Commit 9e654f9a authored by Felix Hilsky's avatar Felix Hilsky
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document my understanding of coordinates

should be removed in final paper, it's just in the document in order to compile it simply
parent 164725d5
@book{IntroRiemannLee,
doi = {10.1007/978-3-319-91755-9},
year = {2018},
publisher = {Springer International Publishing},
author = {John M. Lee},
title = {Introduction to Riemannian Manifolds}
}
.maindir/.latexmkrc
\ No newline at end of file
%! TEX program = lualatex
\input{.maindir/tex/header/preamble-section}
% inputs the preamble only if necessary
\docStart
\section{Differential in coordinate representation}
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}.
We have given
\begin{itemize}
\item two smooth manifolds $M$ ($\dim M = m$) and
$N$ ($\dim N = n$),
\item a smooth map $F \colon M → N$,
\item a point $p ∈ M$,
\item charts $φ = (x^i)_{1 \leq i \leq m} \colon p ∈ U → ℝ^m$
and $ψ = (y^j)_{1 \leq j \leq n} \colon F(p) ∈ V → ℝ^n$,
\item the coordinate representation $\hat F$ of $F$
\begin{equation*}
\hat{F} = ψ^{-1} ∘ F ∘ φ^{-1}
\end{equation*}
\item any vector $v = v^i \tanvecat{x^i}{p} ∈ T_p M$,
where we use Einstein summation convention
\item any $f ∈ C^{}(N)$.
\end{itemize}
We want to show
\begin{equation*}
dF_p \left(v^i \tanvecat{x^i}{p} \right)
= \tanvec[\hat F^j]{x^i}(p) v^i \tanvecat{y^i}{F(p)}
\end{equation*}
$dF_p$ is defined by application to $f$, namely
$dF_p(v) f = v(f ∘ F)$. Hence
\begin{align*}
dF_p \left(v^i \tanvecat{x^i}{p} \right) f
&= 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}}) \\
\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 $\hat F \colon^m \supseteq φ(U) → ℝ^n$}
&= v^i \left( D_{\hat F ∘ φ(p)} (f ∘ ψ^{-1}) ∘ D \hat F(φ(p)) \right)_i \\
\intertext{
Here $D \hat F$ is a matrix $n$ rows and $m$ columns
and $D(f ∘ ψ^{-1})$ is a row vector of length $n$,
so the dimensions match.
The $i$th entry is the product of the row and the $i$th column
of $D \hat F$ which is $\tanvec[\hat F]{x^i}$ with the entries $\tanvec[\hat F^j]{x^i}$. Hence}
&= v^i \rest{_j}{\hat F ∘ φ(p)} (f ∘ ψ^{-1}) ∂_i \hat F^j(p)
\intertext{
Since the charts are parts of the functions differentiated we
can use the definition of the tangent vectors again to write
it in differential geometry terms in $M$ and $N$.
Reordering the terms is possible since they are all numbers.}
&= v^i \tanvecat[f]{y^j}{F(p)} \tanvec[F^j]{x^i}(p)
= \tanvec[F^j]{x^i} v^i \tanvecat{y^j}{F(p)} f
\end{align*}
Since $f$ is arbitrary, we are done.
For me it is (currently) important to write all steps out.
If you are more familiar with those calculations it is probably not
necessary to go down to the flat spaces but are are comfortable to
work with the manifolds as in $^n$.
\docEnd
...@@ -9,6 +9,7 @@ ...@@ -9,6 +9,7 @@
\newpage \newpage
\begin{appendices} \begin{appendices}
% \textinput{some-appendix-section} % \textinput{some-appendix-section}
\textinput{scratchpad/chainrule}
\end{appendices} \end{appendices}
\newpage \newpage
\textinput{declaration} \textinput{declaration}
......
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