Update

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

2-tangentbdl.tex

@@ -363,14 +363,14 @@ \section{Germs and derivations}
   Thus, $\left\{\frac{\partial}{\partial x^i}\Big|_p\;\mid\; 1\leq i\leq n\right\}$ is a basis of $T_p M$.
 \end{proposition}
 \begin{proof}
-  We may assume without loss of generality that $\varphi(p) = 0$ and, thanks to Corollary~\ref{cor:tgsubspace}, that $U$ is star-shaped.
+  We may assume without loss of generality that $\varphi(p) = 0$ and, thanks to Corollary~\ref{cor:tgsubspace}, that $V$ is star-shaped\sidenote[][1em]{Exercise: explain in details why it is the case.}.
   Let $f\in C^\infty(U)$.
   By Lemma~\ref{lem:Taylor} with $h = f \circ \varphi^{-1}$ we get
   \begin{equation}
     f = f(p) + x^i (g_i \circ \varphi),
     \quad g_i(0) = D_i (f \circ \varphi^{-1})(0) = \frac{\partial}{\partial x_i}\Big|_p(f).
   \end{equation}
-  \marginnote{If we are careful with the meaning of our notation, we could write more succintly $\frac{\partial f}{\partial x_i}(p)$ in place of $\frac{\partial}{\partial x_i}\big|_p(f)$ in the same fashion as in Example~\ref{ex:partialderivative}.}
+  \marginnote{If we are careful with the meaning of our notation, we could write more succinctly $\frac{\partial f}{\partial x_i}(p)$ in place of $\frac{\partial}{\partial x_i}\big|_p(f)$ in the same fashion as in Example~\ref{ex:partialderivative}.}
   Thus, for any derivation $v$, we obtain
   \begin{equation}
     v(f) = v(f(p)) + v(x^i)g_i(0) + x^i(p) v(g_i\circ\varphi) = v(x^i)  \frac{\partial}{\partial x_i}\Big|_p(f).
@@ -540,11 +540,11 @@ \section{The differential of a smooth map}\label{sec:diffsmooth}
   The proof follows from the following direct computation after observing that the number $D_j(r^i \circ \psi \circ F \circ \varphi^{-1})(\varphi(p))$ is the $(i,j)$ entry of the Jacobian matrix $D(\psi\circ F \circ\varphi^{-1})(\varphi(p))$. For any $j=1,\ldots,m$, Remark~\ref{rmk:chg_coords} implies
   \begin{align}
     dF_p \left(\frac{\partial}{\partial x^j}\Big|_p\right)
-     & =                                                                                                   %\sum_{i=1}^n 
+     & =                                                                                                   %\sum_{i=1}^n
     dF_p \left(\frac{\partial}{\partial x^j}\Big|_p\right) (y^i) \frac{\partial}{\partial y^i}\Big|_{F(p)} \\
-     & =                                                                                                   %\sum_{i=1}^n 
+     & =                                                                                                   %\sum_{i=1}^n
     \frac{\partial}{\partial x^j}\Big|_p (y^i \circ F) \frac{\partial}{\partial y^i}\Big|_{F(p)}           \\
-     & =                                                                                                   %\sum_{i=1}^n 
+     & =                                                                                                   %\sum_{i=1}^n
     D_j(r^i \circ \psi \circ F \circ \varphi^{-1})(\varphi(p)) \frac{\partial}{\partial y^i}\Big|_{F(p)}.
   \end{align}
 \end{proof}

aom.tex

@@ -275,7 +275,7 @@ \chapter*{Introduction}
 
 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, Aiva Misieviciute, 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, Lars Wieringa and Federico Zadra.
+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, Anna de Bruijn, Daniel Cortlid, Harry Crane, Fionn Donogue, Jordan van Ekelenburg, Brian Elsinga, Hanneke van Harten, Martin Daan van IJcken, Mollie Jagoe Brown, Remko de Jong, Aron Karakai, Wietze Koops, Henrieke Krijgsheld, Valeriy Malikov, Mar\'ia Diaz Marrero, Aiva Misieviciute, Levi Moes, Nicol\'as Moro, Jard Nijholt, Magnus Petz, Jorian Pruim, Luuk de Ridder, Lisanne Sibma, Marit van Straaten, Bo Tielman, Dave Verweg, Ashwin Vishwakarma, Lars Wieringa, Federico Zadra and Jesse van der Zeijden.
 
 \mainmatter