\documentclass[12pt]{article}
\usepackage{latexsym}
\usepackage{epsfig}
\input spec_frac
\begin{document}
\title{311-TEST III}
\author{Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ }
\maketitle
\maketitle

INSTRUCTIONS:\\
ALL PROBLEMS ARE WEIGHTED EQUALLY!\\
DO ALL FOUR PROBLEMS!\\
One page of notes is allowed.

\input table
\newpage
{\bf (1 -- 25 pts)}
Here $\phi, psi$ are continuously differentiable scalar fields.  
If $C$ is a simple closed loop and $S$ is a smooth surface bounded by $C$: 
\begin{enumerate} 
\item (5 pts) Show that 
\[ \oint_C \psi \nabla \phi \cdot d{\bf R}
= -\oint_C \phi \nabla \psi \cdot d{\bf R } \ .\]
\ \ \\[1in]
\item (10 pts) Show that 
\[ \int \int_S \nabla \phi \times \nabla \psi \cdot d{\bf S}
= \oint_C \phi \nabla \psi \cdot d{\bf R } \ .\]
\ \ \\[1in]
\item (10 pts) 
Show that
\[ \oint_C \phi d{\bf R } = -\int \int_S \nabla \phi \times d{\bf S} \ .\]
\end{enumerate}
\newpage
$\ $
\newpage
{\bf (2 -- 25pts)}
Use Stokes' theorem to evaluate
\[ I := \oint_C \left[ x \sin{y}{\bf i} - y\sin{x}{\bf j} + 
                      (2x+y)z^2{\bf k}\right] \cdot d{\bf R} \]
along the path consisting of straight line segments successively joining
the points $P_0=(0,0,0)$ to $P_1=(\pi/2,0,0)$ to $P_2=(\pi/2,0,1)$ to 
    $P_3=(0,0,1)$ $P_4=(0,\pi/2,1)$ to $P_5=(0,\pi/2,0)$ and back to $P_0$. \\
({\bf NOTE:} you are to compute this integral by converting to
appropriate surface integral(s); do not evaluate the line integral directly!)


\newpage
$\ $
\newpage
{\bf (3 -- 25 pts)}
Find $I := \oint_C {\bf F}\cdot d{\bf R}$ where $C$ is the ellipse of 
intersection of the cylinder $x^2+y^2 = 1$ with the plane $z = x$ and 
${\bf F} = (x+y){\bf i} + y{\bf j} + (x + y + z){\bf k}$: \\
{\bf (a -- 12 pts)} By direct evaluation of the line integral. \\[3.5in]
{\bf (b -- 13 pts)} By applying Stoke's theorem to 
convert to a surface integral over an appropriate surface.
\newpage
$\ $
\newpage

{\bf (4 -- 25pts)} 
A torus (donut) has major radius $A$ and minor radius $a$. 
The parametrization ${\bf R}(u,v)$ in terms of the toroidal angle
$u$ and the poloidal angle $v$, is given by
\begin{eqnarray*}
x & = & A\cos{u} + a \cos{u} \cos{v}  \ ,\\
y & = & A\sin{u} + a \sin{u} \cos{v}  \ ,\\
z & = & a \sin{v}  \ ,
\end{eqnarray*}
with $0 \le u \mbox{\ , } v \le 2\pi$.
Compute the area element
\[ d{\bf S} = \frac{\partial {\bf R}}{\partial u}\times
\frac{\partial {\bf R}}{\partial v} \ .\]
Show that the area of the torus is $4\pi^2 A a$.\\

%Find the volume of the region bounded by the surface \\
%\[z = \cos{(x^2+y^2)}\ ,\]
%the cylinder \\
%\[ x^2 + y^2 = \frac{\pi}{4} \ ,\] 
%and the $x-y$ plane.
\end{document}
