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\usepackage{enumerate}
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% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\del}{\partial}
\newcommand{\lra}{\longrightarrow}

\begin{document}
\pagestyle{fancy} \lhead{1300F Geometry and Topology, Assignment 5}
\rhead{Due date: December 16, 2016} \cfoot{} 


%\begin{exercise}
%Let $z$ be the standard complex coordinate on $\CC$, i.e. $z=x+iy$, and form the complex differential form $\tfrac{dz}{z}$. Where is this well-defined?  Decompose it into real and imaginary parts. Are these closed forms? Compute the integrals
%\[
%\int_{S^1} \iota^*\mu,
%\]
%for $\iota:S^1\lra \CC$ the standard inclusion and $\mu$ the real or imaginary part of $\tfrac{dz}{z}$.  Explain, using Stokes' theorem, why the value of the integral does not depend on the radius of the circle.\end{exercise}




%
%\begin{exercise}
%  Consider the pair of vector fields $V = \partial_y$ and $W =
%  y\partial_x - \partial_z$ on $\RR^3$, where we use coordinates
%  $(x,y,z)$.  Is it possible to find a 2-dimensional submanifold of
%  $\RR^3$ with the property that both $V$ and $W$ are tangent to it at
%  all its points? If so, construct one; if not, why not?
%  \end{exercise}


%\begin{exercise}
%  Let $\alpha = dz - xdy$ and $\beta = dx - wdy$ be 1-forms on $\RR^4$
%  (or you can think of them as functions on $T\RR^4$).  Both $\alpha$ and $\beta$ have 
%3-dimensional kernel on each tangent space.  
%\begin{enumerate}
%\item Prove that $\ker \alpha \cap \ker \beta$ has dimension 2.
%\item Give a local basis $(V,W)$ for the above intersection.
%\item Compute $[V,W]$. Is it linearly dependent on $(V,W)$?
%\item Is it possible to find a 3-dimensional submanifold such that
%  $V$, $W$, and $[V,W]$ are everwhere tangent to it? If so, construct one; if not, show why not.
%\end{enumerate}
%\end{exercise}




\begin{exercise}
Determine the flow of the vector field $E = \sum_{i=1}^n x_i \tfrac{\del}{\del x_i}$ on $\RR^n$.  Deduce that this vector field is independent of the choice of linear coordinates.
\end{exercise}

\begin{exercise}
  Consider $S^n$ and its two stereographic coordinate charts
  $\varphi_{S},\varphi_N$ to $\RR^n$.  Using standard coordinates on
  $\RR^n$, write down the coordinate expressions for a smooth,
  nowhere-vanishing $n$-form on $S^n$.
\end{exercise}

\begin{exercise}
Let $X$ be a vector field. Define the operator (the \emph{interior product}) $i_X: \Omega^k(M)\to\Omega^{k-1}(M)$ by the formula 
\[
(i_X\rho) (X_1,\ldots X_{k-1}) = \rho(X,X_1,\ldots, X_{k-1}).
\]
\begin{enumerate}
\item Prove that $i_X$ is a graded derivation of degree $-1$ of the algebra of differential forms. 
\item Prove that the graded commutator $[D_1, D_2] = D_1D_2 - (-1)^{d_1d_2} D_2D_1$ of the graded derivations $D_1,D_2$ of degree $d_1,d_2$ is itself a graded derivation of degree $d_1+d_2$.
\item Define $L_X = d i_X + i_X d$ (the \emph{Lie derivative}) and prove it is a graded derivation of degree zero. 
\item Prove the identity $i_{[X,Y]} = [L_X, i_Y]$, where the left bracket is the Lie bracket of vector fields and the right bracket is the graded commutator.  Note that this identity expresses the intimate relationship between the Lie bracket of vector fields (on the left hand side) and the exterior derivative (on the right hand side).  
\item Prove the identity $L_{[X,Y]} = [L_X,L_Y]$. 

\end{enumerate}
\end{exercise}


\begin{exercise}
Let $\varphi:\RR^3\to \RR^3$ be given by 
\[
(r,\phi,\theta)\mapsto (r\sin\phi\cos\theta, r\sin\phi\sin\theta,r\cos\phi),
\]
where $(r,\phi,\theta)$ are standard Cartesian coordinates on $\RR^3$.
\begin{itemize}
 \item Compute $\varphi^*dx, \varphi^*dy, \varphi^*dz$ where $(x,y,z)$ are Cartesian coordinates for $\RR^3$.
\item Compute  $\varphi^*(dx\wedge dy\wedge dz)$.
\item  Compute the integral
\[
\int_{S^2_r} i_X (dx\wedge dy\wedge dz),
\] 
for the vector field $X = \varphi_* \tfrac{\del}{\del r}$, where $S^2_r$ is the sphere of radius $r$.
\end{itemize}
\end{exercise}

\pagebreak 
\begin{exercise} Use Stokes' theorem if necessary:
    \noindent
    \begin{enumerate}
    \item Let $M$ be a compact orientable smooth $n$-manifold (without
      boundary) and let $\mu\in\Omega^{n-1}(M)$.  Prove there exists a
      point $p\in M$ with $d\mu(p)=0$.
    \item For any sphere $S^k$, let $\iota:S^k\to \RR^{k+1}$ be the
      usual inclusion, and let $v_k\in \Omega^k(S^k)$ be given by
        \[
        v_k = \iota^*\sum_{i=0}^k (-1)^i x^i dx^0\wedge\cdots \wedge
        dx^{i-1}\wedge dx^{i+1}\wedge\cdots\wedge dx^k.
        \]
        Show that $v_k$ is closed and that $[v_k]\neq 0$ in the top de
        Rham cohomology group $H^k(S^k)$.
       \end{enumerate}
\end{exercise}

\begin{exercise}
Compute the de Rham cohomology groups (Using Mayer-Vietoris if necessary) of the following spaces, for all degrees.  
 \begin{itemize}
 \item $\RR^3 -\{p\}$, for $p\in\RR^3$ a point.
\item   $\RR^3-\{p_1\cup p_2\}$ where $p_i$ are distinct points?
\item  $\RR^3-\{\ell_1\cup \ell_2\}$ where $\ell_i$ are non-intersecting lines?
\item $\RR^3-\{\ell_1\cup \ell_2\}$, assuming that $l_1$ intersects $l_2$ in exactly one  point?
\end{itemize}
This question is a slightly easier version of the one John Nash asked in class in the movie ``A Beautiful Mind''.  
\end{exercise}

\end{document}
