Commit e4c3a88d authored by Felix Hilsky's avatar Felix Hilsky
Browse files

include my numbering into bibrefs

parent babc08fb
@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},
......
......@@ -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 $f ∘ ψ^{-1} \colon^n \supseteq ψ(V) → ℝ$
and $\hat F \colon^m \supseteq φ(U) → ℝ^n$}
......
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