
\subsection{\bf $\frac{d}{dx}$ as an Operator and Its}
\subsection{\bf Scalar Products}

\begin{slide}

\begin{itemize}
\item Consider the Integral

\begin{displaymath}
I = \int^b_a g(x) \frac{d}{dx} f(x) dx = \langle g\vert D\vert f\rangle
\end{displaymath}
where $D$ is operator $\frac{d}{dx}$.  

\item Integrate by parts.

\begin{eqnarray*}
I  = & - & \int^b_a f(x)\, \frac{d}{dx}\, g(x)\, dx \\
     & + & \big\lbrack f(x) g(x) \big\rbrack^b_a \\
     & = & f(b)\, g(b) - f(a)\, g(a) \\
\end{eqnarray*}

\item The last term 
\begin{displaymath}
\big\lbrack f(x) g(x) \big\rbrack^b_a  = f(b)\, g(b) - f(a)\, g(a)
\end{displaymath} 

\item We need to get rid of this surface (endpoint) term.

\end{itemize}

\end{slide}

