Differentiability of functions in vector bundles #
Characterization of differentiable functions into a vector bundle. Version at a point within a set
Characterization of differentiable functions into a vector bundle. Version at a point
Characterization of differentiable sections of a vector bundle at a point within a set in terms of the preferred trivialization at that point.
Characterization of differentiable sections of a vector bundle at a point within a set in terms of the preferred trivialization at that point.
Characterization of differentiable functions into a vector bundle in terms of any trivialization. Version at a point within at set.
Characterization of differentiable functions into a vector bundle in terms of any trivialization. Version at a point.
Characterization of differentiable sections a vector bundle in terms of any trivialization. Version at a point within at set.
Characterization of differentiable functions into a vector bundle in terms of any trivialization. Version at a point.
Differentiability of a section on s
can be determined
using any trivialisation whose baseSet
contains s
.
For any trivialization e
, the differentiability of a section on e.baseSet
can be determined using e
.
Consider a differentiable map v : M β Eβ
to a vector bundle, over a basemap bβ : M β Bβ
, and
another basemap bβ : M β Bβ
. Given linear maps Ο m : Eβ (bβ m) β Eβ (bβ m)
depending
differentiably on m
, one can apply Ο m
to g m
, and the resulting map is differentiable.
Note that the differentiability of Ο
cannot be always be stated as differentiability of a map
into a manifold, as the pullback bundles bβ *α΅ Eβ
and bβ *α΅ Eβ
only make sense when bβ
and bβ
are globally smooth, but we want to apply this lemma with only local information.
Therefore, we formulate it using differentiability of Ο
read in coordinates.
Version for MDifferentiableWithinAt
. We also give a version for MDifferentiableAt
, but no
version for MDifferentiableOn
or MDifferentiable
as our assumption, written in coordinates,
only makes sense around a point.
Consider a differentiable map v : M β Eβ
to a vector bundle, over a basemap bβ : M β Bβ
, and
another basemap bβ : M β Bβ
. Given linear maps Ο m : Eβ (bβ m) β Eβ (bβ m)
depending
differentiably on m
, one can apply Ο m
to g m
, and the resulting map is differentiable.
Note that the differentiability of Ο
cannot be always be stated as differentiability of a map
into a manifold, as the pullback bundles bβ *α΅ Eβ
and bβ *α΅ Eβ
only make sense when bβ
and bβ
are globally smooth, but we want to apply this lemma with only local information.
Therefore, we formulate it using differentiability of Ο
read in coordinates.
Version for MDifferentiableAt
. We also give a version for MDifferentiableWithinAt
,
but no version for MDifferentiableOn
or MDifferentiable
as our assumption, written
in coordinates, only makes sense around a point.