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\begin{document}
\title{Solutions, 311-XXVI}
\maketitle

Sec. 5.(5) Schaum's, p.132(40*,41,42*,52*,54*), Text,p.247,(16*)\\

\section{Problem S6.40, p.132}
Evaluate $\oint (x^2-2xy)sx +(x^2y+3)dy$ around the boundary of the region
defined by $y^2 = 8x$ and $x=2$ (a) directly (b) by Green's theorem.\\
({\em Ans.} $128/5$)\\
{\bf Solution:}\\
\newpage
\section{Problem S6.42}
Evaluate $\oint (3x^2+2y)dx - (x+3\cos{y}) dy$ around the parallelogram having
vertices at $(0,0)$, $(2,0)$, $(3,1)$ and $(1,1)$.
\\ 
({\em Ans.} $-6$)\\
{\bf Solution:}\\
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\newpage

\section{Problem S6.52, p.133}
Evaluate $\int \int_S {\bf F}\cdot d{\bf S}$ with
${\bf F} = 2xy{\bf i} + yz^2 {\bf j} + xz {\bf k}$ and $S$ is:
\begin{enumerate}
\item The parallelepiped $x=0$, $y=0$, $z=0$, $x=2$, $y=1$, $z=3$.
\item The surface of the region bounded by the planes
$x=0$, $y=0$, $z=0$, $y=3$ and $x+2z = 6$.
\end{enumerate}
({\em Ans.} (a) $30$, (b) $351/2$)\\
{\bf Solution:}\\

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\newpage
\section{Problem S6.54, p.133}
Evaluate $\int \int_S {\bf r}\cdot {\bf n} dS $ where 
\begin{enumerate}
\item $S$ is the sphere of radius 2 with center at $(0,0,0)$. 
\item $S$ is the surface of the cube 
bounded by the planes $x= \pm 1$, $y= \pm 1$, $z= \pm 1$.
\item $S$ is the surface bounded by the paraboloid $z=4-(x^2+y^2)$ and
the $xy$ plane.
\end{enumerate}
({\em Ans.} (a) $32\pi$, (b) $24$, (c) $24\pi$)\\
{\bf Solution:}\\
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\newpage

\section{Problem 4.7.16, p.247}
A torus (donut) has major radius $A$ and minor radius $a$ (see figure).
Derive the parametrization ${\bf R}(u,v)$ in terms of the toroidal angle
$u$ and the poloidal angle $v$, where
\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*}
Show that the area of the torus is $4\pi^2 A a$.\\
{\bf Solution:}\\
\end{document}

