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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{11:00pm Tuesday, May 14, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 4}
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\begin{document}

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%\firstpagefooter{}{}{(This exam is worth {\numpoints} points.)}
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\begin{questions}
    
%%Marsden 7.5.16
\question A metallic surface $S$ is in the shape of a hemisphere \ $z = \sqrt{R^2 - x^2 - y^2}$, \ where $(x,y)$ satisfies \ $x^2 + y^2 \leq R^2$. \ The mass density (mass per unit area) at \ $(x,y,z) \in S$ \ is given by \ $m(x,y,z) = x^2 + y^2$. \ Find the total mass of $S$.

\bigskip

%%Marsden 7.5.19 (amended)
\question Find the average value of \ $f(x,y,z) = x + z^2$ \ on the truncated cone \ $z^2 = x^2 + y^2$, \ with \ $3 \leq z \leq 4$.

\bigskip

%%Marsden 7.5.20
\question Evaluate the integral \ $\displaystyle \iint_S (1-z) \, dS$, \ where $S$ is the graph of \ $z = 1 - x^2 - y^2$, \ with \ $x^2 + y^2 \leq 1$.

\bigskip

%%based on Rogawski 16.4.5
\question Evaluate \ $\displaystyle \iint_S \mathbf{F} \boldsymbol{\cdot} \, d\mathbf{S}$, \ with \ $\mathbf{F}(x,y,z) = (x,y,z)$, \ and $S$ the part of the plane \ $x + y + z = 1$

with \ $x \geq 0$, \ $y \geq 0$, \ and \ $z \geq 0$.

\bigskip

%%Rogawski 16.4.22
\question Let $\mathcal{S}$ be the ellipsoid \ $\displaystyle \left(\frac{x}{4}\right)^2 + \left(\frac{y}{3}\right)^2 + \left(\frac{z}{2}\right)^2 = 1$. \ Compute the flux of \ $\mathbf{F} = (0, 0, z)$ \ over the

portion of $\mathcal{S}$ where \ $x \leq 0$, $y \leq 0$, $z \leq 0$ with upward-pointing normal.

\bigskip

%%Rogawski 16.4.23
\question Let $\mathbf{v} = (0, 0, z)$ be the velocity field (in meters per second) in $\mathbb{R}^3$.  Compute the volume flow rate (in cubic meters per second) through the upper upper hemisphere $(z \geq 0)$ of the unit sphere \ $x^2 + y^2 + z^2 = 1$.

\bigskip

%%Rogawski 16.4.25
\question A net with surface described by \ $y=0$ \ with \ $x^2 + z^2 \leq 1$ \ is dipped into a river in which the water flows according to the velocity field \ $\mathbf{v} = (x-y, \ z+y+4, \ z^2)$. \ Determine the volume flow rate across the net.

\bigskip

%%Based on Rogawski 16.5.20 and 16.5.21
\question The electric field $\mathbf{E}$ due to a point charge located at the origin in $\mathbb{R}^3$ is given by $\displaystyle\mathbf{E} = k \frac{\mathbf{e}_r}{r^2}$, where $\displaystyle \mathbf{e}_r = \frac{\mathbf{r}}{r}$ is the unit radial vector, $r = \|\mathbf{r}\| = \sqrt{x^2 + y^2 + z^2}$ is distance from the origin, and $k$ is a constant. (Note: The radial vector $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$.)
\begin{parts}
\part Compute the flux of $\mathbf{E}$ out of a sphere of radius $R$ centered at the origin.
\part Notice that the flux does not depend on the radius of the sphere. Explain why this is true.
\end{parts}

\end{questions}

\end{document}

