Merge pull request #6 from Martijnkluitenberg/master

Fixed some typos

1-manifolds.tex

@@ -280,6 +280,11 @@ \section{Differentiable manifolds}
   We may write $M^n$ when we want to emphasize that the dimension of $M$ is $n$.
 \end{notation}
 
+\begin{exercise}
+	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 $U_\alpha$ such that $p\in U_\alpha\subset U$.
+\end{exercise}
+
 \begin{exercise}
   Show that on a second countable differentiable manifold it is always possible to find a countable atlas.
 \end{exercise}
@@ -352,10 +357,7 @@ \section{Differentiable manifolds}
   In the previous example, show that the corresponding transition functions are smooth.
 \end{exercise}
 
-\begin{exercise}
-  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 $U_\alpha$ such that $p\in U_\alpha\subset U$.
-\end{exercise}
+
 
 \begin{exercise}
   Let $f: \R^n \to \R^m$ be a smooth map.
@@ -767,7 +769,7 @@ \section{Smooth maps and differentiability}\label{sec:smoothfn}
   \begin{enumerate}
     \item $F$ is smooth;
     \item $N$ has an atlas $\cA$ such that for all the charts $(V, \psi) = (V, y^1, \ldots, y^n)\in \cA$, the components $y^i \circ F: F^{-1}(V) \to \R$ of $F$ relative to the chart are all smooth;
-    \item for every chat $(V, \psi) = (V, y^1, \ldots, y^n)$ on $N$, the components $y^i \circ F: F^{-1}(V) \to \R$ of $F$ relative to the chart are all smooth.
+    \item for every chart $(V, \psi) = (V, y^1, \ldots, y^n)$ on $N$, the components $y^i \circ F: F^{-1}(V) \to \R$ of $F$ relative to the chart are all smooth.
   \end{enumerate}
   Note that this, in particular, holds for $N=\R^n$.
 \end{exercise}

2-tangentbdl.tex

@@ -5,7 +5,7 @@
 
 \section{Let the fun begin!}
 \newthought{It now remains to define derivatives} of functions between smooth manifolds.
-And, since we saw that euclidean spaces are manifolds, we must mke sure that our definition coincides with the usual one in euclidean spaces.
+And, since we saw that euclidean spaces are manifolds, we must make sure that our definition coincides with the usual one in euclidean spaces.
 
 \begin{marginfigure}[7em]
   \includegraphics{2_1-embedded-sphere-tangent.pdf}
@@ -600,7 +600,7 @@ \section{The differential of a smooth map}\label{sec:diffsmooth}
     & T_{F(x)}N
   \end{tikzcd},
 \end{equation}
-where $T_x$ and $T_{F(p)}$ are defined as above.
+where $T_x$ and $T_{F(x)}$ are defined as above.
 
 \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.

2b-submanifolds.tex

@@ -76,7 +76,7 @@
   We say that $M$ is an \emph{embedded (or regular) submanifold} of $N$ if the inclusion $M\hookrightarrow N$ is an embedding. If the inclusion is just an immersion, we say that $M$ is an \emph{immersed submanifold}.
 \end{definition}
 
-This definition already hints to the fact that smooth maps are going to be usefulin providing ways to nicely include a manifold into the another and in giving new ways to construct manifolds in the first place.
+This definition already hints to the fact that smooth maps are going to be useful in providing ways to nicely include a manifold into the another and in giving new ways to construct manifolds in the first place.
 In the rest of this chapter we will try to give an answer to the following questions:
 \begin{itemize}
   \item if $F$ is an immersion, what can we say about its image $F(M)$ as a subset of $N$?

2c-vectorbdl.tex

@@ -163,7 +163,7 @@
   \end{enumerate}
 \end{exercise}
 
-There are various useful generalization of vector bundles.
-The \emph{fiber bundles} are bundles in which $\R$ is replaced by a more general manifold and are rather pervasive in mathematics and physics.
+There are various useful generalizations of vector bundles.
+The \emph{fiber bundles} are bundles in which $\R^n$ is replaced by a more general manifold and are rather pervasive in mathematics and physics.
 A special class of fiber bundles, the \emph{principal bundles}, have this manifold to be also a group with a well-defined action on the bundle.
 Even if we will not discuss these examples in the notes, we will take a brief detour to discuss group actions, Lie groups and Lie algebras in the next chapter.

3-vectorfields.tex

@@ -191,7 +191,7 @@ \section{Vector fields}
 Any vector field $X\in\fX(W)$ defines a derivation $\cX$ via $\cX(f) = Xf$. In fact the opposite is also true:
 \begin{proposition}
   Let $M$ be a smooth manifold and $\emptyset\neq W\subset M$ an open set.
-  The set of derivation on $W$ and $\fX(W)$ are isomorphic as $C^\infty(W)$-modules.
+  The set of derivations on $W$ and $\fX(W)$ are isomorphic as $C^\infty(W)$-modules.
 \end{proposition}
 \begin{proof}
   Suppose $\cX$ is a derivation on $C^\infty(W)$ and fix $p\in W$. Then $\cX$ defines a derivation on $C^\infty(W)$ \emph{at $p$}, which we casually denote by $X_p$, via the formula

3b-liegroups.tex

@@ -552,9 +552,9 @@ \section{The exponential map}
 
 \begin{remark}
   \marginnote[1em]{The complete formula is called Baker-Campbell-Hausdorff formula and its use appears all over the place in mathematics and physics~\cite{book:bonfigliolifulci}.}
-  Note that it we have not shown $\exp(X + Y) = (\exp X)(\exp Y)$.
+  Note that we have not shown $\exp(X + Y) = (\exp X)(\exp Y)$.
   In fact, this is \emph{false} in general.
-  As a matter of fact, $\exp X \exp Y = exp Z$ where
+  As a matter of fact, $\exp X \exp Y = \exp Z$ where
   \begin{equation}
     Z = X + Y + \frac{1}2 [X,Y] + \frac1{12}[X,[X,Y]] - \frac1{12}[Y,[X,Y]] + \ldots,
   \end{equation}

5-tensors.tex

@@ -188,7 +188,7 @@ \section{Tensors}
 \end{example}
 
 \begin{exercise}\label{exe:iso_vs_endo}
-  Let $V$ be a vector space with an inner product.
+  Let $V$ be a finite-dimensional vector space.
   \begin{enumerate}
     \item Show that the space $T^1_1(V)$ is canonically isomorphic to the space of endomorphisms of $V$, that is, of linear maps $L:V\to V$.
 
@@ -197,7 +197,7 @@ \section{Tensors}
     \item Of course, given the previous example, $T^1_1(V)$ is also canonically isomorphic to the space of endomorphisms of $V^*$, that is, of linear maps $\Lambda:V^*\to V^*$.
           Prove the claim by explicitly constructing the mapping $\ell \leftrightarrow \Lambda$.
   \end{enumerate}
-  \textit{\small Hint: definitions can look rather tautological when dealing with tensors... think carefully about domains and codomains, remember the musical isomorphisms and the tensor pairing.}
+  \textit{\small Hint: definitions can look rather tautological when dealing with tensors... think carefully about domains and co-domains.}
 \end{exercise}
 
 We are now in a good place to discuss how tensors are affected by changes of basis.

6-differentiaforms.tex

@@ -36,7 +36,7 @@ \section{Differential forms}
 \section{The exterior product}
 
 \marginnote{You can find an interesting explanation of the exterior product, based on Penrose's book ``The road to reality'', \href{https://twitter.com/LucaAmb/status/1289244374996406273?s=20}{on a thread by @LucaAmb on Twitter}.}
-If you remember, we said that the determinant was an example of a $T_n^0(R^n)$ tensor: an antisymmetric tensor nonetheless.
+If you remember, we said that the determinant was an example of a $T_n^0(\R^n)$ tensor: an antisymmetric tensor nonetheless.
 At the same time, the determinant of a $n\times n$ matrix, is the signed volume of the parallelotope spanned by the $n$ vectors composing the matrix.
 We also saw that tensors can be multiplied with the tensor product, which gives rise to a graded algebra on the free sum of tensor spaces.
 This leads naturally to the following definition.
@@ -516,7 +516,7 @@ \section{Exterior derivative}
   \begin{enumerate}
     \item $d y \wedge d z$;
     \item $y\; d y \wedge d z$;
-    \item $x^2 + y^2 \; d x \wedge d y$;
+    \item $(x^2 + y^2) \; d x \wedge d y$;
     \item $\cos(x)\; d x \wedge d z$.
   \end{enumerate}
 \end{exercise}

7-integration.tex

@@ -230,7 +230,7 @@ \section{Orientation on manifolds}
 \end{example}
 
 \begin{exercise}
-  Check that the Jacobian determinant $\det(D(\varphi_2\circ \varphi_1^{-1}))$ of the transition chart from Exercise~\ref{exe:orientsphere} is negative, while $\det(D(\widetilde\varphi_2\circ \varphi_1^{-1}))$ is positive.
+  Check that the Jacobian determinant $\det(D(\varphi_2\circ \varphi_1^{-1}))$ of the transition chart from Example~\ref{exe:orientsphere} is negative, while $\det(D(\widetilde\varphi_2\circ \varphi_1^{-1}))$ is positive.
 \end{exercise}
 
 \begin{marginfigure}
@@ -804,6 +804,6 @@ \section{Stokes' Theorem}
     \item Show that $\omega = \iota_X \Omega$ is a closed non-vanishing form on $\partial N$.
     \item Show that $f^*\omega$ is closed.
     \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. }
+    \item Prove Theorem~\ref{thm:bfp1}.\\\textit{Hint: by contradiction, use the half line from $g(p)$ to $p$ to construct a function for which every point in the boundary is fixed. }
   \end{enumerate}
 \end{exercise}