Fix typo

Note for self: search at replace at 1.30 am is always a bad idea

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

Author: mseri

2-tangentbdl.tex

@@ -565,7 +565,7 @@ \section{The differential of a smooth map}\label{sec:diffsmooth}
 The construction that we employed forced us to fix a basis for the spaces, if this was 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 1]}]\label{ex:tg_curve_iso}
+\begin{exercise}[\textit{[homework 2]}]\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
@@ -1212,7 +1212,7 @@ \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 1]}]
+\begin{exercise}[\textit{[homework 2]}]
   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}[(a)]
@@ -1274,7 +1274,7 @@ \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 1]}]\label{exe:onsubmanifold}
+\begin{exercise}[\textit{[homework 2]}]\label{exe:onsubmanifold}
   Show that the orthogonal matrices $O(n) := \{ Q\in GL(n) \mid Q^TQ=\id \}$ form a $n(n-1)/2$-dimensional submanifold of the $n^2$-manifold $\mathrm{Mat}(n)$ of $n\times n$-matrices.
 
   Show also that

3-vectorfields.tex

@@ -29,7 +29,7 @@ \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 1]}]
+\begin{exercise}[\textit{[homework 2]}]
   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}
 
@@ -261,7 +261,7 @@ \section{Lie brackets}
         [X,Y] = \left(X^i\frac{\partial Y^j}{\partial x^i} - Y^i\frac{\partial X^j}{y^i}\right)\frac{\partial}{\partial x^j}.
     \end{equation}
 \end{proposition}
-\begin{exercise}[\textit{[homework 1]}]
+\begin{exercise}[\textit{[homework 2]}]
   Prove the proposition.
 \end{exercise}