\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{Limits and Derivatives (MATH 53)}
%	
%	\maketitle

\section*{The Chain Rule}
\bigskip

\begin{problem}[jS, Exercises~14.5.53]
	If $z = f(x,y)$, where $x = r \cos \theta$ and $y = r \sin \theta$, show that
	\[
		\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = \frac{\partial^2 z}{\partial r^2} + \frac 1{r^2} \frac{\partial^2 z}{\partial \theta^2} + \frac 1r \frac{\partial z}{\partial r}.
	\]
\end{problem}

\begin{problem}[jS, Exercises~14.5.54]
	Suppose that $z = f(x,y)$, where $x = g(s,t)$ and $y = h(s,t)$.
	Show that
	\[
		\frac{\partial^2 z}{\partial t^2} = \frac{\partial^2 z}{\partial x^2} \bigg(\frac{\partial x}{\partial t}\bigg)^2 + 2 \frac{\partial^2 z}{\partial x \partial y}\frac{\partial x}{\partial t} \frac{\partial y}{\partial t} + \frac{\partial^2 z}{\partial y^2}\bigg(\frac{\partial y}{\partial t} \bigg)^2 + \frac{\partial z}{\partial x} \frac{\partial^2 x}{\partial t^2} + \frac{\partial z}{\partial y} \frac{\partial^2 y}{\partial t^2}.
	\]
\end{problem}

\begin{problem}[UC Berkeley midterm]
	Assume that the two equations $f(x,y,z) = 0$ and $g(x,y,z) = 0$ together implicitly define $y$ and $z$ as functions of $x$. Find formulas for $y' = dy/dx$ and $z' = dz/dx$ in terms of the partial derivatives of $f$ and $g$. [Hint: Use chain rule with respect to $x$ to differentiate both equations $f(x,y(x),z(x)) = 0$ and $g(x,y(x),z(x)) = 0$ and then solve the $2 \times 2$ remaining system.]
\end{problem}
	
\vspace{20pt}

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