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\title{MAT 239: Homework 8}

\begin{document}
	\maketitle
	
	\begin{enumerate}
		\item Let $f:\R^3\to \R^3$ be a diffeomorphism. Put coordinates $(x_1,x_2,x_3)$ on the domain and $(y_1,y_2,y_3)$ on the co-domain. Show directly from the definition of the pull-back (compute explicitly) that
		$$f^*(dy_1\wedge dy_2\wedge dy_3)_p = \det(df_p) dx_1\wedge dx_2\wedge dx_3.$$
		
		\item Let $f:X\to Y$ be a smooth map between manifolds, and let $\phi:Y\to \R$ be a smooth function (i.e. a zero form on $Y$). Then the derivative $d\phi$ can be viewed as a $1$-form on $Y$. Prove directly using the definiions that
		$$f^*(d\phi) = d(f^*\phi).$$
		
		\item Let $c:[a,b]\to X$ be a smooth curve, and let $c(a)=p$ and $c(b)=q$. Show that if $\omega=df$ where $f:X\to \R$ is a smooth function then 
		$$\int_a^b c^*\omega = f(q)-f(p)$$
		
		\item Let $\omega \in \Omega^1(\R^2\setminus\{0\})$ be defined by
		$$\omega_{(x,y)} = \left( \frac{-y}{x^2+y^2}\right) dx + \left( \frac{x}{x^2+y^2} \right) dy.$$
		Let $C_R\subset \R^2\setminus \{0\}$ the the circle of radius $R$. Calculate $\int_{C_R}\omega$.
		
		\item Let $\alpha = dz-\sum_{i=1}^n y_idx_i$ on $\R^{2n+1}$. Prove that the $2n+1$ form $\omega := \alpha \wedge (d\alpha)^n$ is non-zero at every point in $\R^{2n+1}$.
		
		\item Prove the following properties for the exterior derivative $d$ defined on a smooth manifold (you may use the corresponding properties for the exterior derivative on $\R^n$):
			\begin{enumerate}
				\item $d(\omega \wedge \eta) = (d\omega)\wedge \eta +(-1)^p\omega \wedge(d\eta)$ where $\omega\in \Omega^p(X)$.
				\item $d(d\omega)=0$.
				\item $d\circ g^* = g^*\circ d$.
			\end{enumerate}
		
		\item (Not graded, but highly recommended) Work through the exercises in Guillemin Pollack Chapter 4 Section 6 (Cohomology wih Forms).
		
	\end{enumerate}
	
\end{document}