\documentclass[12pt]{article}
\usepackage{fullpage,pandora,amsmath,euler}

\begin{document}
\title{Mathematics 226, Spring 2000}
\author{Final Exam}
\date{May 3, 2000}
\maketitle

\begin{enumerate}
\def\grad{{\boldsymbol\nabla}}
\item Simplify the following vector expressions as much as possible.
  \begin{enumerate}
  \item $\displaystyle
         \boldsymbol{v} \times (\boldsymbol{v} - \boldsymbol{w}) -
         \boldsymbol{w} \times (\boldsymbol{v} + \boldsymbol{w})$
%        $\displaystyle
%         \boldsymbol{u} \times (\boldsymbol{u} - \boldsymbol{v}) -
%         \boldsymbol{v} \times (\boldsymbol{u} + \boldsymbol{v})$
  \item $\displaystyle \grad \cdot \grad (f^2) - 2 \, f \, \grad \cdot
         \grad f + \grad \times \grad g$
%        $\displaystyle \grad \times \grad f + 
%         \grad \cdot \grad (g^2) - 2 \, g \, \grad \cdot \grad g$  
  \end{enumerate}
\hfill(5+5)


\item Let $z = \ln\dfrac{x}{y}$.
  \begin{enumerate}
  \item Find the maximum rate of change of $z$, and the direction in
  which it occurs, at the point
  $(1,-1)$.
%  $(1,1)$.
  \item Find an equation for the line tangent to the level curve of
  $z$ at   
  $(1,1)$.
  \end{enumerate}
\hfill(5+5)


\item Evaluate the following integrals.
  \begin{enumerate}
  \item $\displaystyle \int_0^1 \int_{\sqrt x}^1 \sin y^3 \, dy \, dx$
        %$\displaystyle \int_0^1 \int_{\sqrt y}^1 \sin x^3 \, dx \, dy$
  \item $\displaystyle \iiint_B (x^2 + y^2 + z^2) \, dx \, dy \, dz$
  
        where $B$ is the unit sphere.
  \item $\displaystyle \int_C (e^{\sin x} + y) \, dx -
         (e^{\cos y} - x) \, dy$
         
        where $C$ is the perimeter of the square with vertices
        $(\pm 2, \pm 2)$.
%        $(\pm 3, \pm 3)$.
        Is the vector field $\boldsymbol{F} = (e^{\sin x} + y) \,
         \boldsymbol{i} - 
         (e^{\cos y} - x) \, \boldsymbol{j}$ conservative? Explain.
  \end{enumerate}
\hfill(10+10+10)


\item
\begin{enumerate}
\item
Find all local maximum and minimum values of the function
\[
  f(x,y) = x^2 - 2 \, y^2 
%  f(x,y) = 2 \, x^2 - 2 y^2 
\]
on the disk $x^2 + y^2 \leq 1$.
\item Use a method of your choice to compute the distance of the plane
\[
  x - 2 \, y + 3 \, z = 4
%  3 \, x - 2 \, y + z = 4
\]
to the origin.
\end{enumerate}
\hfill(10+10)


\item Use the Divergence Theorem to compute the flux of the vector
field
\[
  \boldsymbol{F} = (xy, \ln (1+z^2), z^2) 
%  \boldsymbol{F} = (xz, y^2, \cos(1+y^2)) 
\]
across the surface of the tetrahedron with vertices $(0,0,0)$,
$(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.
\hfill(10)


\item
\begin{enumerate}
\item State Stokes' Theorem and explain \emph{briefly} the meaning of
  every letter that appears in the equation.
  
\item Show that Stokes' Theorem reduces to Green's Theorem when
  restricted to the $xy$-plane (i.e.\ if none of the quantities
  involved has any dependence on $z$).

\item
Show that the area of a plane region with boundary $C$ is given by
\[
  A = \frac12 \int_C x \, dy - y \, dx.
\]
\item Evaluate this formula  when $C$ is a circle of radius $R$.
\hfill(5+5+5+5)

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


