Update

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

Author: mseri

1-manifolds.tex

@@ -346,7 +346,7 @@ \section{Differentiable manifolds}
   For any open set $U\subseteq M$ and any point $p\in U$, prove the existence of a coordinate open set $U_\alpha$ such that $p\in U_\alpha\subset U$.
 \end{exercise}
 
-\begin{exercise}[\textit{[homework 1]}]
+\begin{exercise}
   Let $f: \R^n \to \R^m$ be a smooth map.
   Show that its graph
   \begin{equation}
@@ -523,7 +523,7 @@ \section{Differentiable manifolds}
   \textit{\small Hint: use Exercise~\ref{exe:RPSN}.}
 \end{exercise}
 
-\begin{exercise}[Stereographic projections \textit{[homework 1]}]\label{ex:stereo}
+\begin{exercise}[Stereographic projections]\label{ex:stereo}
   Let $N$ denote the north pole $(0,\ldots,0,1)\in\bS^n\subset\R^{n+1}$ and let $S$ denote the south pole $(0,\ldots,0,-1)$.
   Define the \emph{stereographic projections} $\sigma:\bS^n\setminus\{N\}\to \R^n$ by
   \begin{marginfigure}
@@ -661,7 +661,7 @@ \section{Smooth maps and differentiability}
   \end{proposition}
 \end{exercise}
 
-\begin{exercise}[\textit{[homework 1]}]
+\begin{exercise}
   Prove that $\R^2\setminus\{(0,0)\}$ is a two-dimensional manifold and construct a diffeomorphism from this manifold to the circular cylinder
   \begin{equation}
     C := \{ (x,y,z)\in\R^3 \mid x^2+y^2 = 1\}\subset\R^3.
@@ -699,7 +699,7 @@ \section{Smooth maps and differentiability}
   Show that there exists a diffeomorphism between the smooth structures $(\R, \id_\R)$ and $(\R, \psi)$ from the previous example.
 \end{exercise}
 
-\begin{exercise}[\textit{[homework 1]}]
+\begin{exercise}
   For $r>0$, let $\phi_r:\R\to\R$ be the map given by
   \begin{equation}
     \phi_r(t) := \begin{cases}
@@ -838,8 +838,7 @@ \section{Partitions of unity}
 
 \section{Manifolds with boundary}\label{sec:mbnd}
 
-\newthought{The definition of manifolds has a serious limitation}, even though it is perfectly good to describe curves\footnote{E.g. the circle seen in Example~\ref{ex:S1emb}.} and surfaces\footnote{E.g. the $
-2$-spheres $\bS^2$ from the homework sheet.}, it fails to describe many natural objects like a \emph{closed} interval $[a,b]\in\R$ or the \emph{closed} disk $D_1(0)$ of Example~\ref{ex:uball}.
+\newthought{The definition of manifolds has a serious limitation}, even though it is perfectly good to describe curves\footnote{E.g. the circle seen in Example~\ref{ex:S1emb}.} and surfaces\footnote{E.g. the $2$-spheres $\bS^2$.}, it fails to describe many natural objects like a \emph{closed} interval $[a,b]\in\R$ or the \emph{closed} disk $D_1(0)$ of Example~\ref{ex:uball}.
 Note that in each of these cases, both the interior and the boundary are smooth manifolds and their dimension differ by one\footnote{In the first case the interior $(a,b)$ is a $1$-manifold and the boundary, the set $\partial[a,b] = \{a,b\}$, is a $0$-manifold. In the second case the interior of $D_1(0)$ is the open unit ball, a $2$-manifold, and the boundary $\partial D_1(0)$ is the $1$-manifold $\bS^1$.}.
 
 Let's do a step back and think about topological manifolds: since both the closed interval and the closed disk are closed sets, we have problems to make them locally euclidean in neighbourhoods of their boundaries.
@@ -1008,7 +1007,7 @@ \section{Manifolds with boundary}\label{sec:mbnd}
   Why is $\varphi_1$ not appearing in $\partial \cC$?
 \end{exercise}
 
-\begin{exercise}[\textit{[homework 1]}]
+\begin{exercise}
   Let $M = D_1\subset \R^n$ be the $n$-dimensional closed unit ball from Example~\ref{ex:uball}.
   \begin{enumerate}
     \item Show that $M$ is a topological manifold with boundary in which each point of $\partial M = \bS^{n-1}$ is a boundary point and each point in $\mathring M = \{x\in\R^n\mid\|x\|<1\}$ is an interior point.
@@ -1020,4 +1019,4 @@ \section{Manifolds with boundary}\label{sec:mbnd}
 \begin{tcolorbox}
   Differentiable manifolds without boundary (cf. Definition~\ref{def:diffmanifold}) can be thought as a special case of differentiable manifolds with boundary (cf. Definition~\ref{def:diffmanifoldwb}) where the boundary happens to be empty.
   Therefore, with the exception of the beginning of Chapter~\ref{ch:2}, we will no-longer distinguish the two concepts: from now on, a manifold may have or may not have a boundary.
-\end{tcolorbox}
+\end{tcolorbox}

2-tangentbdl.tex

@@ -574,7 +574,7 @@ \section{The differential of a smooth map}\label{sec:diffsmooth}
 \newthought{An aspect of the construction above is particularly disturbing}: it forced us to fix a basis on the spaces; if this were truly necessary it would defeat the purpose of this whole chapter.
 Fortunately for us, the following exercise shows that, at any given point, the tangent space to a vector space is \emph{canonically}\footnote{That is, independently of the choice of basis.} identified with the vector space itself.
 
-\begin{exercise}[\textit{[homework 2]}]\label{ex:tg_curve_iso}
+\begin{exercise}\label{ex:tg_curve_iso}
   Let $V$ and $W$ be finite-dimensional vector spaces, endowed with their standard smooth structure (see Exercise~\ref{exe:subsetsmanifolds}).
   \begin{enumerate}
     \item Fix $a\in V$. For any vector $v\in V$ define a map $\cT_a(v) : C^\infty(V) \to \R$ by
@@ -1268,7 +1268,7 @@ \section{Submanifolds}
   Show that $g$ is a smooth embedding and, therefore, that $g(U)$ is a smooth embedded $n$-dimensional submanifold\footnote{$g(U)$ is the the \emph{graph} of $f$!} of $\R^{n+1}$.
 \end{exercise}
 
-\begin{exercise}[\textit{[homework 2]}]\label{exe:onsubmanifold}
+\begin{exercise}\label{exe:onsubmanifold}
   Show that the orthogonal matrices
   \begin{equation}
     O(n) := O(n, \R) = \{ Q\in \mathrm{Mat}(n, \R) \mid Q^TQ=\id \}
@@ -1293,7 +1293,7 @@ \section{Submanifolds}
   If for all $p\in M$, the intersection $F_p := F\cap E_p$ is a $k$-dimensional subspace of the vector space $E_p$ and $\pi|_F : F \to M$ defines a rank-$k$ vector bundle, then $\pi|_F: F \to M$ is called a \emph{subbundle} of $E$.
 \end{definition}
 
-\begin{exercise}[\textit{[homework 2]}]
+\begin{exercise}
   Let $M$ be a smooth $m$-manifold and $N$ a smooth $n$-manifold.
   Let $F:M\to N$ be an embedding and denote $\widetilde M = F(M)\subset N$.
   \begin{enumerate}

3-vectorfields.tex

@@ -30,7 +30,7 @@ \section{Vector fields}
 \end{equation}
 This defines $n$ functions $X^i: U\to\R$, called \emph{component functions of $X$} in the chart.
 
-\begin{exercise}[\textit{[homework 2]}]
+\begin{exercise}
   Show that, in the notation above, the restriction of $X$ to $U$ is smooth if and only if its component functions with respect to the chart are smooth.
 \end{exercise}
 
@@ -273,7 +273,7 @@ \section{Lie brackets}
         [X,Y] = \left(X^i\frac{\partial Y^j}{\partial x^i} - Y^i\frac{\partial X^j}{\partial x^i}\right)\frac{\partial}{\partial x^j}.
     \end{equation}
 \end{proposition}
-\begin{exercise}[\textit{[homework 2]}]
+\begin{exercise}
   Prove the proposition.
 \end{exercise}
 

3b-liegroups.tex

@@ -301,7 +301,7 @@ \section{Lie algebras}
   Let $G$ be a Lie group and $v\in\fX_L(G)$.
   Then $v$ is complete.
 \end{proposition}
-\begin{exercise}\textit{[homework 3]}
+\begin{exercise}
   Prove the proposition.\\  
   \textit{\small Hint: extend a curve starting at $e$ to a curve starting at $g$.}
 \end{exercise}
@@ -650,4 +650,4 @@ \section{The exponential map}
   \end{enumerate}
 \end{exercise}
 
-We will come back to discuss Lie groups and Lie algebras in the appendix on distribution theory and Frobenius theorem.
+We will come back to discuss Lie groups and Lie algebras in the appendix on distribution theory and Frobenius theorem.

4-cotangentbdl.tex

@@ -260,7 +260,7 @@ \section{One-forms and the cotangent bundle}
   %   \left(\varphi_i, (\varphi_i^{-1})^*\right) : T^* U_i &\to T^*\varphi(U_i),\\
   %   (p, \omega) &\mapsto \left(\varphi_i(p), d(\varphi_i^{-1})^*)_p\omega \right).
   % \end{align}
-\begin{exercise}\label{exe:prooftstarmbld}\textit{[homework 3]}
+\begin{exercise}\label{exe:prooftstarmbld}
   Mimicking what we did for Theorem~\ref{thm:tgbdlsmoothmfld}, complete the proof of Theorem~\ref{thm:starmbld}.
 \end{exercise}
 %\end{proof}
@@ -320,7 +320,7 @@ \section{One-forms and the cotangent bundle}
   \end{equation}
 \end{definition}
 
-\begin{exercise}[\textit{[homework 3]}]\label{ex:propdiff}
+\begin{exercise}\label{ex:propdiff}
   Let $F:M\to N$ be a smooth map between smooth manifolds.
   Suppose $f$ is a continuous real valued function on $N$ and $\omega\in\fX^*(N)$ is a covector field on $N$.
   \begin{enumerate}
@@ -474,7 +474,7 @@ \section{Line integrals}
   \end{align} 
 \end{example}
 
-\begin{exercise}[\textit{[homework 3]}]\label{exe:FTC}
+\begin{exercise}\label{exe:FTC}
   Let $M$ be a smooth manifold, $\gamma: I = [a,b]\subset \R \to M$ a smooth curve and $\omega\in\fX^*(M)$ a $1$-form.
   Show the following properties.
   \begin{enumerate}

5-tensors.tex

@@ -453,7 +453,7 @@ \section{Tensor bundles}
   An interesting, not really surprising though (right?), property is the following: $F_* df = d(F_* f)$.
 \end{example}
 
-\begin{exercise}[\textit{[homework 3]}]
+\begin{exercise}
   Let $F:M\to N$ and $G:N\to P$ two diffeomorphisms of smooth manifolds.
   \begin{enumerate}
     \item Show that the chain rule $(G\circ F)_* = G_* \circ F_*$ holds.

6-differentiaforms.tex

@@ -249,7 +249,7 @@ \section{The interior product}
     \end{equation}
   \end{enumerate}
 \end{lemma}
-\begin{exercise}[\textit{[homework 4]}]
+\begin{exercise}
   Prove the Lemma.
 \end{exercise}
 
@@ -299,7 +299,7 @@ \section{Differential forms on manifolds}
     F^*\left(\omega_J dx^J\right) = (\omega_{j_1,\ldots, j_k}\circ F) d(x^{j_1}\circ F)\wedge\cdots\wedge d(x^{j_k}\circ F).
   \end{equation}
 \end{theorem}
-\begin{exercise}%[\textit{[homework 4]}]
+\begin{exercise}
   Prove the theorem.
 \end{exercise}
 
@@ -351,7 +351,7 @@ \section{Differential forms on manifolds}
   where $DF$ represents the Jacobian matrix of $F$ in these coordinates.
 \end{proposition}
 
-\begin{exercise}\textit{[homework 4]}
+\begin{exercise}
   Prove the Proposition~\ref{prop:wedgeToJDet}.\\
   \textit{\small Hint: look at Theorem~\ref{thm:pullbacksdifferentialforms}.}
 \end{exercise}
@@ -744,7 +744,7 @@ \section{De Rham cohomology and Poincar\'e lemma}
 This is the statement of the so-called Poincar\'e lemma.
 We are going to prove it in two slightly different flavours: its classical version and a slight generalization.
 
-\begin{exercise}[\textit{[homework 4]}]
+\begin{exercise}
   Let $M=\R^2\setminus\{0\}$, $\{x,y\}$ denote the standard euclidean coordinates in $\R^2$ and $\omega$ be the one-form on $M$ from Example~\ref{ex:li} given by
   \begin{equation}
     \omega = \frac{xdy - ydx}{x^2+y^2}.
@@ -934,4 +934,4 @@ \section{De Rham cohomology and Poincar\'e lemma}
 \end{corollary}
 \begin{proof}
 Every point in a $n$-manifold has a neighbourhood which is homeomorphic to $\R^n$ and so is contractible.
-\end{proof}
+\end{proof}

7-integration.tex

@@ -198,7 +198,7 @@ \section{Orientation}
   %In particular, the unit $n$-sphere $\bS^n\subset\R^{n+1}$ is orientable.
 \end{exercise}
 
-\begin{exercise}[\textit{[homework 4]}]\label{ex:TMorient}
+\begin{exercise}\label{ex:TMorient}
   Let $M$ be a smooth manifold without boundary and $\pi: TM \to M$ its tangent bundle.
   Show that if $\{(U_\alpha,\varphi_\alpha)\}$ is any atlas on $M$, then the corresponding\footnote{Remember Theorem~\ref{thm:tgbdlsmoothmfld}.} atlas $\{(TU_\alpha, \widetilde\varphi_\alpha)\}$ on $TM$ is oriented.
   This, in particular, proves that the total space $TM$ of the tangent bundle is always orientable, regardless of the orientability of $M$.
@@ -425,7 +425,7 @@ \section{Integrals on manifolds}
   \end{equation}
 \end{example}
 
-\begin{exercise}[Fubini's theorem \textit{[homework 4]}]\label{exe:fubini}
+\begin{exercise}[Fubini's theorem]\label{exe:fubini}
   Let $M^m$ and $N^n$ be oriented manifolds. Endow $M\times N$ with the product orientation, that is\footnote{An equivalent way is to say that if $v_1,\ldots,v_m\in T_pM$ and $w_1, \ldots, w_n\in T_q N$ are positively oriented bases in the respective spaces, then \begin{equation}
     (v_1,0),\ldots,(v_n,0),(0,w_1), \ldots, (0,w_n)\in T_{(p,q)}(M\times N)
   \end{equation}is defined to be a positively oriented basis in the product.}, if $\pi_M : M\times N \to M$ and $\pi_N: M\times N \to N$ are the canonical projections on the elements of the product, and $\omega$ and $\eta$ respectively define orientations on $M$ and $N$, then the orientation on $M\times N$ is defined to be the orientation defined by $\pi_M^* \omega \wedge \pi_N^* \eta$.
@@ -736,4 +736,4 @@ \section{Stokes' Theorem}
   \item Prove Theorem~\ref{thm:bfp2}.\\\textit{Hint: assume there is $f$ such that $f(p)=p$ for all $p\in \partial N$ and use integration to get a contradiction.}
   \item Prove Theorem~\ref{thm:bfp1}.\\\textit{Hint: by contradiction, use the half line from $p$ to $g(p)$ to construct a function for which every point in the boundary is fixed. }
 \end{enumerate}
-\end{exercise}
+\end{exercise}

aom.tex

@@ -192,7 +192,7 @@
 }
 \author{Marcello Seri}
 \publisher{Bernoulli Institute\\ \noindent
-A.Y. 2020--2021\\ \noindent 
+A.Y. 2021--2022\\ \noindent 
 \MakeLowercase{\texttt{m.seri@rug.nl}}
 }
 
@@ -208,7 +208,7 @@
     \setlength{\parskip}{\baselineskip}
     Copyright \copyright\ \the\year\ \thanklessauthor
 
-    \par Version 0.11 -- \today
+    \par Version 0.12 -- \today
 
     \vfill
     \small{\doclicenseThis}
@@ -222,7 +222,7 @@
 \chapter*{Introduction}
 \addcontentsline{toc}{chapter}{Introduction}
 
-At the entry for \emph{Mathematical analysis}, our modern source of truth -- Wikipedia -- says
+At the entry for \emph{Mathematical analysis}, our modern source of truth---Wikipedia---says
 
 \begin{quotation}
   \emph{Mathematical analysis} is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.
@@ -347,4 +347,4 @@ \chapter{Vector bundles and connections}
 
 \printbibliography
 \addcontentsline{toc}{chapter}{Bibliography}
-\end{document}
+\end{document}