\documentclass[]{article}
\headheight=8pt     \topmargin=0pt
\textheight=624pt   \textwidth=432pt
\oddsidemargin=18pt \evensidemargin=18pt

\usepackage{amssymb,amsmath,amsthm}
\usepackage{epsfig}
\usepackage{subfigure}
\usepackage{enumerate}
\usepackage{color}
\usepackage{graphicx}
\usepackage{enumerate}

\DeclareMathOperator{\Tr}{Tr}
\pagestyle{plain}

\theoremstyle{plain}
\newtheorem{definition}{Definition}

\begin{document}

\title{MTH 352/652\\
Classwork \#1}
\date{}
\maketitle
\begin{enumerate}
\item Suppose $z=e^{x(s,t)}\sin(y(s,t))$ and $x(s,t)=st^2$, $y(s,t)=s^2t$.
\begin{enumerate}
\item Directly substitute in $x(s,t)$ and $y(s,t)$ and then compute $\displaystyle{\frac{\partial z}{\partial s}}$.
\vspace{2in}
\item Compute $\displaystyle{\frac{\partial z}{\partial s}}$ using the chain chain rule and then substitute in $x(s,t)$ and $y(s,t)$.
\vspace{2in}

\end{enumerate}

\newpage
\item If $g(s,t)=f(s^2-t^2,t^2-s^2)$ and $f$ is differentiable, show that $g$ satisfies the following equation:
\begin{equation*}
t\frac{\partial g}{\partial s}+s\frac{\partial g}{\partial t}=0.
\end{equation*}

\newpage
\item Suppose $z=f(x,y)$ has continuous second-order partial derivatives and $x=r^2+s^2$ and $y=2rs$.
\begin{enumerate}
\item Find $\displaystyle{\frac{\partial z}{\partial r}}$
\vspace{2in}
\item Find $\displaystyle{\frac{\partial^2 z}{\partial r^2}}$
\end{enumerate}
\newpage 
\item Suppose $z=f(x,y)$, where $x=g(s,t)$ and $y=h(s,t)$. Show that
\begin{equation*}
\frac{\partial^2 z}{\partial t^2}=\frac{\partial^2z}{\partial x^2}\left(\frac{\partial x}{\partial t}\right)^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}\left(\frac{\partial y}{\partial t}\right)^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{equation*}
\end{enumerate}
\end{document}