
\subsection{\bf Equivalence of Integral and Differential Formulation of
Laplace's Equation}

\begin{slide}

\begin{itemize}
\item Therefore, if $F$ is stationary, i.e.
\[
\frac{\partial F}{\partial \left( \eta (\underline{x}) \right)} = 0
\]
for all $\eta (\underline{x})$ consistent with boundary
conditions, then $\nabla^2 \varphi = 0$

\item Thus, Laplace's equation is entirely equivalent to
VARIATIONAL PRINCIPLE
\begin{eqnarray*}
\delta F & = & 0 \\
F & = & \int_{\Omega} \frac{1}{2} \left( \nabla \varphi \right)^2
d^3 \underline{x} \\
\varphi (\underline{x}) & = & \mbox{known}\ f(x)\ \mbox{on}\
\partial\Omega
\end{eqnarray*}

We have converted differential equation to an integral . . .
\end{itemize}

\end{slide}


