Minor improvements and typos

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

1-manifolds.tex

@@ -608,7 +608,7 @@ \section{Smooth maps and differentiability}\label{sec:smoothfn}
 
 \begin{exercise}
   Prove the proposition.\\
-  \textit{\small Hint: go cyclic, for example show $(i)\Rightarrow(ii)$, $(ii)\Rightarrow(iii)$, $(iii)\Rightarrow(i)$.}
+  \textit{\small Hint: go cyclic, for example show $(i)\Rightarrow(iii)$, $(iii)\Rightarrow(ii)$, $(ii)\Rightarrow(i)$.}
 \end{exercise}
 
 At this point, the generalization of smooth functions to smooth maps between manifolds should not come as a surprise.
@@ -1084,9 +1084,9 @@ \section{Manifolds with boundary}\label{sec:mbnd}
   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.
-    \item Give a smooth structure to $M$ such that every smooth interior chart is a smooth chart for the standard smooth structure on $\mathring M$.\\
-      \textit{\small Hint: consider the map $\pi\circ\sigma^{-1}:\R^n\to\R^n$ where $\sigma:\bS^n\to\R^n$ is the stereographic projection from Exercise~\ref{ex:stereo} and $\pi:\R^{n+1}\to\R^n$ is a projection that omits one of the first $n$ coordinates.}
+    \item Give a smooth structure to $M$ such that every smooth interior chart is a smooth chart for the standard smooth structure on $\mathring M$.
   \end{enumerate}
+  \textit{\small Hint: consider the map $\pi\circ\sigma^{-1}:\R^n\to\R^n$ where $\sigma:\bS^n\to\R^n$ is the stereographic projection from Exercise~\ref{ex:stereo} and $\pi:\R^{n+1}\to\R^n$ is a projection that omits one of the first $n$ coordinates.}
 \end{exercise}
 
 \begin{tcolorbox}

2-tangentbdl.tex

@@ -1119,7 +1119,7 @@ \section{Submanifolds}
 \end{exercise}
 
 
-In fact, also analogues of the implicit functions theorem carry over.
+In fact, also analogues of the implicit function theorem carry over.
 We will state them without going into the details of the proofs.
 
 \begin{marginfigure}
@@ -1213,16 +1213,16 @@ \section{Submanifolds}
 \begin{proposition}\label{prop:submanifolds_and_R}
   The following assertions are equivalent.
   \begin{enumerate}[(i)]
-    \item $P^k\subset N^n$ is a $k$-dimensional submanifold\footnote{So, $k \leq n$.}.
-    \item $P$ is locally the image of an embedding of a subset of $\R^k$.
-    That is, for every $p\in P$ there exists $V\subset P$ open neighbourhood of $p$, an open set $U\subset\R^k$ and an embedding
+    \item $P^n\subset M^m$ is a $n$-dimensional submanifold\footnote{So, $n \leq m$.}.
+    \item $P$ is locally the image of an embedding of a subset of $\R^n$.
+    That is, for every $p\in P$ there exists $g\subset P$ open neighbourhood of $p$, an open set $U\subset\R^n$ and an embedding
     \begin{equation}
-      \varphi : U \to N \quad\mbox{such that}\quad \varphi(U)=V.
+      \varphi : U \to M \quad\mbox{such that}\quad \varphi(U)=V.
     \end{equation}
-    \item $P$ is locally a level set of a submersion into $\R^{n-k}$.
-    That is, for every $p\in P$ there exists $V\subset P$ open neighbourhood of $p$ and a submersion $\psi: V \to\R^{n-k}$ such that
+    \item $P$ is locally a level set of a submersion into $\R^{m-n}$.
+    That is, for every $p\in P$ there exists $V\subset P$ open neighbourhood of $p$ and a submersion $\psi: V \to\R^{m-n}$ such that
     \begin{equation}
-      N\cap V = \{q\in V \;\mid\; \psi(q) = 0\}.
+      M\cap V = \{q\in V \;\mid\; \psi(q) = 0\}.
     \end{equation}
   \end{enumerate}
 \end{proposition}

aom.tex

@@ -213,7 +213,7 @@
   \setlength{\parskip}{\baselineskip}
   Copyright \copyright\ \the\year\ \thanklessauthor
 
-  \par Version 0.20 -- \today
+  \par Version 0.21 -- \today
 
   \vfill
   \small{\doclicenseThis}