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

\noindent{\bf 1.} Let $f\colon \R\to\R$ be given by $y=x^3$. 
Calculate $f_*(\p/\p x)$ and $f^*\,dy$. Is $f_*(\p/\p x)$ smooth?



\bigskip

\noindent{\bf 2.} Let $f$ be a smooth function on a manifold $M$.
Show that $df=f^*dy$, where $y$ is the natural coordinate on
$\R$. Furthermore, let $(x_1,\ldots,x_n)$ be a system of local
coordinates on $M$.  Prove that in local coordinates
$$
df=\sum_{i=1}^n \frac{\p f}{\p x_i}dx_i .
$$
\bigskip

\noindent Remark. Here one has to be careful with definitions.  In
this problem we define the one-form $df$ by $df(v)=L_v f$. The
one-forms $dx_1,\ldots,dx_n$ are defined as the basis in $T_x^* M$
dual to the basis $\p/\p x_1,\ldots \p/\p x_n$ in $T_x M$. (Note that
alternatively one could define $dx_i$ by $dx_i(v)=L_v x_i$, where
$x_i$ is now thought of as a function defined on an open subset of
$M$. It follows from the assertion of the problem that this definition
is equivalent to the one through the dual basis.)


\bigskip


\noindent{\bf 3.} Let $\alpha$ be a one-form on a smooth connected
manifold $M$.  Recall that $\alpha$ is said to be exact if there
exists a function $f$ such that $\alpha=df$. Prove that

\begin{itemize}

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

\end{itemize}

\bigskip

\noindent Remark. Here is a hint for part $\Leftarrow)$. The goal is to construct
$f$ such that $\alpha=df$.  Fix a reference point $x_0\in M$. For
$x\in M$, set $f(x)=\int_\eta \alpha$, where $\eta$ is curve
connecting $x_0$ to $x$. First show that the function $f$ is well
defined,  i.e., $f(x)$
is really independent of the curve $\eta$\ connecting $x_0$ and $x$,
using the assumption that $\int_\gamma\alpha=0$.
Then prove that $df=\alpha$. (For instance, one can use local coordinates
to this end.)

\bigskip

\noindent{\bf 4}  Find three different differential forms $\alpha$  in the plane for which  $d \alpha = dx \wedge dy$.

\noindent{\bf 5.} 
Let $\alpha=x\,dy$ on the plane $\R^2$ with coordinates
$(x,y)$. Denote by $S^1_R$ the circle of radius $R>0$ centered at the
origin, oriented counter clockwise. Evaluate
$$
\int_{S^1_R}\alpha.
$$
Conclude from the result that $\alpha$ is not exact. More generally, let $\gamma$ be a closed
simple curve in $\R^2$. What is the geometrical meaning of 
$\int_\gamma x\,dy$?

\bigskip


\noindent{\bf 5.} Problem 6-7 (Chapter 6) on page 152 of the textbook.






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