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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{Tuesday, November 14, 2022}
\newcommand{\myAssignmentTitle}{Homework Assignment 5}
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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 Marsden 7.5.15
\question Find the mass of a spherical surface $S$ of radius $R$ with the property that at each point $(x,y,z) \in S$ the mass density is equal to a constant $\rho$ times the distance between $(x,y,z)$ and some fixed point $(x_0,y_0,z_0) \in S$.

\bigskip

%%based on Rogawski 16.5.5
\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

%%Based on Rogawski 16.5.19
\question Let \ $r = \sqrt{x^2 + y^2 + z^2}$ \ and let \ $\mathbf{e}_r = \left(\frac{x}{r}, \frac{y}{r}, \frac{z}{r}\right)$ \ be the unit radial vector field.  Compute the integral of \ $\mathbf{F} = e^{-r} \, \mathbf{e}_r$ \ over the upper hemisphere \ $x^2 + y^2 + z^2 = 9$ \ with outward-pointing normal.

\bigskip

%%Rogawski 16.5.22
\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 16.5.23
\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.

%%Based on Rogawski 16.R.61
\question Calculate the flux of the vector field \ $\mathbf{E}(x,y,z) = (0,0,x)$ \ through the part of the ellipsoid
\begin{displaymath} 4 x^2 + 9 y^2 + z^2 = 36\end{displaymath}
where \ $z \geq 3$, $x \geq 0$, and $y \geq 0$. \newline
(Hint: Use the parametrization $\Psi(r,\theta) = \left( 3 r \cos(\theta), \; 2 r \sin(\theta), \; 6 \sqrt{1 - r^2} \right)$ with an appropriate domain.)

\end{questions}

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