Fix typos pointed out from students:

Signed-off-by: Marcello Seri <[email protected]>

1-manifolds.tex

@@ -71,7 +71,7 @@ \section{Topological manifolds}\label{sec:top_manifolds}
   A map $f: X \to Y$ between two topological spaces $(X,\cT)$ and $(Y, \cU)$ is called:
   \begin{itemize}
     \item \emph{continuous} if $U\in\cU$ implies that $f^{-1}(U)\in\cT$, that is, preimages of open sets under $f$ are open;
-    \item \emph{homeomorphism} if it is bijective\footnote{I.e., a one to one correspondence. Formally it means that it is both injective and surjective.} and continuous with continuous inverse.\marginnote{The existence of a homeomorphism between two spaces can be thought as those spaces being equivalent in a loose sense: they can be deformed continuously into each other.}
+    \item  a \emph{homeomorphism} if it is bijective\footnote{I.e., a one to one correspondence. Formally it means that it is both injective and surjective.} and continuous with continuous inverse.\marginnote{The existence of a homeomorphism between two spaces can be thought as those spaces being equivalent in a loose sense: they can be deformed continuously into each other.}
   \end{itemize}
 \end{definition}
 
@@ -195,15 +195,15 @@ \section{Differentiable manifolds}
   Then, the following holds.
   \begin{enumerate}[(i)]
     \item\label{thm:chainrule1} The function $g\circ f: U\subset\R^n \to\R^m$ is continuously differentiable and its total derivative~\eqref{eq:jacobian} at a point $x\in U$ is given by
-    \begin{equation}
-      D(g\circ f)(x) = (Dg)(f(x)) \circ Df(x).
-    \end{equation}
+          \begin{equation}
+            D(g\circ f)(x) = (Dg)(f(x)) \circ Df(x).
+          \end{equation}
     \item\label{thm:chainrule2} Denote $x=(x^1, \ldots, x^n)\in\R^n$ and $y=(y^1,\ldots,y^k)\in\R^k$ the coordinates on the respective euclidean spaces and $f=(f^1,\ldots,f^k)$ and $g=(g^1,\ldots,g^m)$ the components of the functions. Then the partial derivatives of $g\circ f$ are given by\footnote{Using Einstein's notation, this could be written as \begin{equation}\frac{\partial (g^i\circ f)}{\partial x^j}(x) = \frac{\partial g^i}{\partial y^r}(f(x)) \frac{\partial f^r}{\partial x^j}(x).\end{equation}}
-    \begin{equation}
-      \frac{\partial g^i\circ f}{\partial x^j}(x)
-      = \sum_{r=1}^k \frac{\partial g^i}{\partial y^r}(f(x)) \frac{\partial f^r}{\partial x^j}(x),
-      \qquad 1\leq i \leq m,\; 1\leq j\leq n.
-    \end{equation}
+          \begin{equation}
+            \frac{\partial g^i\circ f}{\partial x^j}(x)
+            = \sum_{r=1}^k \frac{\partial g^i}{\partial y^r}(f(x)) \frac{\partial f^r}{\partial x^j}(x),
+            \qquad 1\leq i \leq m,\; 1\leq j\leq n.
+          \end{equation}
   \end{enumerate}
 \end{theorem}
 
@@ -281,7 +281,7 @@ \section{Differentiable manifolds}
 \end{notation}
 
 \begin{exercise}
-	Let $\{(U_\alpha, \varphi_\alpha)\}$ be the maximal atlas on a manifold $M$.
+  Let $\{(U_\alpha, \varphi_\alpha)\}$ be the maximal atlas on a manifold $M$.
   For any open set $U\subseteq M$ and any point $p\in U$, prove the existence of a coordinate open set\footnote{Cf. Notation~\ref{def:coords_obj}.} $U_\alpha$ such that $p\in U_\alpha\subset U$.
 \end{exercise}
 
@@ -357,8 +357,6 @@ \section{Differentiable manifolds}
   In the previous example, show that the corresponding transition functions are smooth.
 \end{exercise}
 
-
-
 \begin{exercise}
   Let $f: \R^n \to \R^m$ be a smooth map.
   Show that its graph
@@ -391,7 +389,6 @@ \section{Differentiable manifolds}
   Given two manifolds $(M_1, \cA_1)$ and $(M_2, \cA_2)$, we can define the \emph{product manifold} $M_1 \times M_2$ by equipping $M_1 \times M_2$ with the product topology\footnote{Open sets in the product are generated by products of open sets from the respective topological spaces.} and covering the space with the atlas $\{ (U_1\times U_2, (\varphi_1, \varphi_2)) \;\mid\; (U_1, \varphi_1)\in\cA_1, (U_2, \varphi_2)\in \cA_2\}$.
 \end{example}
 
-
 Instead of always constructing a topological manifold and then specify a smooth structure, it is often convenient to combine these steps into a single construction.
 This is especially useful when the initial set is not equipped with a topology.
 In this respect, the following lemma provides a welcome shortcut: in brief it says that given a set with suitable ``charts'' that overlap smoothly, we can use those to define both a topology and a smooth structure on the set.
@@ -466,7 +463,7 @@ \subsection{Quotient manifolds}\label{sec:quotient}
 Before moving on, below we are going to look at a couple of simpler, notable, examples of quotient manifolds.
 
 \begin{example}
-  Let $\RP^n$ denote the $n$-dimensional real projective space, that is, the space of lines in $\R^{n+1}$ passing through the origin.
+  Let $\RP^n$ denote the $n$-dimensional \emph{real projective space}, that is, the space of lines in $\R^{n+1}$ passing through the origin.
   This is a notable example of quotient manifold: we are going to show that $\RP^n$ is a smooth manifold of dimension $n$.
 
   We can define an equivalence relation on $\R^{n+1}_0:=\R^{n+1}\setminus\{0\}$ by declaring that for any $x,y\in \R^{n+1}_0$
@@ -514,17 +511,17 @@ \subsection{Quotient manifolds}\label{sec:quotient}
   This map is well--defined because its value is unchanged by multiplying $x$ by a non-zero constant.
   Moreover, $\varphi_i$ is continuous and invertible, the inverses can be computed explicitly\footnote{Check that the composition $\varphi_i \circ \varphi_i^{-1} = \id$. What happens if you define $\widetilde\varphi^{-1}_i(y^1, \ldots, y^n) := \left[y^1, \ldots, y^{i-1}, 42, y^{i+1}, \ldots, y^n\right]$, what would the corresponding $\widetilde\varphi_i$ be?} as
   \begin{equation}
-    \varphi_i^{-1}(y^1,\ldots,y^n) = \left[y^1, \ldots, y^{i-1}, 1, y^{i+1}, \ldots, y^n\right].
+    \varphi_i^{-1}(y^1,\ldots,y^n) = \left[y^1, \ldots, y^{i}, 1, y^{i+1}, \ldots, y^n\right].
   \end{equation}
-  Since $\{U_0, \ldots, U_n\}$ is an open covering of $\RP^n$, this shows tht $\RP^n$ is locally euclidean of dimension $n$.
+  Since $\{U_0, \ldots, U_n\}$ is an open covering of $\RP^n$, this shows that $\RP^n$ is locally euclidean of dimension $n$.
 
   \newthought{Let's equip $\RP^n$ with a smooth structure}.
   We are already half-way through: we are going to show that the coordinate charts $(U_i, \varphi_i)$ defined above are, in fact, all smoothly compatible.
   Without loss of generality, let's assume $i>j$.
   Then, a brief computation shows
   \begin{align}
-    \varphi_j\circ\varphi_i^{-1} & (y^1, \ldots, y^n)                                                                                                                                                 \\
-                                 & = \left(\frac{y^1}{y^j},\ldots,\frac{y^{j-1}}{y^j},\frac{y^{j+1}}{y^j},\ldots,\frac{y^{i-1}}{y^j},\frac1{y^j},\frac{y^{i+1}}{y^j}, \ldots, \frac{y^n}{y^j}\right),
+    \varphi_j\circ\varphi_i^{-1} & (y^1, \ldots, y^n)                                                                                                                                             \\
+                                 & = \left(\frac{y^1}{y^j},\ldots,\frac{y^{j-1}}{y^j},\frac{y^{j+1}}{y^j},\ldots,\frac{y^i}{y^j},\frac1{y^j},\frac{y^{i+1}}{y^j}, \ldots, \frac{y^n}{y^j}\right),
   \end{align}
   which is a diffeomorphism from $\varphi_i(U_i\cap U_j)$ to $\varphi_j(U_i\cap U_j)$ since $x^j\neq 0$ on $U_j$.
   The atlas defined by the collection $\{(U_i, \varphi_i)\}$ is called \emph{standard atlas} and makes $\RP^n$ a smooth manifold.
@@ -682,7 +679,6 @@ \section{Smooth maps and differentiability}\label{sec:smoothfn}
   \end{enumerate}
 \end{exercise}
 
-
 The following corollary is just a restatement of Proposition~\ref{prop:smoothlocal}, but provides a useful perspective on the construction of smooth maps.
 
 \begin{proposition}[Gluing lemma for smooth maps]
@@ -889,7 +885,7 @@ \section{Partitions of unity}\label{sec:partition_of_unity}
   Let $M$ be a smooth manifold. A \emph{partition of unity} is a collection $\{\rho_\alpha \mid \alpha\in A\}$ of functions $\rho_\alpha:M\to\R$ such that
   \begin{enumerate}[(i)]
     \item $0 \leq \rho_\alpha \leq 1$ for all $p\in M$ and $\alpha\in A$;
-          \item\label{def:pou.2} the collection $\{\rho_\alpha \mid \alpha\in A\}$ is \emph{locally finite}, that is, for any $p\in M$ there are at most finitely many $\alpha\in A$ such that $p\in\supp(\rho_\alpha)$;
+    \item\label{def:pou.2} the collection $\{\rho_\alpha \mid \alpha\in A\}$ is \emph{locally finite}, that is, for any $p\in M$ there are at most finitely many $\alpha\in A$ such that $p\in\supp(\rho_\alpha)$;
     \item for all $p\in M$ one has $\sum_{\alpha\in A} \rho_\alpha(p) = 1$.
           \marginnote{For any $p$, $\sum_{\alpha\in A} \rho_\alpha(p)$ is a finite sum by~\ref{def:pou.2}. Thus, the function defined by the sum $\rho := \sum \rho_\alpha$ is a well define smooth function on $M$. We call such sum a \emph{locally finite} sum.}
   \end{enumerate}
@@ -1011,7 +1007,7 @@ \section{Manifolds with boundary}\label{sec:mbnd}
 
 \begin{proof}[Alternative proof of Proposition~\ref{prop:bdwelldef}]
   Let $p\in M$ be an interior point, that is, there exists $\varphi: U\subset M \to \cH^n$, $p\in U$, such that $x^n(p) = r^n\circ\varphi(p) > 0$.
-  
+
   Consider a different chart $\psi:V\subset M\to \cH^n$, $p\in V$.
   We want to show that $\psi^n(p) > 0$. Since
   \begin{equation}\label{eq:trans-int-chart}
@@ -1097,8 +1093,8 @@ \section{Manifolds with boundary}\label{sec:mbnd}
   \includegraphics{1_5_9-cyl1}
 \end{marginfigure}
 \begin{marginfigure}
-    \includegraphics{1_5_9-cyl2}
-    \caption{Compare $\varphi$ with the stereographic projections from Exercise~\ref{ex:stereo}. Do you notice any similarity?}
+  \includegraphics{1_5_9-cyl2}
+  \caption{Compare $\varphi$ with the stereographic projections from Exercise~\ref{ex:stereo}. Do you notice any similarity?}
 \end{marginfigure}
 
 \begin{example}
@@ -1122,7 +1118,7 @@ \section{Manifolds with boundary}\label{sec:mbnd}
           then $x = \varphi_2(p) = (\sigma\circ\psi)(p)$ and $\varphi_2(U_2) = \R\times[0,1/2) \subset\cH^2$.
     \item $U_3 = \{p\in\cC\mid 1/2 < p^3 \leq 1, \; (p^1, p^2) \neq (0, -p^3) \}$ and $\varphi_3$ defined similarly as in the previous point.
   \end{itemize}
-    The boundary is given by $\partial\cC = \varphi_2^{-1}(\R\times\{0\}) \cup \varphi_3^{-1}(\R\times\{0\})$.
+  The boundary is given by $\partial\cC = \varphi_2^{-1}(\R\times\{0\}) \cup \varphi_3^{-1}(\R\times\{0\})$.
 \end{example}
 
 \begin{exercise}

aom.tex

@@ -3,7 +3,7 @@
 \setcounter{secnumdepth}{3}
 \setcounter{tocdepth}{2}
 \usepackage{microtype, ifluatex, ifxetex}
-%Next block avoids bug, from  http://tex.stackexchange.com/a/200725/1913 
+%Next block avoids bug, from  http://tex.stackexchange.com/a/200725/1913
 \ifx\ifxetex\ifluatex\else % if lua- or xelatex http://tex.stackexchange.com/a/140164/1913
   \usepackage{fontspec}
   \setmainfont[Renderer=Basic, Scale=0.90]{OpenDyslexic}
@@ -40,15 +40,15 @@
 \usepackage{tikz,tikz-cd,quiver,bbm,mathbbol}
 \DeclareSymbolFontAlphabet{\mathbbl}{bbold} %let's you use \mathbbl{k} for a field k
 \hypersetup{colorlinks} %puts color to hyperlinks
-\setcounter{secnumdepth}{2} 
+\setcounter{secnumdepth}{2}
 \usepackage{enumerate}
 \usepackage{mathabx}
 \usepackage{mathtools}
 \mathtoolsset{showonlyrefs, mathic}
 \usepackage[english]{babel}
 \usepackage{hyperref}
 \hypersetup{
-  colorlinks=true,    
+  colorlinks=true,
   urlcolor=Cerulean,
   linkcolor = ForestGreen,
 }
@@ -198,7 +198,7 @@
 }
 \author{Marcello Seri}
 \publisher{Bernoulli Institute\\ \noindent
-  %A.Y. 2022--2023\\ \noindent 
+  %A.Y. 2022--2023\\ \noindent
   \MakeLowercase{\texttt{[email protected]}}
 }
 
@@ -249,7 +249,7 @@ \chapter*{Introduction}
 Some of these topics are already present in appendices to the notes, other will be progressively added in due course.
 Throughout the course and these notes, I will try to give particular emphasis on the usefulness of these topics in the mathematics of mechanics and their relevance in certain aspects of topology and field theory.
 
-The course relies \emph{heavily} on your knowledge of linear and multilinear algebra, multivariable analysis and dynamical systems. 
+The course relies \emph{heavily} on your knowledge of linear and multilinear algebra, multivariable analysis and dynamical systems.
 This should not come as a surprise: differential geometry studies the natural space in which analysis, in the sense of derivation and integration, can be performed, and was born together with classical mechanics, somehow as unique discipline, before these started diverging on their own paths.
 
 An old mathematical joke says that
@@ -271,11 +271,11 @@ \chapter*{Introduction}
 
 The idea for the cut that I want to give to this course was inspired by the online \href{https://www.video.uni-erlangen.de/course/id/242}{Lectures on the Geometric Anatomy of Theoretical Physics} by Frederic Schuller, by the lecture notes of Stefan Teufel's Classical Mechanics course~\cite{lectures:teufel} (in German), by the classical mechanics book by Arnold~\cite{book:arnold} and by the Analysis of Manifolds chapter in~\cite{book:thirring}.
 In some sense I would like this course to provide the introduction to geometric analysis that I wish was there when I prepared my \href{https://www.mseri.me/lecture-notes-hamiltonian-mechanics/}{first edition} of the Hamiltonian mechanics course (see also my lecture notes for that course \cite{lectures:seri:hm}).
-In addition to the reference above, these lecture notes have found deep inspiration from~\cite{lectures:merry,lectures:hitchin} (all freely downloadable from the respective authors' websites), and from the book~\cite{book:abrahammarsdenratiu}.
+In addition to the reference above, these lecture notes have found deep inspiration from~\cite{lectures:merry,lectures:hitchin} (all freely downloadable from the respective authors' websites), and from the book~\cite{book:abrahammarsdenratiu}.\medskip
 
-I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his invaluable suggestions, comments and corrections, and to Bram Brongers\footnote{You can also have a look at \href{https://fse.studenttheses.ub.rug.nl/25344/}{his bachelor thesis} to learn more about some interesting advanced topics in differential geometry.} for his comments, corrections and the appendices that he contributed to these notes.
+I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his invaluable suggestions, comments and corrections, and to Bram Brongers\footnote{You can also have a look at \href{https://fse.studenttheses.ub.rug.nl/25344/}{his bachelor thesis} to learn more about some interesting advanced topics in differential geometry.} for his comments, corrections and the appendices that he contributed to these notes.\medskip
 
-Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Daniel Cortlid, Harry Crane, Anna de Bruijn, Remko de Jong, Luuk de Ridder, Brian Elsinga, Mollie Jagoe Brown, Aron Karakai, Wietze Koops, Henrieke Krijgsheld, Valeriy Malikov, Mar\'ia Diaz Marrero, Levi Moes, Nicol\'as Moro, Magnus Petz, Jorian Pruim, Lisanne Sibma, Bo Tielman, Jesse van der Zeijden, Jordan van Ekelenburg, Hanneke van Harten, Martin Daan van IJcken, Marit van Straaten, Dave Verweg and Federico Zadra.
+Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Jamara Admiraal, Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Daniel Cortlid, Harry Crane, Anna de Bruijn, Remko de Jong, Luuk de Ridder, Brian Elsinga, Mollie Jagoe Brown, Aron Karakai, Wietze Koops, Henrieke Krijgsheld, Valeriy Malikov, Mar\'ia Diaz Marrero, Levi Moes, Nicol\'as Moro, Jard Nijholt, Magnus Petz, Jorian Pruim, Lisanne Sibma, Bo Tielman, Jesse van der Zeijden, Jordan van Ekelenburg, Hanneke van Harten, Martin Daan van IJcken, Marit van Straaten, Dave Verweg and Federico Zadra.
 
 \mainmatter