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{\bf Math 222 in class problems} \hspace{.125in} \textbf{Week: March 8, 2021} \hspace{.5in}  \textbf{Name:} %outlier malcom
\begin{enumerate}

%DO CHANGE OF COORDINATES IN WEEK 7

%\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 Folland, Jones 10.F (on HW 7)\\
Calculate $\iint_S (x+y)^4(x-y)^{-5}dA$ where $S$ is the square $-1 \leq x+y \leq 1$, $1\leq x-y \leq 3$.  Answer: $\frac{4}{81}$
  
  
\item Warm-Up \\
Prove that the area of an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} \leq 1$ is $\pi ab$.  

\item Stewart (on HW 7) \\
Evaluate the integral by making an appropriate change of variables: 
\[
\iint_E \sin(9x^2+4y^2) dA, 
\]
where $E$ is the region in the first quadrant bounded by the ellipse $9x^2+4y^2=1$.  
Answer : $\frac{\pi}{24}(1-\cos1)$.

\item Jones 10-41 \\
UPDATED so that $E$ is actually an ellipsoid:  Let $E$ be the ellipsoid
\[
\frac{x_1^2}{a_1^2}+\frac{x_2^2}{a_2^2}+ ... + \frac{x_n^2}{a_n^2} \leq 1,
\]
where the semi-axes $a_1,...,a_n$ are any positive integers.  Prove that
\[
\mbox{vol}(E) = a_1 \cdots a_n \mbox{vol}_n(B(0,1))
\]
Sanity check: The unit 2-ball is the unit disk, which has area $\pi$.  Phew!

\item Folland \& Jones 10:29 \\
Use ``double polar coordinates" in $\R^4$:
\[
\begin{array}{lcl}
x&=& r \cos \theta, \\
y&= &r \sin \theta, \\ 
z& = & s \cos \varphi, \\
w & = & s \sin \varphi \\
\end{array}
\]
to compute the 4-dimensional volume of the ball $x^2+y^2+z^2+w^2 \leq R^2$.  Answer: $\frac{\pi^2}{2}R^4$

\item (Challenging) Jones 10:29 \\
Using ``$n \ \times$ polar coordinates"  in $\R^{2n}$, prove that the $2n$-dim volume of the ball $
x_1^2+....+x_{2n}^2 \leq 1$ is
\[
\mbox{vol}_{2n}(B(0,1)):=\dfrac{\pi^{n}}{n!}.
\] 
(Extra Spicy) Modifying this choice of coordinates appropriately for odd dimensions (one possibility is coming up with the analogue of higher dimensional spherical coordinates):
\[
\mbox{vol}_{2n+1}(B(0,1)):=\dfrac{2 (n!) (4\pi)^{n}}{(2n+1)!}.
\]
Suggestions: Using $\varphi_1,\varphi_2,...,\varphi_{2n-1}\in[0,\pi]$, and $\varphi_{2n} \in [0, 2\pi]$, set  $x_1=\rho \cos (\varphi_1)$ and for $i\in[1,n]$,
\[
\begin{array}{rcl}
x_{2i} &= & \rho \sin(\varphi_{1}) \cdots \sin (\varphi_{2i-1})\cos ( \varphi_{2i}) \\
x_{2i+1} & = & \rho \sin (\varphi_{1} ) \cdots \sin( \varphi_{2i-1}) \sin  ( \varphi_{2i} ) \\
\end{array}
\]
You then should get
\[
\mbox{Jac}_{2n+1} = \rho \sin ( \varphi_1 )\cdots \sin (\varphi_{2n-1}) \mbox{Jac}_{2n}
\]
You can then use this to prove by induction that closed form expression for the volume element in $(2n+1)$-spherical coordinates is:
\[
d^nV = \left| \det \frac{\partial(x_i)}{\partial(\rho,\varphi_j)} \right| = \rho^{2n} \sin^{2n-1}(\varphi_1) \sin^{2n-2}(\varphi_2) \cdots \sin (\varphi_{2n-1})   d\rho \ d\varphi_1 \cdots d\varphi_n
\]
Sanity check: When $n=1$, how do these hyperspherical coordinates compare with our 3-D spherical coordinates?  If you get stuck, see also \S 1-3 of \url{https://en.wikipedia.org/wiki/N-sphere}.  \\
The other means of attack is to prove the recursive formula using some tricks:
\[
\mbox{vol}_{n}(B(0,1)):=\dfrac{2\pi}{n}\mbox{vol}_{n-2}(B(0,1)).
\]
(Frank does this in the book.)
\end{enumerate}




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