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{\bf Math 401 HW\#3, due Wednesday 9/18/24} \hspace{.5in} \textbf{NAME:} \\\hspace{4in} \\

Do not upload solutions to the \textcolor{blue}{\textbf{In Class Warm Up Problems}} to gradescope.  %\\
%\textbf{UPDATED: Moved the last exercise about the Jacobian to HW \#4}.
%The problems to upload to gradescope this week are all on page 2. \\

 \textcolor{red}{\textbf{UPDATE 9/10: The parametrization is a diffeo exercise was moved to HW 4.}}


\begin{enumerate}



\item[\textcolor{blue}{$\star$}] \textcolor{blue}{\textbf{Multi Warm Up}}  \\
Let $(x,y) = F(u,v) = (u^2-v^2, 2uv)$. 
\begin{enumerate}
\item Restricting to $u>0$, sketch the image of the $xy$-plane of the coordinate grid.
\item Draw the curves in the $u>0$ half-plane that map to the coordinate grid in the $xy$-plane.
\end{enumerate}



\item \textbf{Spherical coordinates} \\
 Let $f: \R^3 \to \R^3$ be the spherical coordinate map,
 \[
 (x,y,z) = f(\rho, \varphi, \theta) = (\rho \sin \varphi \cos \theta, \rho \sin \varphi \sin \theta, \rho \cos \varphi).
  \]
\begin{enumerate}
\item Describe and attempt to sketch by hand the surfaces in $xyz$-space that are the images of  $\rho = $ positive constant, $\varphi =$ c (check  $c\in (0,\frac{\pi}{2})$, $c=\frac{\pi}{2}$, $c\in(\frac{\pi}{2}, \pi)$ separately), and $\theta = k \in [0,2\pi) $.
\item Compute the derivative $Df$ and show the Jacobian is $\det Df(\rho, \varphi, \theta)=\rho^2 \sin \varphi$.
\item What is the condition on the point $(\rho_0, \varphi_0, \theta_0)$ for $f$ to be locally invertible about this point?  What is the corresponding condition on $(x_0,y_0,z_0) =f(\rho_0, \varphi_0, \theta_0)$ ?
\end{enumerate}



\item \textbf{Folland \S 3.4}: Transformations and Coordinate Systems \\
 Let $(u,v) = F(x,y) = (x-y, xy)$. 
\begin{enumerate}
\item Sketch some curves $x-y = \mbox{constant}$ and $xy = \mbox{constant}$ in the $xy$-plane.  Which regions in the $xy$-plane map onto the rectangle in the $uv$-plane given by $0\leq u \leq 1, \ 1\leq v\leq 4$?  There are two of them; draw a picture of them.
\item Compute the derivative $DF$ and the Jacobian $J=\det DF$
\item When $J$ vanishes at a point $p$, what can be said about the gradients $\nabla u(p)$ and $\nabla v(p)$?  
  
\item Notice that $F(2,-3) = (5,-6)$.  Compute explicitly the local inverse $G$ of $F$ such that $G(5,-6)=(2,-3)$ and compute its derivative $DG$.
\item Note that $DF(2,-3)$ and $DG(5,-6)$ are matrix inverses of each other.

\end{enumerate}


%\item \textbf{Exercise \S 2 (4) }\\
% Let $S$ be a compact surface, and let there be a differentiable function $f: S \to \R$ with at most three critical points. Show that $S$ is connected. \\
 %\emph{Let Prof Jo or Leo know if you need a hint.}

\item \textbf{Exercise 2.24} \\
 Prove that one-sheeted cone is not a surface: $ C = \{ (x,y,z) \in \R^3 \ | \ z^2=x^2+y^2, z \geq 0 \} $

\item \textbf{First part of Exercise 2 (7)} \\
Let $
S=\{ (x,y,z) \in \R^3 \ | \ e^{x^2} + e^{y^2} + e^{z^2} =a \}, $
with $a>3$.  Prove that $S$ is a surface.


%\item Exercise 2.71 (a) \\
 %Let $S$ be a surface and let $R$ be a straight line of $\R^3$.  Prove that if $S$ is compact, then there is a point of $S$ whose normal line intersects the line $R$ perpendicularly.



\item[*] Assignment Reflections\\
 How difficult was this assignment?  How many hours did you spend on it?  Which problems did you find helpful?

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\end{enumerate}

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