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\title{Solutions, 311-XXVII}
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Sec. 5.2 (1*,2*,8a*), S6.74*, p.134 \\

\section{Problem 5.2.1, p.284}
Evaluate
\[ \int \int_S \left[
\frac{1}{R} \nabla \phi - \phi \nabla\left(\frac{1}{R}\right)
\right] \cdot d{\bf S} \]
over the surface of the sphere $(x-3)^2 + y^2 + z^2 = 25$,
where $\phi = xyz + 5$.
{\bf Solution:}\\
\newpage
\section{Problem 5.2.2, p.284}
Evaluate
\[ \int \int_S \left[
\phi \nabla\left(\frac{1}{R}\right) - \frac{1}{R} \nabla \phi
\right] \cdot d{\bf S} \]
\begin{enumerate}
\item over the surface of the ellipsoid 
$x^2/9 + y^2/16 + z^2/25 = 1$,
where $\phi = x^2+y^2-2z^2+4$.
\item over the surface of the cylindrical pillbox bounded by
$x^2+y^2 = 25$ and $z=\pm10$, 
where $\phi = x^2-z^2 + 5$.
\end{enumerate}
{\bf Solution:}\\
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\newpage

\section{Problem 5.2.8a, p.285}
Let the domain $D$ be bounded by the surface $S$ as in the divergence theorem,
and assume that all fields satisfy the appropriate differentiability conditions.
Prove the identity:
\[ \int \int \int_D \nabla\phi\cdot \nabla\times{\bf F} dV
= \int \int_S {\bf F}\times\nabla\phi\cdot d{\bf S} \ .\]
{\bf Solution:}\\

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\section{Problem S6.74, p.134}
If a region $V$ bounded by a surface $S$ has a continuous charge (or mass)
distribution of density $\rho$, the potential $\phi(P)$ at a point
$P$ is defined by
\[\phi = \int \int \int_V \frac{\rho dV}{r} \ .\]
Deduce the following under suitable assumptions:
\begin{enumerate}
\item $\int \int_S {\bf E}\cdot d{\bf S} = 4\pi \int \int \int_V \rho dV$,
where ${\bf E} = -\nabla \phi$.
\item $\nabla^2 \phi = -4\pi \rho$ (Poisson's equation) at all points
$P$ where charges exist, and $\nabla^2 \phi = 0$ (Laplace's equation)
where no charges exist.
\end{enumerate}
{\bf Solution:}\\
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