Add more exercises

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

1-manifolds.tex

@@ -503,7 +503,7 @@ \section{Differentiable manifolds}
 
   \begin{marginfigure}
     \includegraphics{1_2_25-sphere}
-    \caption{The identification $\sim$ of antipodal points maps the sphere to a disk. Embedding $\bS^n/\!\sim$ in $\R^{n+1}$, one can define a map $\pi_D$ that projects the representative of $[x]$ in the north hemisphere orthogonally to the disk $D^2 = \{x\in\R^{n+1} \mid \|x\|\leq 1, \; x^{n+1}=0\}$ (the equator is mapped to itself). }
+    \caption{The identification $\sim$ of antipodal points maps the sphere to a disk. Embedding $\bS^n/\!\sim$ in $\R^{n+1}$, one can define a map $\pi_D$ that projects the representative of $[x]$ in the north hemisphere orthogonally to the disk $D^n = \{x\in\R^{n+1} \mid \|x\|\leq 1, \; x^{n+1}=0\}$ (the equator is mapped to itself). }
   \end{marginfigure}
   There is a nice interpretation of this construction in terms of flattening spheres.
   Observe that a line through the origin always intercepts a sphere $\bS^n$ at two antipodal points and, conversely, each pair of antipodal point determines a unique line through the center.

7-integration.tex

@@ -621,6 +621,16 @@ \section{Stokes' Theorem}
   concluding the proof.
 \end{proof}
 
+\begin{exercise}
+  Let $D^n := \{x=(x^1, \ldots, x^n)\in\R^n \mid \|x\| \leq 1\}$ denote the unit disk in $\R^n$ centred at $0$. Recall that $\partial D^n = \bS^{n-1}$.
+  \begin{enumerate}
+    \item Compute $\int_{\bS^1} \nu$ where $\nu$ is the following 1-form on $\R^2$: $\nu = -x^2 dx^1 + x^1 dx^2$.
+    \item Compute $\int_{\bS^2} \omega$ where $\omega$ is the following 2-form on $\R^3$: $\omega = -x^1 dx^1\wedge dx^3 - x^2 dx^1\wedge dx^3 + x^3 dx^1\wedge dx^2$.
+    \item Show that $\eta$ and $\omega$ above are closed but not exact (as differential forms on $\bS^1$ and $\bS^2$ respectively).
+  \end{enumerate}
+  \textit{\small Hint: if you look carefully, you may notice that you don't really need to write anything down in coordinates.}
+\end{exercise}
+
 \begin{example}
   Consider the annulus $M=\{(x,y)\in\R^2 \mid 1/2\leq x^2+y^2 \leq 1\}$ and the $1$-form $\omega = \frac{-y dx + x dy}{x^2 + y^2} = d\theta$ where $(x,y) = (\rho\cos\theta, \rho\sin\theta)$.
 

aom.tex

@@ -207,7 +207,7 @@
     \setlength{\parskip}{\baselineskip}
     Copyright \copyright\ \the\year\ \thanklessauthor
 
-    \par Version 0.9 -- \today
+    \par Version 0.9.1 -- \today
 
     \vfill
     \small{\doclicenseThis}