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\begin{document}
\title{Solutions, 311-XXI}
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{\bf 21( 4/ 8) Introduction to Divergence and Stokes theorems\\
          Sec. 4.(9)  p.262(3(a,b*),5,9,10*,11*,17*)}\\
\section{Problem 4.9.3b}
Use the divergence theorem to solve 4.7.5:\\
 Given ${\bf F}= x{\bf i}+y{\bf j}+(z^2-1){\bf k}$ find $\int \int 
{\bf F}\cdot {\bf n} dS $ over the closed surface bounded by the planes
$z=0$, $z=1$, and the cylinder $x^2+y^2=a^2$, where ${\bf n}$ is the unit
outward normal.\\ 
{\bf Solution:}\\
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\newpage

\section{Problem 4.9.10}
By means of Stokes' theorem, find $\int {\bf F}\cdot d{\bf r}$ around the
ellipse $x^2+y^2=1$, $z=y$, 
where ${\bf F}= x{\bf i}+(x+y){\bf j}+(x+y+z){\bf k}$.\\
{\bf Solution:}\\
\begin{center}
\psfig{figure=4_9_10.ps,width=4.in}
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\newpage

\section{Problem 4.9.11}
Evaluate $\int \int \left(\nabla \times {\bf F}\right)\cdot d{\bf S} $,
where  ${\bf F}= y{\bf i}+(x-2x^3z){\bf j}+xy^3{\bf k}$ and $S$ is the
surface of a sphere $x^2+y^2+z^2 = a^2$ above the $xy$ plane.\\
{\bf Solution:}\\
\begin{center}
\psfig{figure=4_9_11.ps,width=4.in}
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\newpage

\section{Problem 4.9.17}
Given $\phi(x,y,z) = xyz + 5$, find the surface integral of the normal component
of $\nabla \phi$ over $x^2+y^2+z^2 = 9$.\\
{\bf Solution:}\\


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