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\noindent
\parbox{2.25in}{\bf Math 3510}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Nov. $9^\mathrm{th}$, 2018}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
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\begin{description}
\item[\#1 The Unit Circle is a Manifold] Let $M=S^1 = \{ (x,y) \in \mathbb{R} \;|\; x^2+y^2=1 \}$ (i.e., the unit circle). 
\begin{enumerate}[(a)]
\item Show $S^1$ is a manifold using Theorem 3.2 (the cheaty/easy way).

\item As we did with the unit sphere, define maps $\varphi_{x}^{\pm}$ and $\varphi_{y}^{\pm}$. For example: $\varphi_{x}^{-}:\{ (x,y) \in S^1 \;|\; y < 0 \} \to ???$ (you figure out the proper open subset of $\mathbb{R}$) is defined by $\varphi^{-}_{x}(x,y)=x$.

Show $S^1$ is a manifold with these maps forming an atlas. 

Don't forget to explain why $\varphi_{x}^{+},\varphi_{x}^{-}\varphi_{y}^{+},\varphi_{x}^{-}$ are valid charts. Also, you should carefully write down transition functions (complete with domains/ranges) and explain why they are smooth.

\item Consider $\tau^{-1} : (0,2\pi) \to S^1 - \{ (1,0) \}$ defined by $\tau^{-1}(\theta) = (\cos(\theta),\sin(\theta))$. Explain why $\tau$ is a possible chart. Then show that $\tau$ is compatible with your atlas of $\varphi$'s defined in part (b).

\item Given an atlas such that the determinants of all of the Jacobian's of transition functions are all always positive, we have an {\it oriented atlas} for our {\it oriented manifold}. Find such an oriented atlas for $S^1$. 

\item Show $f:S^1 \to \mathbb{R}$ defined by $f(x,y)=x^2+3xy$ is a smooth function using the atlas structure defined in part (b).
\end{enumerate}


\item[\#2 The Torus is a Manifold] We showed that the torus $T = \{ (x,y,z) \;|\; (3-\sqrt{x^2+y^2})^2+z^2=1 \}$ is an embedded submanifold of $\mathbb{R}^3$ using Theorem 3.2 in class. The calculation was a bit involved. It is much easier to show that the torus $T = \{ (x,y,z,t) \;|\; x^2+y^2=1 \mbox{ and } z^2+t^2=1 \} = S^1 \times S^1$ is a embedded submanifold of $\mathbb{R}^4$ using Theorem 3.2. Show this.

\end{description}

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