% Simplified version of an UMTYMP template.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% If the text is too small, change 10pt to 12pt
\documentclass[11pt]{article}
\RequirePackage{amsmath,amsthm,amssymb,latexsym,amsthm,verbatim,xspace,amstext,xspace,setspace}

% Here's where you edit the class, section and title information

% Set a fixed margin along all sides
\usepackage[margin=1in, nohead, pdftex]{geometry}
\usepackage{enumerate,graphicx}
% Re-adjust the top margin if desired
\topmargin -0.2in
% These packages used for spacing in the header and throughout


% This next command affects the page numbering.  "empty" is best for short handouts
\pagestyle{empty}

% Uncomment one of the following to set the spacing to single, 1.5 or double spacing
\singlespacing
%\onehalfspacing
%\doublespacing

% This prevents paragraphs from begin indented.  Comment it out to have indentation.
\parindent 0ex
\setlength{\textheight}{9.5in}
% Finally we start!
\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is the "header" at the top.  You shouldn't have to edit these lines!
% Instead, chance the Class, Section and Title above.
\begin{center}{ \Large{Worksheet}  \\\large{11/9/2017} }\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% At this point, you might have some introductory text, or jump straight
% into an ``enumerate'' environment for the problem statements.
\bigskip
\begin{enumerate}
\item \begin{enumerate}\item Compute the Jacobian determinant $\dfrac{\partial(x,y)}{\partial(u,v)}$ of the transformation $x=u+v$, $y=-u+2v$.\vspace{2in}
\item Compute the Jacobian determinant $\dfrac{\partial(x,y)}{\partial(u,v)}$ of the transformation $u=xy$, $v=y/x$.\\(\emph{Note: you need to solve for $x$  and $y$ in terms of $u$, $v$.})\vspace{3in}
\item Compute the Jacobian determinant $\dfrac{\partial(x,y,z)}{\partial(u,v,w)}$ of the transformation $x=uv+w$, $y=w$, $z=v$.\vspace{3in}
\end{enumerate}
\newpage
\item Use the polar coordinates to compute\[\iint_{R}\cos(x^2+y^2)dA\] where $R$ is the annulus $1\le x^2+y^2\le 4$.\vspace{2.5in}
\item Determine the new region that we get by applying the given transformation to the region $R$.
\begin{enumerate}
\item $R$ is the region bounded by $y=x$, $y=3x$, $x+y=4$, and the tranformation is $x=u-v$, $y=u+v$.\vspace{3in}
\item $R$ is the region bounded by $x=0$, $x=1$, $y=0$, $y=\sqrt{x}$, and the tranformation is $x=u^2$, $y=v$.\vspace{3in}
\end{enumerate}
\item Use the cylindrical coordinates to compute\[\iiint_{E}(8+x+y)dV\] where $E$ is the solid bounded by $z=16-x^2-y^2$ and $z=0$.\vspace{3.5in}
\item Use the spherical coordinates to compute\[\iiint_{U}\sqrt{x^2+y^2+z^2}dV\] where $U$ is the unit ball $x^2+y^2+z^2=1$.\vspace{3.5in}
\item Use a suitable change of variables to compute the area of the ellipse \[\frac{x^2}{9}+\frac{y^2}{4}=1\]
\vspace{3.5in}
\item Compute \[\iint_{R}\sqrt{xy}dA\] where $R$ is the region bounded by $xy=1$, $xy=9$, $y=x$, $y=2x$, using the change of variables $u=xy$, $y=vx$.
\end{enumerate}

\end{document}











