\documentclass[reqno,11pt]{amsart}
%\documentstyle[amssymb 12pt]{article}

\usepackage{xypic}
\input xy
\xyoption{all}
\usepackage{epsfig}
\usepackage{color}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amscd}
\usepackage{amsopn}

\def    \R      {{\mathbb R}}
\def    \reals      {{\mathbb R}}
\def \p {\partial }

\setlength{\textheight}{9.5in}

%\voffset=-1in

\begin{document}
\pagestyle{empty}
%\input{mssymb}



\begin{center}\vspace{1in}{\bf MATH 209, MANIFOLDS II, WINTER 2011}\\
\bigskip
{\bf Homework Assignment I: One-forms and integration}\\
\end{center}
\bigskip



{\bf 1}.  A one-form  $\alpha$   on a smooth connected
manifold $M$ is called  {\it exact} if and only if  there
exists a function $f$ such that $\alpha=df$. Prove 

\begin{itemize}

\item
$\alpha$ is exact $\Longleftrightarrow$ $\int_\gamma\alpha=0$ for any smooth closed 
                                        curve $\gamma$.

\end{itemize}

\bigskip


{\bf 2}. [HAND IN]   Let $\alpha=x\,dy$ on the plane $\R^2$ with coordinates
$(x,y)$. Let   $S^1_R$ denote  the circle of radius $R>0$ centered at the
origin, oriented counter clockwise. Evaluate
$$
\int_{S^1_R}\alpha.
$$
Conclude  that $\alpha$ is not exact. 

If  $\gamma$ is any  closed
simple curve in $\R^2$ what is the geometrical meaning of 
$\int_\gamma x\,dy$?

\bigskip
 


{\bf 3}.  Find three different one-forms $\alpha$ in the plane for which $d \alpha = dx \wedge dy$.  
\bigskip

{\bf 4}.  Find a function $S= S(x,y)$ such that $x dy = \frac{1}{2} (xdy - ydx) + dS$.
\bigskip

{\bf 5}.  Let  $\alpha=(x\,dy - ydx)/(x^2 + y^2)$.    

a) Show that $d \alpha = 0$.

b)  Show that $\int_{S^1_R} \alpha = 2 \pi$ where $S^1_R$ is as per problem {\bf 2}. 

c) Show that in polar coordinates $(r, \theta)$ we have 
$\alpha = d \theta$.   

d) Explain why (c) does not contradict problem {\bf 1}.





\end{document}






