Add precisation

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

3-vectorfields.tex

@@ -67,7 +67,7 @@ \section{Vector fields}
 As it turns out, we can employ differentials of diffeomorphisms to map vector fields to vector fields.
 
 \begin{definition}
-  Let $F:M\to N$ be a diffeomorphism between smooth manifolds.
+  Let $F:M\to N$ be a diffeomorphism\footnote{While to define the pushforward it is enough to have a smooth bijection, if you take a map with non-smooth inverse you may end up with a non-smooth vector field. Try it out: take $M=N=\mathbb{R}$, $X = \partial/\partial x$ and $F(x) = x^3$, and compute $F_* X$.} between smooth manifolds.
   Then, the \emph{pushforward} $F_*$ of $F$, defined by\footnote{In coordinates, this reads\begin{equation}
       (F_* X)_q = dF_{F^{-1}(q)}(X_{F^{-1}(q)}).
     \end{equation}}

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.
 
-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, Wietze Koops, Henrieke Krijgsheld, Levi Moes, Nicol\'as Moro, 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 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, Daniel Cortlid, Harry Crane, Anna de Bruijn, Remko de Jong, Luuk de Ridder, Brian Elsinga, Mollie Jagoe Brown, Aron Karakai, Wietze Koops, Henrieke Krijgsheld, Levi Moes, Nicol\'as Moro, 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 and Federico Zadra.
 
 \mainmatter