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{ \large \textbf{\textsc{Math 5588 -- Homework 9 (Due Thursday April 6)}}}
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\begin{enumerate}
\item Prove the Leibniz integral rule
\[\frac{d}{dx}\left( \int_{a(x)}^{b(x)} f(x,t) \, dt  \right) = f(x,b(x))b'(x) - f(x,a(x))a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x}(x,t)\, dt.\]
[Hint: For $a,b,x \in \R$ define
\[F(a,b,x) = \int_{a}^{b} f(x,t) \, dt,\]
and apply the multivariate chain rule
\[\frac{d}{dx} F(a(x),b(x),x) = \frac{\partial F}{\partial a}a'(x) +\frac{\partial F}{\partial b}b'(x) + \frac{\partial F}{\partial x}\cdot 1.\]]
\item Solve the wave equation in three dimensions $n=3$ with initial data $u(x,0) =0$ and $u_t(x,0)=x_2$ using Kirchoff's formula.
\item Let $u(x,t)$ be a solution of the damped wave equation 
\[ \left\{ \begin{aligned}
u_{tt} + \gamma u_t - \Delta u &= 0&& \text{in } \R^n\times (0,\infty)\\
u&=f&&\text{on } \R^n\times \{t=0\}\\
u_t&=g&&\text{on } \R^n\times \{t=0\},\end{aligned}\right.\]
where $\gamma\geq 0$.
Fix $x_0 \in \R^n$, $t_0 > 0$, and define the backwards wave cone 
\[K(x_0,t_0):=\Big\{(x,t) \, : \, 0 \leq t \leq t_0 \ \text{ and } \ |x-x_0| \leq t_0-t\Big\}.\]
Prove that if $f\equiv g\equiv 0$ in $B(x_0,t_0)\times \{t=0\}$, then $u\equiv 0$ in the cone $K(x_0,t_0)$.
[Hint: Mimic the proof from class, in particular use the same energy.]
\item Repeat problem 3 for the nonlinear wave equation
\[u_{tt} - \Delta u + u^3 = 0.\]
[Hint: You will need to modify the energy to account for the $u^3$ term.]
\item Sketch a triangulation of the following domains so that all triangles have side length at most $\tfrac{1}{2}$:
\begin{enumerate}
\item A unit square.
\item An isosceles triangle with vertices $(-\tfrac{1}{2},0)$,$(\tfrac{1}{2},0)$, and $(0,1)$.
\item The square $[-2,2]^2$ with the hole $[-1,1]^2$ removed.
\item The unit disk.
\item The annulus $1 \leq |x| \leq 2$.
\end{enumerate}
\item For a given vertex $v\in \R^2$ of a triangulation, the corresponding \emph{vertex polygon} is the union of all triangles for which $v$ is a vertex. Describe the vertex polygons for a triangulation that uses regular equilateral triangles. 
\end{enumerate}
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