' + g')
theorem
mdifferentiable_at.add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f g : M → E'}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f z)
(hg : mdifferentiable_at I (model_with_corners_self 𝕜 E') g z) :
mdifferentiable_at I (model_with_corners_self 𝕜 E') (f + g) z
theorem
mdifferentiable.add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f g : M → E'}
(hf : mdifferentiable I (model_with_corners_self 𝕜 E') f)
(hg : mdifferentiable I (model_with_corners_self 𝕜 E') g) :
mdifferentiable I (model_with_corners_self 𝕜 E') (f + g)
theorem
mfderiv_add
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f g : M → E'}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f z)
(hg : mdifferentiable_at I (model_with_corners_self 𝕜 E') g z) :
mfderiv I (model_with_corners_self 𝕜 E') (f + g) z = mfderiv I (model_with_corners_self 𝕜 E') f z + mfderiv I (model_with_corners_self 𝕜 E') g z
theorem
has_mfderiv_at.const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f : M → E'}
{f' : tangent_space I z →L[𝕜] E'}
(hf : has_mfderiv_at I (model_with_corners_self 𝕜 E') f z f')
(s : 𝕜) :
has_mfderiv_at I (model_with_corners_self 𝕜 E') (s • f) z (s • f')
theorem
mdifferentiable_at.const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f : M → E'}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f z)
(s : 𝕜) :
mdifferentiable_at I (model_with_corners_self 𝕜 E') (s • f) z
theorem
mdifferentiable.const_smul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : M → E'}
(s : 𝕜)
(hf : mdifferentiable I (model_with_corners_self 𝕜 E') f) :
mdifferentiable I (model_with_corners_self 𝕜 E') (s • f)
theorem
const_smul_mfderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f : M → E'}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f z)
(s : 𝕜) :
mfderiv I (model_with_corners_self 𝕜 E') (s • f) z = s • mfderiv I (model_with_corners_self 𝕜 E') f z
theorem
has_mfderiv_at.neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f : M → E'}
{f' : tangent_space I z →L[𝕜] E'}
(hf : has_mfderiv_at I (model_with_corners_self 𝕜 E') f z f') :
has_mfderiv_at I (model_with_corners_self 𝕜 E') (-f) z (-f')
theorem
has_mfderiv_at_neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f : M → E'}
{f' : tangent_space I z →L[𝕜] E'} :
has_mfderiv_at I (model_with_corners_self 𝕜 E') (-f) z (-f') ↔ has_mfderiv_at I (model_with_corners_self 𝕜 E') f z f'
theorem
mdifferentiable_at.neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f : M → E'}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f z) :
mdifferentiable_at I (model_with_corners_self 𝕜 E') (-f) z
theorem
mdifferentiable_at_neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f : M → E'} :
mdifferentiable_at I (model_with_corners_self 𝕜 E') (-f) z ↔ mdifferentiable_at I (model_with_corners_self 𝕜 E') f z
theorem
mdifferentiable.neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f : M → E'}
(hf : mdifferentiable I (model_with_corners_self 𝕜 E') f) :
mdifferentiable I (model_with_corners_self 𝕜 E') (-f)
theorem
mfderiv_neg
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
(f : M → E')
(x : M) :
mfderiv I (model_with_corners_self 𝕜 E') (-f) x = -mfderiv I (model_with_corners_self 𝕜 E') f x
theorem
has_mfderiv_at.sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f g : M → E'}
{f' g' : tangent_space I z →L[𝕜] E'}
(hf : has_mfderiv_at I (model_with_corners_self 𝕜 E') f z f')
(hg : has_mfderiv_at I (model_with_corners_self 𝕜 E') g z g') :
has_mfderiv_at I (model_with_corners_self 𝕜 E') (f - g) z (f' - g')
theorem
mdifferentiable_at.sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f g : M → E'}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f z)
(hg : mdifferentiable_at I (model_with_corners_self 𝕜 E') g z) :
mdifferentiable_at I (model_with_corners_self 𝕜 E') (f - g) z
theorem
mdifferentiable.sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{f g : M → E'}
(hf : mdifferentiable I (model_with_corners_self 𝕜 E') f)
(hg : mdifferentiable I (model_with_corners_self 𝕜 E') g) :
mdifferentiable I (model_with_corners_self 𝕜 E') (f - g)
theorem
mfderiv_sub
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{z : M}
{f g : M → E'}
(hf : mdifferentiable_at I (model_with_corners_self 𝕜 E') f z)
(hg : mdifferentiable_at I (model_with_corners_self 𝕜 E') g z) :
mfderiv I (model_with_corners_self 𝕜 E') (f - g) z = mfderiv I (model_with_corners_self 𝕜 E') f z - mfderiv I (model_with_corners_self 𝕜 E') g z
theorem
has_mfderiv_within_at.mul'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
{z : M}
{F' : Type u_11}
[normed_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
{p' q' : tangent_space I z →L[𝕜] F'}
(hp : has_mfderiv_within_at I (model_with_corners_self 𝕜 F') p s z p')
(hq : has_mfderiv_within_at I (model_with_corners_self 𝕜 F') q s z q') :
has_mfderiv_within_at I (model_with_corners_self 𝕜 F') (p * q) s z (p z • q' + p'.smul_right (q z))
theorem
has_mfderiv_at.mul'
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{z : M}
{F' : Type u_11}
[normed_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
{p' q' : tangent_space I z →L[𝕜] F'}
(hp : has_mfderiv_at I (model_with_corners_self 𝕜 F') p z p')
(hq : has_mfderiv_at I (model_with_corners_self 𝕜 F') q z q') :
has_mfderiv_at I (model_with_corners_self 𝕜 F') (p * q) z (p z • q' + p'.smul_right (q z))
theorem
mdifferentiable_within_at.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
{z : M}
{F' : Type u_11}
[normed_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
(hp : mdifferentiable_within_at I (model_with_corners_self 𝕜 F') p s z)
(hq : mdifferentiable_within_at I (model_with_corners_self 𝕜 F') q s z) :
mdifferentiable_within_at I (model_with_corners_self 𝕜 F') (p * q) s z
theorem
mdifferentiable_at.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{z : M}
{F' : Type u_11}
[normed_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
(hp : mdifferentiable_at I (model_with_corners_self 𝕜 F') p z)
(hq : mdifferentiable_at I (model_with_corners_self 𝕜 F') q z) :
mdifferentiable_at I (model_with_corners_self 𝕜 F') (p * q) z
theorem
mdifferentiable_on.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
{F' : Type u_11}
[normed_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
(hp : mdifferentiable_on I (model_with_corners_self 𝕜 F') p s)
(hq : mdifferentiable_on I (model_with_corners_self 𝕜 F') q s) :
mdifferentiable_on I (model_with_corners_self 𝕜 F') (p * q) s
theorem
mdifferentiable.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{F' : Type u_11}
[normed_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
(hp : mdifferentiable I (model_with_corners_self 𝕜 F') p)
(hq : mdifferentiable I (model_with_corners_self 𝕜 F') q) :
mdifferentiable I (model_with_corners_self 𝕜 F') (p * q)
theorem
has_mfderiv_within_at.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
{z : M}
{F' : Type u_11}
[normed_comm_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
{p' q' : tangent_space I z →L[𝕜] F'}
(hp : has_mfderiv_within_at I (model_with_corners_self 𝕜 F') p s z p')
(hq : has_mfderiv_within_at I (model_with_corners_self 𝕜 F') q s z q') :
has_mfderiv_within_at I (model_with_corners_self 𝕜 F') (p * q) s z (p z • q' + q z • p')
theorem
has_mfderiv_at.mul
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{z : M}
{F' : Type u_11}
[normed_comm_ring F']
[normed_algebra 𝕜 F']
{p q : M → F'}
{p' q' : tangent_space I z →L[𝕜] F'}
(hp : has_mfderiv_at I (model_with_corners_self 𝕜 F') p z p')
(hq : has_mfderiv_at I (model_with_corners_self 𝕜 F') q z q') :
has_mfderiv_at I (model_with_corners_self 𝕜 F') (p * q) z (p z • q' + q z • p')
Model with corners #
@[protected]
theorem
model_with_corners.has_mfderiv_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{x : H} :
has_mfderiv_at I (model_with_corners_self 𝕜 E) ⇑I x (continuous_linear_map.id 𝕜 (tangent_space I x))
@[protected]
theorem
model_with_corners.has_mfderiv_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{s : set H}
{x : H} :
has_mfderiv_within_at I (model_with_corners_self 𝕜 E) ⇑I s x (continuous_linear_map.id 𝕜 (tangent_space I x))
@[protected]
theorem
model_with_corners.mdifferentiable_within_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{s : set H}
{x : H} :
mdifferentiable_within_at I (model_with_corners_self 𝕜 E) ⇑I s x
@[protected]
theorem
model_with_corners.mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{x : H} :
mdifferentiable_at I (model_with_corners_self 𝕜 E) ⇑I x
@[protected]
theorem
model_with_corners.mdifferentiable_on
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{s : set H} :
mdifferentiable_on I (model_with_corners_self 𝕜 E) ⇑I s
@[protected]
theorem
model_with_corners.mdifferentiable
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H) :
mdifferentiable I (model_with_corners_self 𝕜 E) ⇑I
theorem
model_with_corners.has_mfderiv_within_at_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{x : E}
(hx : x ∈ set.range ⇑I) :
has_mfderiv_within_at (model_with_corners_self 𝕜 E) I ⇑(I.symm) (set.range ⇑I) x (continuous_linear_map.id 𝕜 (tangent_space (model_with_corners_self 𝕜 E) x))
theorem
model_with_corners.mdifferentiable_on_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H) :
mdifferentiable_on (model_with_corners_self 𝕜 E) I ⇑(I.symm) (set.range ⇑I)
theorem
mdifferentiable_at_atlas
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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 ∈ charted_space.atlas H M)
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
mdifferentiable_at I I ⇑e x
theorem
mdifferentiable_on_atlas
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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 ∈ charted_space.atlas H M) :
mdifferentiable_on I I ⇑e e.to_local_equiv.source
theorem
mdifferentiable_at_atlas_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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 ∈ charted_space.atlas H M)
{x : H}
(hx : x ∈ e.to_local_equiv.target) :
mdifferentiable_at I I ⇑(e.symm) x
theorem
mdifferentiable_on_atlas_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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 ∈ charted_space.atlas H M) :
mdifferentiable_on I I ⇑(e.symm) e.to_local_equiv.target
theorem
mdifferentiable_of_mem_atlas
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 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 ∈ charted_space.atlas H M) :
theorem
mdifferentiable_chart
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
(x : M) :
theorem
tangent_map_chart
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{p q : tangent_bundle I M}
(h : q.proj ∈ (charted_space.chart_at H p.proj).to_local_equiv.source) :
tangent_map I I ⇑(charted_space.chart_at H p.proj) q = ⇑((bundle.total_space.to_prod H E).symm) (⇑(charted_space.chart_at (model_prod H E) p) q)
The derivative of the chart at a base point is the chart of the tangent bundle, composed with the identification between the tangent bundle of the model space and the product space.
theorem
tangent_map_chart_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{p : tangent_bundle I M}
{q : tangent_bundle I H}
(h : q.proj ∈ (charted_space.chart_at H p.proj).to_local_equiv.target) :
tangent_map I I ⇑((charted_space.chart_at H p.proj).symm) q = ⇑((charted_space.chart_at (model_prod H E) p).symm) (⇑(bundle.total_space.to_prod H E) q)
The derivative of the inverse of the chart at a base point is the inverse of the chart of the tangent bundle, composed with the identification between the tangent bundle of the model space and the product space.
Differentiable local homeomorphisms #
theorem
local_homeomorph.mdifferentiable.symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e) :
@[protected]
theorem
local_homeomorph.mdifferentiable.mdifferentiable_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
mdifferentiable_at I I' ⇑e x
theorem
local_homeomorph.mdifferentiable.mdifferentiable_at_symm
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
{x : M'}
(hx : x ∈ e.to_local_equiv.target) :
mdifferentiable_at I' I ⇑(e.symm) x
theorem
local_homeomorph.mdifferentiable.symm_comp_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
(mfderiv I' I ⇑(e.symm) (⇑e x)).comp (mfderiv I I' ⇑e x) = continuous_linear_map.id 𝕜 (tangent_space I x)
theorem
local_homeomorph.mdifferentiable.comp_symm_deriv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M'}
(hx : x ∈ e.to_local_equiv.target) :
@[protected]
noncomputable
def
local_homeomorph.mdifferentiable.mfderiv
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
tangent_space I x ≃L[𝕜] tangent_space I' (⇑e x)
The derivative of a differentiable local homeomorphism, as a continuous linear equivalence
between the tangent spaces at x and e x.
theorem
local_homeomorph.mdifferentiable.mfderiv_bijective
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
function.bijective ⇑(mfderiv I I' ⇑e x)
theorem
local_homeomorph.mdifferentiable.mfderiv_injective
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
function.injective ⇑(mfderiv I I' ⇑e x)
theorem
local_homeomorph.mdifferentiable.mfderiv_surjective
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
function.surjective ⇑(mfderiv I I' ⇑e x)
theorem
local_homeomorph.mdifferentiable.ker_mfderiv_eq_bot
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
linear_map.ker (mfderiv I I' ⇑e x) = ⊥
theorem
local_homeomorph.mdifferentiable.range_mfderiv_eq_top
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
linear_map.range (mfderiv I I' ⇑e x) = ⊤
theorem
local_homeomorph.mdifferentiable.range_mfderiv_eq_univ
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
{x : M}
(hx : x ∈ e.to_local_equiv.source) :
theorem
local_homeomorph.mdifferentiable.trans
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{E'' : Type u_8}
[normed_add_comm_group E'']
[normed_space 𝕜 E'']
{H'' : Type u_9}
[topological_space H'']
{I'' : model_with_corners 𝕜 E'' H''}
{M'' : Type u_10}
[topological_space M'']
[charted_space H'' M'']
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e)
{e' : local_homeomorph M' M''}
[smooth_manifold_with_corners I M]
[smooth_manifold_with_corners I' M']
[smooth_manifold_with_corners I'' M'']
(he' : local_homeomorph.mdifferentiable I' I'' e') :
local_homeomorph.mdifferentiable I I'' (e.trans e')
Differentiability of ext_chart_at #
theorem
has_mfderiv_at_ext_chart_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{x y : M}
(h : y ∈ (charted_space.chart_at H x).to_local_equiv.source) :
has_mfderiv_at I (model_with_corners_self 𝕜 E) ⇑(ext_chart_at I x) y (mfderiv I I ⇑(charted_space.chart_at H x) y)
theorem
has_mfderiv_within_at_ext_chart_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
{x y : M}
(h : y ∈ (charted_space.chart_at H x).to_local_equiv.source) :
has_mfderiv_within_at I (model_with_corners_self 𝕜 E) ⇑(ext_chart_at I x) s y (mfderiv I I ⇑(charted_space.chart_at H x) y)
theorem
mdifferentiable_at_ext_chart_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{x y : M}
(h : y ∈ (charted_space.chart_at H x).to_local_equiv.source) :
mdifferentiable_at I (model_with_corners_self 𝕜 E) ⇑(ext_chart_at I x) y
theorem
mdifferentiable_on_ext_chart_at
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
(I : model_with_corners 𝕜 E H)
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{x : M} :
mdifferentiable_on I (model_with_corners_self 𝕜 E) ⇑(ext_chart_at I x) (charted_space.chart_at H x).to_local_equiv.source
Unique derivative sets in manifolds #
theorem
unique_mdiff_on.unique_mdiff_on_preimage
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{s : set M}
[smooth_manifold_with_corners I' M']
(hs : unique_mdiff_on I s)
{e : local_homeomorph M M'}
(he : local_homeomorph.mdifferentiable I I' e) :
unique_mdiff_on I' (e.to_local_equiv.target ∩ ⇑(e.symm) ⁻¹' s)
If a set has the unique differential property, then its image under a local diffeomorphism also has the unique differential property.
theorem
unique_mdiff_on.unique_diff_on_target_inter
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
(hs : unique_mdiff_on I s)
(x : M) :
unique_diff_on 𝕜 ((ext_chart_at I x).target ∩ ⇑((ext_chart_at I x).symm) ⁻¹' s)
If a set in a manifold has the unique derivative property, then its pullback by any extended chart, in the vector space, also has the unique derivative property.
theorem
unique_mdiff_on.unique_diff_on_inter_preimage
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{E' : Type u_5}
[normed_add_comm_group E']
[normed_space 𝕜 E']
{H' : Type u_6}
[topological_space H']
{I' : model_with_corners 𝕜 E' H'}
{M' : Type u_7}
[topological_space M']
[charted_space H' M']
{s : set M}
(hs : unique_mdiff_on I s)
(x : M)
(y : M')
{f : M → M'}
(hf : continuous_on f s) :
unique_diff_on 𝕜 ((ext_chart_at I x).target ∩ ⇑((ext_chart_at I x).symm) ⁻¹' (s ∩ f ⁻¹' (ext_chart_at I' y).source))
When considering functions between manifolds, this statement shows up often. It entails the unique differential of the pullback in extended charts of the set where the function can be read in the charts.
theorem
unique_mdiff_on.smooth_bundle_preimage
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
{F : Type u_8}
[normed_add_comm_group F]
[normed_space 𝕜 F]
(Z : M → Type u_9)
[topological_space (bundle.total_space F Z)]
[Π (b : M), topological_space (Z b)]
[Π (b : M), add_comm_monoid (Z b)]
[Π (b : M), module 𝕜 (Z b)]
[fiber_bundle F Z]
[vector_bundle 𝕜 F Z]
[smooth_vector_bundle F Z I]
(hs : unique_mdiff_on I s) :
unique_mdiff_on (I.prod (model_with_corners_self 𝕜 F)) (bundle.total_space.proj ⁻¹' s)
In a smooth fiber bundle, the preimage under the projection of a set with unique differential in the basis also has unique differential.
theorem
unique_mdiff_on.tangent_bundle_proj_preimage
{𝕜 : Type u_1}
[nontrivially_normed_field 𝕜]
{E : Type u_2}
[normed_add_comm_group E]
[normed_space 𝕜 E]
{H : Type u_3}
[topological_space H]
{I : model_with_corners 𝕜 E H}
{M : Type u_4}
[topological_space M]
[charted_space H M]
[smooth_manifold_with_corners I M]
{s : set M}
(hs : unique_mdiff_on I s) :