
\subsection{\bf Example for Two-Dimensional Triangular Elements}

\begin{slide}
\begin{itemize}

\item Show that integrating over one triangle gives
\begin{eqnarray*}
F & = & \int_{\Delta} \frac{1}{2} \left( \nabla\varphi \right)^2 d^2 x \\
& = & \frac{1}{4} \left\lbrack \varphi_0\, \varphi_1\, \varphi_2
\right\rbrack \left\lbrack \begin{array}{ccc}
c_1 + c_2 & -c_2 & -c_1 \\
-c_2 & c_2 + c_0 & -c_0 \\
-c_1 & -c_0 & c_0 + c_1 \\ \end{array} \right\rbrack \quad
\left\lbrack \begin{array}{c}
\varphi_0 \\
\varphi_1 \\
\varphi_2 \\ \end{array} \right\rbrack \\
\end{eqnarray*}
where $c_j = \cot\theta_j$
\end{itemize}

\end{slide}

