Small typos fixed

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

2b-submanifolds.tex

@@ -103,15 +103,15 @@ \section{Inverse function theorem}
   our choice of coordinates after the above observations implies that
   $\det \left( \frac{\partial Q^i}{\partial x^j} \right) \neq 0$ at $(x,y) = (0,0)$.
 
-  Since the gradient of $Q$ with respect to $z$ is regular, we are going to extend
+  Since the gradient of $Q$ with respect to $x$ is regular, we are going to extend
   the mapping with the identity on the rest of the coordinates to get a regular map
   on the whole neighbourhood.
   Let $\varphi : U \to \R^m$ be defined by $\varphi(x,y) = (Q(x,y), y)$. Then,
   \begin{equation}
     D\varphi(0,0) = 
     \begin{pmatrix}
       \frac{\partial Q^i}{\partial x^j}(0,0) & \frac{\partial Q^i}{\partial y^j}(0,0) \\
-      0 & \id_{\R^k}
+      0 & \id_{\R^{m-k}}
     \end{pmatrix}
   \end{equation}
   has nonvanishing determinant by hypothesis.
@@ -185,6 +185,12 @@ \section{Inverse function theorem}
   concluding the proof.
 \end{proof}
 
+\begin{exercise}
+  Formulate and prove a version of the Rank theorem for a map $F : M^m \to N^n$ of constant rank $k$,
+  where $M$ is a smooth manifold with boundary, $N$ is a smooth manifold without boundary
+  and $\ker dF_p \not\subseteq T_p\partial M$.
+\end{exercise}
+
 \section{Embeddings, submersions and immersions}
 
 Looking at the statement of the Rank Theorem, one can already see that there can be different possibilities depending on the relation between, $m$, $n$ and $k$. This warrants a definition.
@@ -261,7 +267,7 @@ \section{Embeddings, submersions and immersions}
 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 $N$?
-  \item if $F$ is a submersion, what can we say about its levelsets $f^{-1}(q) \subset M$?
+  \item if $F$ is a submersion, what can we say about its levelsets $F^{-1}(q) \subset M$?
 \end{itemize}
 And what can we say about the corresponding tangent spaces?