\documentclass[12pt]{article}
\usepackage{physics,graphicx,multicol}
\pagenumbering{gobble}

%%% MACROS
\renewcommand{\vec}[1]{\ensuremath{\mathbf{#1}}}
\newcommand{\pts}[1]{(#1 points)}

\newif\ifkey
\keytrue %comment to hide key

\newcommand{\key}[1]{
\ifkey
	\fbox{
	\begin{minipage}{0.875\textwidth}
	#1
	\end{minipage}
	}
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}

\begin{document}
\begin{large}
\noindent
\makebox[\textwidth]{
	\textbf{Quiz 8 | MA261 | July 25, 2017}
}
\makebox[\textwidth]{Christina Jamroz, Alden Bradford}
\begin{enumerate}
\ifkey\else
\itemsep2em
\fi

%%% PROBLEMS

%16.4.9
\item \pts{8} Use Green's theorem to evaluate $\displaystyle \int_C y^3\;dx-x^3\;dy$, where $C$ is the positively-oriented circle $x^2+y^2=4$.

\key{
Here $P=y^3$, $Q=-x^3$, so $\pdv{Q}{x}-\pdv{P}{y}=-3x^2-3y^2=-3r^2$. Do the integral in polar coordinates:
\begin{align*}
\int_0^{2\pi}\int_0^2(-3r^2)r\;dr\;d\theta&=-6\pi\int_0^2 r^3\;dr\\
&=-6\pi\frac{r^4}{4}\Big|_0^2\\
&=-24\pi
\end{align*}
}

%16.5.2
\item \pts{12} Find (a) the divergence and (b) the curl of $\vec{F}(x,y,z)=xy^2z^3\vec{i}+x^3yz^2\vec{j}+x^2y^3z\vec{k}$.

\key{
$\nabla\cdot\vec{F}=y^2z^3+x^3z^2+x^2y^3$\\
$\nabla\times\vec{F}=\\
(3x^2y^2z-2x^3yz)\vec{i}+
(3xy^2z^2-2xy^3z)\vec{j}+
(3x^2yz^2-2xyz^3)\vec{k}$
}

\end{enumerate}
\end{large}
\end{document}

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