

\subsection{\bf 4.3\hspace{1em} The Derivative Method for $\chi^2$---I}

\begin{slide}
\begin{itemize}
\item Let's first generalize the previous discussion to $n$
parameters---getting a vector $F_j$ of first and matrix $M_{jk}$ of
second derivatives
\begin{eqnarray}
F_j = \frac{\partial\chi^2}{\partial\alpha_j} & = & \sum^N_{i=1}
\frac{(t_i - e_i)}{\sigma^2_i}\, \frac{\partial t_i}{\partial\alpha_j}
\nonumber \\
M_{jk} = \frac{1}{2}\, \frac{\partial^2 \chi^2}{\partial\alpha_j
\partial\alpha_k} & = & \sum^N_{j,k=1} \frac{1}{\sigma^2_i}\,
\frac{\partial t_i}{\partial\alpha_j} \,\frac{\partial t_i}{\partial\alpha_k}
\end{eqnarray}

\item Note that---in our approximation---$M$ is positive
semi-definite---which is also a technical advantage.
\end{itemize}
\end{slide}

