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{\bf Bonus Math 222 in class problems} \hspace{.125in} \textbf{Week: April 12, 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[]
\textbf{Volumes of $n$-balls in $\R^{n}$}

\item Folland \& Jones 10:29 \\
Use ``double polar coordinates" in $\R^4$:
\[
\begin{array}{lcl}
x_1&=& r \cos \theta, \\
x_2&= &r \sin \theta, \\ 
x_3& = & s \cos \varphi, \\
x_4 & = & s \sin \varphi \\
\end{array}
\]
to compute the 4-dimensional volume of the ball $x_1^2+x_2^2+x_3^2+x_4^2 \leq R^2$.  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 (as in Jones \S 10):
\[
\mbox{vol}_{n}(B(0,1)):=\dfrac{2\pi}{n}\mbox{vol}_{n-2}(B(0,1)).
\]

\item[]
\textbf{Surface Area of $(n-1)$-spheres in $\R^{n}$:}
\item Find a parametrization for $S^3$ in $\R^4$.  (Use double polar coordinates, then do another polar coordinate transformation to deal with the $(r,s)$ variables $r^2+s^2=1$.  Then compute the Gram matrix)  Show that the 3-D volume of $S^3$ is $2\pi^2$ (and if our 3-sphere has radius $R$ then it has 3-D volume $2\pi^2R^3$).

\item Find a parametrization for $S^4$ in $\R^5$.  (Use the extra spicy odd dimensional spherical coordinates and set $\rho=1$.  You'll need to compute the Gram matrix to find the 4-D volume element - this won't be the Jacobian computed in \#2).  Show that the 4-D volume of $S^4$ is $\frac{8}{3}\pi^2$ (and if our 4-sphere has radius $R$ then it has 4-D volume $\frac{8}{3}\pi^2R^4$.
\end{enumerate}




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