
\subsection{\bf Variation in Integral}

\begin{slide}

\begin{itemize}
\item Consider variation

Let $\varphi \rightarrow \varphi + \eta$ with $\eta = 0$ on
$\partial\Omega$ and $\eta$ a function of $\underline{x}$ over $d$
dimensional $\Omega$

\item Then this change in $\varphi$ induces a change 
$F \rightarrow F + \delta F$ where, to first order in $\eta$, 
\[ 
\delta F \approx \int_{\Omega} \left( \nabla\varphi \cdot
\nabla\eta \right) d^3 \underline{x} 
\]

\item We have Gauss's theorem
\[
\int_{\Omega} \nabla \underline{u} d^3 x = \int_{\partial\Omega}
\underline{u} \cdot d\underline{s}
\]
where $d\underline{s}$ is differential on surface $\partial\Omega$

\item apply to $\underline{u} = \eta\nabla\varphi$
\[
\nabla \underline{u} = \nabla\eta \cdot \nabla\varphi + \eta \nabla^2
\varphi
\]
where $\eta = 0$ on $\partial\Omega$

\item Then
\[
\int_{\, \Omega} \nabla\varphi\, \nabla\eta d^3 \underline{x} = - \int
\eta \nabla^2 \varphi d^3 \underline{x}
\]
relates $\delta F$ to $\nabla^2 \varphi$ where
\[
\nabla\varphi = \left( \frac{\partial\varphi}{\partial x},
\frac{\partial\varphi}{\partial y}, \frac{\partial\varphi}{\partial z}
\right)
\]
and
\[
\left( \nabla\varphi \right)^2 = 
\left( \frac{\partial\varphi}{\partial x}\right)^2 +
\left( \frac{\partial\varphi}{\partial y}\right)^2 +
\left( \frac{\partial\varphi}{\partial z}\right)^2 
\]

\end{itemize}
\end{slide}

