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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, March 2, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 4}
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\begin{document}

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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

\question Compute the area enclosed by the ellipse \ $\displaystyle \left(\frac{x}{c}\right)^2 + \left(\frac{y}{d}\right)^2 = 1$.

\bigskip

\question Find the area of the region between the $x$-axis and the cycloid parametrized by

$\mathbf{r}(t) = \left(t - \sin(t), \ 1 - \cos(t)\right)$ \ with \ $0 \leq t \leq 2\pi$.

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\bigskip

\question Let \ $\displaystyle P(x,y) = \frac{-y}{x^2 + y^2}$ \ and \ $\displaystyle Q(x,y) = \frac{x}{x^2 + y^2}$, \ and let $D$ be the unit disk \ $D = \left\{ (x,y) \mid x^2 + y^2 \leq 1 \right\}$.
\begin{parts}
\part Evaluate the area integral \ $\displaystyle \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dx \, dy$ \ over the unit disk $D$.
\part Evaluate the line integral \ $\displaystyle \int_{\partial D} P \, dx + Q \, dy$ \ around $\partial D$, the unit circle with positive orientation.

\smallskip

\part Briefly explain why Green's theorem failed.
\end{parts}

\end{questions}

\end{document}

