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\begin{document}



\title{ Math 3220-3 Homework 4}
\author{Due April 3 }
\maketitle

Fix $0<b<a$ and define $\Phi:\R^2\to\R^3$  by $\Phi(\phi,\theta)=(x(\phi,\theta),y(\phi,\theta),z(\phi,\theta)) $ where
\begin{eqnarray}
x & = & (a+b\ \cos\phi)\cos\theta \nonumber  \\
y & = & (a+b\ \cos\phi)\sin\theta   \nonumber  \\
z & = & b\ \sin\phi  \nonumber
\end{eqnarray}
This is a parametrized surface in $\R^3$.  Let $T$ denote its image, which is a \emph{torus}.  By periodicity, $T = \Phi(D)$ where $D=I_1\times I_2$ for any two intervals $I_1,I_2$ of length $2\pi$.  We have many choices for $D$.  In each problem we may make a different choice as convenient.
\vskip10pt
\begin{enumerate}  
\item 
\begin{enumerate}
  \item Sketch $T$.  Show the meaning of $a,b,\phi,\theta$ in your sketch.
  \item Prove that $T$ is the same as the set of solutions of $g(x,y,z)=0$ where
  $$
  g(x,y,z)=(x^2+y^2+z^2-a^2-b^2)^2 - 4a^2(b^2-z^2)
  $$
 
  \item Show that the gradient $\nabla_{(x,y,z)}g\ne 0$ at every  $(x,y,z)$  with $g(x,y,z)=0$
  \item What would happen if $0<b=a$?  Would the last statement still be true?
  \item What would happen if $0=b < a$?
  \item $g(x,y,z)$ is an equation of degree $4$ in $x,y,z$.  Is it possible to define $T$ by equations of smaller degree? 
   
\end{enumerate}
\vskip10pt
\item With the same meaning of $\Phi, D$ and $T$, let $f:T\to \R$ be the function $f(x,y,z) = x$ for $(x,y,z)\in T$.  (See Rudin, exercise 9.12, with different notation).
\begin{enumerate}
  \item Use the parametrization $\Phi$ to find the critical points of $f$ on $T$.  In other words, find the points in $D$ (say, take $D=[0,2\pi]\times [0.2\pi ]$) where the gradient of $f\circ \Phi$ vanishes.
  \item For each critical point that you found, show that  it is non-degenerate and decide if it is a local maximum, local minimum, or saddle point. 
  \item Find the critical points of $f$ by applying the method of Lagrange multipliers to $f$ and the equation $g =0$ above.  Check that you got the same critical points as those that you found using the parametrization.
\end{enumerate}
\vskip19pt
\item With the same meaning of $\Phi,D, T$:
\begin{enumerate}
  \item For each $p=(\phi,\theta)\in D$, Find 
  $$
  \frac{\partial \Phi}{\partial \phi}(p) \wedge \frac{\partial \Phi}{\partial \theta}(p) \in \Lambda^2(\R^3)
  $$
  \item Find the norm 
  $ \Big{ | }\frac{\partial \Phi}{\partial \phi} \wedge \frac{\partial \Phi}{\partial \theta}\Big{|}.$
  \item Use the formula  (to be discussed in class) for the area of a parametrized surface $\Phi:D\to\R^n$
 $$
 Area = \int_D \Big{ | }\frac{\partial \Phi}{\partial \phi} \wedge \frac{\partial \Phi}{\partial \theta}\Big{|} d\phi d\theta
 $$ 
  to find the area of $T$.  Any surprises?  Any remarks?
  \item Take $D=[-\pi/2,3\pi/2]\times[-\pi/2,3\pi/2]$, and subdivide $D$ into $D_1$ and $D_2$ by the value of the first coordinate $\phi$, that is, define 
  $$
  D_1=D\cap\{ -\pi/2 \le \phi\le \pi/2\},\ \ \ D_2=D\cap\{ \pi/2 \le \phi\le 3\pi/2\}
  $$ 


  Let 
  $$
  T_1=\Phi(D_1),\ \  T_2=\Phi(D_2)
  $$
   so that $T = T_1\cup T_2$ with $T_1\cap T_2$ one-dimensional, hence area zero.  
   
   Find the areas of $T_1,T_2$, check that they add to the area of $T$.  Finally check with your sketch of $T$ to see if the one you found  to be of larger area corresponds to what you see in your sketch.
   \end{enumerate}
   \vskip10pt
\item Same $\Phi,T,D_1,D_2,T_1,T_2$ as in last problem.  Find 
\begin{enumerate}
  \item $\Phi^*( dy\wedge dz)$
  \item $\int_T\Phi^*(dy\wedge dz)$
  \item For $i=1,2$ subdivide $D_i=D_{i,1}\cup D_{i,2}$ by the value of the second coordinate $\theta$:
  $$
  D_{i,1} = D_i\cap\{-\pi/2\le\theta\le\pi/2\} \text{ and } D_{i,2} =D_i\cap\{\pi/2\le\theta\le 3\pi/2\}
  $$
  
 
  Let $T_{i,j}=\Phi(D_{i,j})$ be the corresponding decomposition of $T$:
  $$
  T =T_{1,1}\cup T_{1,2}\cup  T_{2,1} \cup T_{2,2}
  $$
  with all intersections zero or one-dimensional, hence area zero.
  
  \begin{enumerate}
 \item Find the four values of 
  $$
  \int_{T_{i,j}}\Phi^*(dy\wedge dz)\  \  \text{ for } i,j =1,2
  $$
  by computing the integrals of the form you found in part (a) over $D_{i,j}$
 \item Let  $p_{y,z}:T\to \R^2$ be projection on the $(y,z)$-plane: $ p_{y,z}(x,y,z) = (y,z), $
 
Prove (briefly, say by looking at the sketch) that $p_{y,z}|_{T_{i,j}}:T_{i,j}\to \R^2$ is bijective onto its image, with smooth inverse.
\item From the general theory it follows that 
$$ 
\int_{D_{i,j}}\Phi^*(dy\wedge dz) =\int_{T_{i,j}}(p_{y,z})^*(dy\wedge dz) = \pm Area(p_{y,z}(T_{i,j}))
$$
  Find these projections $p_{y,z}(T_{i,j})$  (look at your sketch) and check this last equation by using elementary geometry, then compare with your answer in (i).
  
  \end{enumerate}
  \end{enumerate}
\end{enumerate}

 \end{document}

