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Felix Hilsky
masterthesis
Commits
e4c3a88d
Commit
e4c3a88d
authored
Jan 13, 2022
by
Felix Hilsky
Browse files
include my numbering into bibrefs
parent
babc08fb
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2
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tex/bibliography.bib
View file @
e4c3a88d
@book
{
IntroRiemannLee
,
@book
{
q3-
IntroRiemannLee
,
doi
=
{10.1007/978-3-319-91755-9}
,
year
=
{2018}
,
publisher
=
{Springer International Publishing}
,
author
=
{John M. Lee}
,
title
=
{Introduction to Riemannian Manifolds}
}
@article
{
Ball2011
,
@article
{
q4-
Ball2011
,
doi
=
{10.1007/s00205-011-0421-3}
,
year
=
{2011}
,
month
=
may
,
...
...
@@ -17,7 +17,7 @@
title
=
{Orientability and Energy Minimization in Liquid Crystal Models}
,
journal
=
{Archive for Rational Mechanics and Analysis}
}
@article
{
Nitschke2018
,
@article
{
q5-
Nitschke2018
,
doi
=
{10.1098/rspa.2017.0686}
,
url
=
{https://royalsocietypublishing.org/doi/10.1098/rspa.2017.0686}
,
year
=
{2018}
,
...
...
tex/scratchpad/chainrule.tex
View file @
e4c3a88d
...
...
@@ -10,7 +10,7 @@ I always struggle with the more-dimensional chainrule
and with coordinates in differential geometry.
Hence I tried to understand the Jacobian matrix
representation of the
\newTerm
{
differential of
$
F
$
at
$
p
$}
as in
\textcite
[Tangent Vectors][377]
{
IntroRiemannLee
}
.
as in
\textcite
[Tangent Vectors][377]
{
q3-
IntroRiemannLee
}
.
We have given
\begin{itemize}
...
...
@@ -40,7 +40,7 @@ $dF_p(v) f = v(f ∘ F)$. Hence
&
= v
^
i
\tanvecat
{
x
^
i
}{
p
}
(f ∘ F)
\\
&
= v
^
i
\tanvecat
{
x
^
i
}{
p
}
(f ∘ ψ
^{
-1
}
∘
\hat
F ∘ φ)
&&
(
\text
{
Def.
}
\hat
F)
\\
&
= v
^
i
\rest
{
∂
_
i
}{
φ(p)
}
(f ∘ ψ
^{
-1
}
∘
\hat
F)
\\
&
\qquad
(
\text
{
Def.
}
\tanvecat
{
x
^
i
}{
p
}
\text
{
\textcite
[A.2][377]
{
IntroRiemannLee
}}
)
\\
&
\qquad
(
\text
{
Def.
}
\tanvecat
{
x
^
i
}{
p
}
\text
{
\textcite
[A.2][377]
{
q3-
IntroRiemannLee
}}
)
\\
\intertext
{
Now we are in a purely flat setting and can apply the
calculus chain rule for
$
f ∘ ψ
^{
-
1
}
\colon
ℝ
^
n
\supseteq
ψ
(
V
)
→ ℝ
$
and
$
\hat
F
\colon
ℝ
^
m
\supseteq
φ
(
U
)
→ ℝ
^
n
$}
...
...
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