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\title{CS 205b / CME 306}
\author{Application Track}
\date{Homework 1}
\maketitle
\begin{enumerate}
\item Use conservation of mass to show that the sum of the outward-facing area-weighted normals of a triangle mesh must be the zero vector.
\item The strong form conservation of mass in an Eulerian frame can be written as $\rho_t + \rho_x u + \rho u_x = 0$. For each of the three terms:
\begin{enumerate}
\item Provide a physical description of what the term means,
\item Describe a physical situation in which that term is identically zero in a region while the other two terms remain nonzero, and
\item Show that the situation can actually occur by finding $\rho$ and $u$ such that the term is identically zero in the region $x,t \in [0,1]$ while the other two terms
are nonzero throughout the entire region.
\end{enumerate}
\item In this sequence of problems, we will construct a kernel function $W(\xx,h)$ for use in the SPH method in 1D, 2D, and 3D.
\begin{enumerate}
\item Since we would like $W(\xx,h)$ to be symmetric about the origin, we take $W(\xx,h) = c_d(h) f(\|x\|/h)$, where $c_d(h)$ is a normalization factor that depends on the dimension
$d$ and the radius of influence $h > 0$. The function $f(r)$ need not be defined for $r < 0$. Find $c_1(h)$, $c_2(h)$, and $c_3(h)$. (Hint: Use polar coordinates in 2D and spherical
coordinates in 3D.)
\item We would like the radius of influence of the kernel $W(\xx,h)$ to be $h$. What conditions does this place on $f(r)$?
\item We further require that $W(\xx,h)$ be have continuous second derivatives everywhere. What conditions does the continuity requirement place on $f(r)$? Be sure the kernel also
satisfies this continuity requirement at the origin. (Hint: it is sufficient to look at 1D with $h=1$.)
\item Find a suitable piecewise cubic function $f(r)$ defined for $r \ge 0$ that satisfies all of these requirements.
\item Evaluate $c_1(h)$, $c_2(h)$, and $c_3(h)$.
\end{enumerate}
\end{enumerate}
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