In this section, the concepts of vector fields, covector fields and tensor fields shall be presented. We will also define what it means that one of those (vector field, covector field, tensor field) is differentiable. Then we will show how suitable restrictions of all these things can be written as sums of the bases of the respective spaces induced by a chart, and we will show a simultaneously sufficient and necessary condition of differentiability based on this sum expression.
Lemma 5.4:
Let
be a
-dimensional manifold of class
and
be contained in its atlas. Then the vector fields
![{\displaystyle p\mapsto \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p},j\in \{1,\ldots ,d\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afa3064acebff113c1252e76e34c9101948491d1)
are differentiable of class
.
Proof:
Let
. Then we have:
![{\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\varphi (p)=\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\phi (p))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3f334694b9caf5426cb45841a960d5b0e6e24c)
Let now
be another chart in the atlas of
. Then the function
![{\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)\varphi {\big |}_{O\cap U\cap V}\circ \theta |_{O\cap U\cap V}^{-1}=\partial _{x_{j}}(\varphi \circ \phi ^{-1})\circ (\phi |_{O\cap U\cap V}\circ \theta |_{O\cap U\cap V}^{-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/858cf16af9b2039e1dbfc271920f11802d0e9cb0)
is smooth, as the composition of smooth functions.
If
is a manifold of class
, we even have that since
is contained in
for all
, if a chart around
is given by
, then for all
![{\displaystyle \mathbf {X} (p)=\sum _{j=1}^{d}x_{j}(p)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}~~~~~(*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fecf6036291f7efa7764bc736a0d5cfe8db82d3)
, where
![{\displaystyle x_{j}:U\cap O\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc06b0a70bbf04e27a04de00d0fee245e135d36b)
are functions from
to
. This follows from chapter 2, where we remarked based on two theorems of the section, that
![{\displaystyle \left\{\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}{\Bigg |}j\in \{1,\ldots ,d\}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3594783d1afe5b6154e2380ead27301023b1f7)
is a basis of
for
.
Theorem 5.5:
Let
be a vector field on the open subset
of the
-dimensional
manifold
, and let
be contained in the atlas of
. Then the vector field
is differentiable of class
iff all the
, as defined by eq.
above, are contained in
.
Proof:
1.) We prove that if all the
defined by
are contained in
, that then
is differentiable of class
.
This is because if
is contained in
, then due to lemma 5.4 and theorem 2.24 all the summands of the function
![{\displaystyle \sum _{j=1}^{d}\mathbf {V} _{\phi ,j}\left({\frac {\partial }{\partial \phi _{j}}}\right)\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/88a4d4a2b7eb539f51b53abfe129b4cb6fca71f6)
are differentiable of class
. Due to theorem 2.23 and induction, we have that the function itself is differentiable of class
. Due to
, the function is identical to
.
2.) We prove that if
is differentiable of class
, then so are the
defined by
.
Due to lemma 2.3, if we write
, the functions
,
are contained in
.
By definition of the differentiability of class
of
, we have that the functions
![{\displaystyle \mathbf {V} \phi _{k},k\in \{1,\ldots ,d\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2148df7fbac57550e9d7e73392ca0363fd5dca)
are contained in
. But due to
and lemma 2.4, we have for all
:
![{\displaystyle {\begin{aligned}\mathbf {V} \phi _{k}(p)&=\mathbf {V} _{\phi ,j}(p)\sum _{j=1}^{d}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})\\&=\mathbf {V} _{\phi ,j}(p)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82eb0af9d66a662b60f16a4b6212f2cf4c20e787)
Hence:
![{\displaystyle \mathbf {V} \phi _{k}=\mathbf {V} _{\phi ,j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d350b516ce7458a3b79fda7b1680ad49ded00956)
, where since the two functions are equal and one of them is differentiable of class
, both of them are differentiable of class
.
Lemma 5.9:
Let
be a manifold of class
and
be contained in its atlas. Then the covector fields
![{\displaystyle p\mapsto (d\phi _{j}),j\in \{1,\ldots ,d\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4965578443563aef14517a97d908e57bbdd59498)
are differentiable of class
.
Proof:
Let
be differentiable of class
, and let
. Due to lemma 2.3, the function
is differentiable of class
. Since
is differentiable of class
, it follows that
![{\displaystyle \mathbf {V} \phi _{j}=d\phi _{j}\mathbf {V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/641818b170a17635407a80c2dd0d616730e9691b)
is differentiable of class
(the latest equation follows from the definition of
).
If
is a manifold of class
, we even have that since
is contained in
for all
, if a chart around
is given by
, then for all
![{\displaystyle \alpha (p)=\sum _{j=1}^{d}x_{j}^{*}(p)(d\varphi _{j})_{p}~~~~~(**)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd9c52b2be9e1c2b5469184761e910d4df1f1e73)
, where
![{\displaystyle x_{j}^{*}:U\cap O\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72acb525059ed03000a174ac53eb7e99b66f39b5)
are functions from
to
. This follows from chapter 2, where we remarked based on two theorems of the chapter, that
![{\displaystyle \left\{(d\phi _{j})_{p}{\big |}j\in \{1,\ldots ,d\}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c12d2be2877186a774c2fde04767718b1c550a)
is a basis of
for
.
Proof:
1.) We show that the differentiability of class
of the
defined by
implies the differentiability of
.
Let
be a vector field on
which is differentiable of class
. Due to lemma 5.9 and theorem 2.24, all the summands of the function
![{\displaystyle \sum _{j=1}^{d}\alpha _{\phi ,j}d\phi _{j}\mathbf {V} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5acf1933b89deb93116b02d1553e99db4cb2edb9)
are contained in
. Therefore, due to theorem 2.23 and induction, also the function itself is contained in
. But due to
, the function is equal to
.
2.) We show that if
is differentiable, then so are the
,
defined by
.
Due to lemma 5.4, we have that for
, the vector field
is differentiable of
. Hence, due to the differentiability of
, the function
![{\displaystyle \alpha \left({\frac {\partial }{\partial \phi _{k}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acffa3399635541c933bc21fd0feca80fe466195)
is contained in
. But due to
, we have
![{\displaystyle {\begin{aligned}\alpha \left({\frac {\partial }{\partial \phi _{k}}}\right)(p)&=\sum _{j=1}^{d}\alpha _{\phi ,j}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)\\&=\alpha _{\phi ,k}(p)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f68246ba440bb996abfa6d83ac6aa513faac43)
Hence:
![{\displaystyle \alpha \left({\frac {\partial }{\partial \phi _{k}}}\right)=\alpha _{\phi ,k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa91497eea0e841da9c93169ee5438e72437cfac)
and hence
.
Definition 5.11:
Let
be a vector space, and
its dual space.