Updates on last chapter

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

4-cotangentbdl.tex

@@ -368,7 +368,7 @@ \section{One-forms and the cotangent bundle}
 In this case you often say that the vector fields are $F$-related\footnote{This is a definition that can be properly formalized, but we will not spend any time on it in during the course.} or that they behave naturally: you can either pull back the function $f$ to $M$ or push forward the vector field $X$ to $N$.
 
 \begin{exercise}
-  Let $\{\rho\alpha\}$ denote a partition of unity on a manifold $M$ subordinate to an open cover $\{U_\alpha\}$.
+  Let $\{\rho_\alpha\}$ denote a partition of unity on a manifold $M$ subordinate to an open cover $\{U_\alpha\}$.
   Let $F:N\to M$ denote a smooth map between smooth manifolds.
   With the definition of pullback of functions given above, prove that
   \begin{enumerate}

6-differentiaforms.tex

@@ -236,14 +236,14 @@ \section{The interior product}
 \end{definition}
 
 In other words, $\iota_v\omega$ is obtained from $\omega$ by inserting $v$ into ``the first slot''.
-By convention $\iota_v\omega = 0$ if $\omega\in\Lambda^0$.
+By convention $\iota_v\omega = 0$ if $\omega\in\Lambda^0(V)$.
 
 \begin{lemma}\label{lemma:intprod}
   Let $V$ be a real $n$-dimensional vector space and $v\in V$.
   Then the following hold.
   \begin{enumerate}
-    \item $\iota_v\circ \iota_v = 0$ and, thus, $\iota_v\circ\iota_u = -\iota_u\circ\iota_v$;
-    \item if $\omega\in\Lambda^k$ and $\eta\in\Lambda^h$,
+    \item $\iota_v\circ \iota_v = 0$ and $\iota_v\circ\iota_u = -\iota_u\circ\iota_v$;
+    \item if $\omega\in\Lambda^k(V)$ and $\eta\in\Lambda^h(V)$,
     \begin{equation}
       \iota_v(\omega\wedge\eta) = (\iota_v\omega)\wedge\eta + {(-1)}^k\omega\wedge(\iota_v\eta).
     \end{equation}