\documentclass[12pt]{amsart}

% Package Being Used:

\usepackage{enumitem}
\usepackage{bigints}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{graphicx}
\usepackage{psfrag}
\usepackage{color}
\usepackage{url}
\usepackage{algpseudocode}
\usepackage{fancyhdr}
\usepackage{xy}
\input xy
\xyoption{all}

%\usepackage{soul} % this is a package used only temporary that allows us to underline comments that we are planning to remove in future versions of the paper.

% Paper Format and Geometry:

\voffset=-1.4mm
\oddsidemargin=17pt
\evensidemargin=17pt
\topmargin=26pt
\headheight=9pt     
\textheight=576pt
\textwidth=440.8pt
\parskip=0pt plus 4pt

% Head Labels:

\pagestyle{fancy}
\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
%\fancyhead[CE]{\fontsize{9}{11}\selectfont F. GOTTI AND M. GOTTI}
%\fancyhead[CO]{\fontsize{9}{11}\selectfont ATOMICITY AND BOUNDEDNESS OF MONOTONE PUISEUX MONOIDS}
%\fancyhead[LE,RO]{\thepage}

% Theorems-like Format and Numbering:


\newtheorem{theorem}{Theorem}%[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}

% Personalized Commands:

\newcommand\cc{\mathbb{C}}
\newcommand\ff{\mathbb{F}}
\newcommand\nn{\mathbb{N}}
\newcommand\qq{\mathbb{Q}}
\newcommand\rr{\mathbb{R}}
\newcommand\zz{\mathbb{Z}}

\newcommand\pval{\mathsf{v}_p}

\newcommand{\twopf}[4]
{
	\left\{
	\begin{array}{ll}
		#1 & \mbox{if } #2 \\
		#3 & \mbox{if } #4
	\end{array}
	\right.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
	
%	\mbox{}
%	\title{Multivariable Calculus}
%	
%	\maketitle
	
\section*{Surface Integrals and Flux}
\bigskip

\begin{problem} [Stewart, Exercises~16.5.(13,15,18)]
	Determine whether or not the following vector fields are conservative. For each conservative, find a function $f$ such that $F = \nabla f$.
	\begin{enumerate}
		\item $F(x,y,z) = \langle y^2z^3, 2xyz^3, 3xy^2z^2 \rangle$.
		\vspace{5pt}
		\item $F(x,y,z) = \langle z \cos y, 2xyz^3, xz \sin y, x \cos y \rangle$.
		\vspace{5pt}
		\item $F(x,y,z) = \langle e^x \sin yz, z e^x \cos yz, ye^x \cos yz \rangle$.
	\end{enumerate}
\end{problem}

\begin{problem}[Stewart, Exercises~16.5.(19,20)]
	Is there a vector field $G$ on $\rr^3$ satisfying the following condition.
	\begin{enumerate}
		\item $\emph{curl}(G) = \langle x \sin y, \cos y, z - xy \rangle$. Explain why?
		\item $\emph{curl}(G) = \langle x,y,z \rangle$. Explain why?
	\end{enumerate}
\end{problem}

The {\bf Laplace operator} $\nabla^2$ is defined by $\nabla^2 f = f_{xx} + f_{yy} + f_{zz}$, where $f$ is a function having continuous second partial derivatives. The Laplace operator can be applied also to a vector field $F = \langle P, Q, R \rangle$ as follows: $\nabla^2 F =  \langle \nabla^2 P, \nabla^2 Q, \nabla^2 R \rangle$.

\begin{problem} [Stewart, Exercises~16.5.(27,28,29)]
	For vector fields $F$ and $G$ on $\rr^3$, argue the following identities:
	\begin{enumerate}
		\item $\emph{div} (F \times G) = G \cdot \emph{curl} \, F - F \cdot \emph{curl} \, G$,
		\vspace{5pt}
		\item $\emph{div} (\nabla F \times \nabla G) = 0$,
		\vspace{5pt}
		\item $\emph{curl}(\emph{curl} \, F) = \emph{grad}(\emph{div} \, F) - \nabla^2 F$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercises~16.5.(30,31)]
	For the vector field $r = \langle x,y,z \rangle$, argue the following identities:
	\begin{enumerate}
		\item $\nabla \cdot (|r| r) = 4|r|$,
		\vspace{5pt}
		\item $\nabla^2 (|r|^3) = 12 r$,
		\vspace{5pt}
		\item $\nabla(1/|r|) = - r/|r|^3$,
		\vspace{5pt}
		\item $\nabla(\ln |r|) = r/|r^2|$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercises~16.5.(33,34)]
	Let $C$ be a positively-oriented, piecewise-smooth, simple closed curve in $\rr^2$ given by $r(t) = \langle x(t), y(t) \rangle$ with $a \le t \le b$, and let $n(t) = 1/|r'(t)| \langle y'(t), - x'(t) \rangle$.
	\begin{enumerate}
		\item Use Green's Theorem to argue that
		\[
			\oint_C F \cdot n \, ds = \iint_{\emph{int}(C)} \emph{div} \, F(x,y) \, dA,
		\]
		for any smooth vector field $F$ on $\rr^2$.
		\vspace{5pt}
		\item Use the previous part to argue Green's First Identity:
		\[
			\iint_{\emph{int}(C)} f \nabla^2g \, dA = \oint_C f(\nabla g) \cdot n \, ds - \iint_{\emph{int}(C)} \nabla f  \cdot \nabla g \, dA,
		\]
		for any functions $f$ and $g$ whose appropriate partial derivatives exist and are continuous.
		\vspace{5pt}
		\item Use the previous part to argue Green's Second Identity:
		\[
			\iint_{\emph{int}(C)} (f \nabla^2g - g \nabla^2f) \, dA = \oint_C (f \nabla g - g \nabla f) \cdot n \, ds,
		\]
		for any functions $f$ and $g$ whose appropriate partial derivatives exist and are continuous.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.6.34]
	Find the equation of the tangent plane to the surface parameterized by $r(u,v) = (u^2 + 1, v^3 + 1, u + v)$ at $(5,2,3)$.
\end{problem}

\begin{problem} [Cal Final, Summer 2018W]
	Find the tangent plane to the parametrized surface $r(u,v) = \langle u^2 -1, uv, v^3 \rangle$ at the point $(3,4,8)$.
\end{problem}

\begin{problem} [Stewart, Exercise~16.6.61]
	Find the area of the part of the sphere $x^2 + y^2 + z^2 = 4z$ that lies inside the paraboloid $z = x^2 + y^2$.
\end{problem}

\begin{problem}
	Deduce the formula for the flux of a vector field $F = \langle P,Q,R \rangle$ across a surface $S$ that is the graph of a smooth function $g \colon U \subset \rr^2 \to \rr$.
\end{problem}

\begin{problem} [Stewart, Exercises~16.7.(15,26)]
	Evaluate the following surface integrals.
	\begin{enumerate}
		\item $\iint_S y \, dS$, where $S$ is the surface $y = x^2 + 4z$ for $(x,z) \in [0,1] \times [0,1]$.
		\vspace{5pt}
		\item $\iint_S F \cdot dS$, where $F(x,y,z) = \langle y, -x, 2z \rangle$ and $S$ is the hemisphere $x^2 + y^2 + z^2 = 4$ and $z \ge 0$ oriented downward.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercise~16.7.43]
	A fluid has density $870~kg/m^3$ and flows with velocity $v(x,y,z) = \langle z, y^2, x^2 \rangle$, where $x,y$, and $z$ are measured in meters and the components of $v$ in meters per second. Find the rate of flow outward through the cylinder $x^2 + y^2 = 4$ for $0 \le z \le 1$. [Hint: the rate of flow outward is the flux $\iint_S (\rho v) \cdot n \, dS$.]
\end{problem}

\begin{problem} [Stewart, Exercise~16.7.49]
	Let $F$ be an inverse square field, that is, $F(r) = cr/|r|^3$ for some constant $c$, where $r = \langle x,y,z \rangle$. Show that the flux of $F$ across a sphere $S$ with center the origin is independent of the radius of $S$.
\end{problem}

\begin{problem} [Cal Final, Fall 09]
	Evaluate the surface integral
	\[
		\int \! \! \int_S z \, dS,
	\]
	where $S$ is the part of the sphere $x^2 + y^2 + z^2 = 4$ that lies inside the cylinder $x^2 + y^2 = 1$ and above the $xy$-plane. [Answer: $2\pi$]
\end{problem}

\begin{problem} [UC Berkeley Final]
	Evaluate the flux
	\[
		\int \! \! \int_T F \cdot dS
	\]
	of the vector field $F(x,y,z) = \langle -x, xy, zx \rangle$ across the triangle $T$ with vertices $(1,0,0)$, $(0,2,0)$, and $(0,0,2)$ with downward orientation. [Answer: $1/3$]
\end{problem}

\begin{problem} [UC Berkeley Final]
	Let $S$ be a surface which is contained in the plane $z = x+y$, oriented upward. Suppose that $S$ has area $2018$. Consider the constant vector field $F = \langle 3,4,5 \rangle$. Calculate the flux of $F$ across the surface $S$. [Answer: $-4036/\sqrt{3}$]
\end{problem}

\vspace{20pt}

\begin{thebibliography}{20}
	
	\bibitem{jS15} J.~Stewart: \emph{Calculus} 8th Edition, Cengage Learning, Boston 2016.
	
\end{thebibliography}
	
\end{document}