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\begin{center}\vspace{1in}{\bf MATH 209, MANIFOLDS II, WINTER 2011}\\
\bigskip
{\bf Homework Assignment V: Orientations and integration}\\

\end{center}
\bigskip

%\noindent Throughout this assignment we assume that $M$ a smooth
%manifold of dimension $n$.

\bigskip

\noindent{\bf 1.} Let $F\colon S^n\to S^n$ be the antipodal map.

\begin{enumerate}

\item[(a)] Prove that $F$ is orientation preserving when $n$ is odd
and orientation reversing when $n$ is even.

\item[(b)] Prove that $\RP^n$ is orientable if and only if $n$ is odd.

\end{enumerate}

\bigskip

\noindent{\bf 2.} Let $F\colon \C\to \C$ be a holomorphic function.
Show that $F$ is necessarily orientation preserving at its regular
points, i.e., $F^* dx\wedge dy=f\, dx\wedge dy$ with $f\geq 0$. 

\bigskip

\noindent{\bf 3.} Let $N$ be a hypersurface in $M$ and let $\omega$
be a volume form on $M$.

\begin{enumerate}

\item[(a)] Let $v$ be a vector field nowhere tangent to $N$. Prove
that $i_v\omega|_N$ is a volume form on $N$.

\item[(b)] Prove that $i_v\omega|_N=i_w\omega|_N$ if $v-w$ is tangent to $N$.

\end{enumerate}

\noindent Remark. Assume that the hypersurface $N$ is the boundary of
$M$. Then the construction of Part (a) gives an alternative description
of the orientation induced on $N$. Indeed, let $v$ point outward and let
an orientation of $M$ be determined by $\omega$. Then the induced orientation
of $N=\p M$ is determined by $i_v\omega$ and is well defined. Note also
that in both (a) and (b) it suffices to have $v$ defined only along $N$. 

\bigskip

\noindent{\bf 4.} Let $M\subset \R^3$ be the graph of a function
$z=f(x,y)$ with $(x,y)$ lying in some bounded closed domain $U$ of
$\R^2$. Let $\vv$ be a vector field in $\R^3$ defined on a neighborhood
of $M$. Recall from vector calculus that the surface integral of $\vv$ over
$M$ is defined as
$$  
\int\!\!\!\int_M \vv\cdot d\mathbf{S}=
\int\!\!\!\int_M \vv\cdot \vn\, dS
=
\int\!\!\!\int_U (\vv\cdot \vn)(x,y)\sqrt{1+(\p_x f)^2+(\p_y f)^2}
\, dx\,dy,
$$
where $\vn$ is the unit upward normal vector field to $M$. Let
$\omega=dx\wedge dy\wedge dz$. 

\begin{enumerate}

\item[(a)]
Prove that $\int\!\!\!\int_M \vv\cdot d\mathbf{S}=\int_M i_{\vv}\omega$,
where the orientation of $M$ is induced by $\omega$ and $\vn$ as in Problem 3.

\item[(b)]
Prove that $F^* i_{\vn}\omega=\sqrt{1+(\p_x f)^2+(\p_y f)^2}\, dx\wedge dy$,
where $F\colon U\to M$ is the natural diffeomorphism 
$(x,y)\mapsto (x,y, f(x,y))$.

\end{enumerate}

\bigskip

\noindent Remark. Since every hypersurface is locally a graph (in some
orthogonal coordinates), this statement indicates that in general the
integral of $\vv$ over a hypersurface in the sense of vector calculus is
equal to the integral of $i_{\vv}\omega$.

\bigskip

\noindent{\bf 5.} Problem 15-3, page 407 of the textbook.

% Let $\omega\in\Omega^2(M)$ and
% $u\colon [0,1]\times [0,1]\to M$ be a smooth map. Prove that
% $u^*\omega=
% \omega\left(\frac{\p u}{\p t}, \frac{\p u}{\p s}\right)\,dt\wedge ds
% $
% and, as a consequence,
% $$
% \int_u\omega=\int_0^1\int_0^1 
% \omega\left(\frac{\p u}{\p t}, \frac{\p u}{\p s}\right)\,dt\,ds.
% $$

\bigskip

\noindent{\bf 6$^*$.} Prove the fundamental theorem of algebra as
outlined in class.


\bigskip

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