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{\bf Math 222 HW\#6, due Thursday 3/25/21} \hspace{.5in} \textbf{NAME:} \hspace{4in}
%outlier malcom
\begin{enumerate}


%\item Multi Warm Up \\
%Reparametrize the below curve by arc length from $(1,1,0)$ in the direction of increasing $t$
%\[
%r(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
%\]
%Then compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, $k(s)$, and $\tau(s)$.


%\item Problem 8-3: Jones\\
%For the record, here is the general principle used above.  Prove that for a general real symmetric $n\times n$ matrix $A$ which is positive semi-definite and for $t\in \R^n$, 
%\[
%At \cdot t = 0 \mbox{ if and only if } At=0.
%\]
%(It will be easiest to do each direction separately.)
\item Folland 3.4, Jones 10.G \& 10.I \\
Let $(u,v) = F(x,y) = (x-y, xy)$. 
\begin{enumerate}
\item Sketch some of the 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 The Jacobian $J$ vanishes at $(a,b)$ precisely when the gradients $\nabla u(a,b)$ and  $\nabla v(a,b)$ are linearly dependent, i.e., when the level sets of $u$ and $v$ passing through $a $ and $b$ are tangent to each other.  Use your sketch of the level sets in (a) to show pictorially that this assertion is correct.  
\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 Show by explicit calculation that the matrices $DF(2,-3)$ and $DG(5,-6)$ are inverses of each other.
\end{enumerate}

\item Folland 3.4, \\
  Let $F: \R^3 \to \R^3$ be the spherical coordinate mapping:
 \[
 (x,y,z) = F(\rho, \varphi, \theta) = (\rho \sin \varphi \cos \theta, \rho \sin \varphi \sin \theta, \rho \cos \varphi).
  \]
\begin{enumerate}
\item Describe (or draw) the surfaces in $xyz$-space that are the images of the planes $\rho = $ positive constant, $\varphi =$ constant (check $(0,\frac{\pi}{2})$, $\frac{\pi}{2}$, $(\frac{\pi}{2}, \pi)$ separately), and $\theta = $ constant (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 Folland, Jones 10.F\\
Calculate $\iint_S (x+y)^4(x-y)^{-5}dA$ where $S$ is the square $-1 \leq x+y \leq 1$, $1\leq x-y \leq 3$.
\item Stewart (on in class problems) \\
Evaluate the integral by making an appropriate change of variables: 
\[
\iint_R \sin(9x^2+4y^2) dA, 
\]
where $R$ is the region in the first quadrant bounded by the ellipse $9x^2+4y^2=1$.




\item[*] Assignment Reflections\\
 How difficult was this assignment?  How many hours did you spend on it?  Which problems did you find to provide a worthwhile learning experience?  Should I be assigning a similar number of problems, fewer problems, or more problems in the future?  Is there a good mix of theory and computations?

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\end{enumerate}

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