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<H2 ALIGN="center"><A NAME="SECTION00000000000000000000">
Math examples</A>
</H2>
<DIV ALIGN="RIGHT">
<IMG
 WIDTH="474" HEIGHT="160" ALIGN="BOTTOM" BORDER="0"
 SRC="img1.gif"
 ALT="\begin{eqnarray}
\phi(\lambda) &amp; = &amp; \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\in...
...\epsilon , \bar{\epsilon} &amp; = &amp; \mbox{actual/average energy loss}
\end{eqnarray}">
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</DIV>

<P>
Since&nbsp;(<A HREF="sampleMathThumb.html#eqn:stress-sr">6</A>) or&nbsp;(<A HREF="sampleMathThumb.html#gdef">7d</A>) should hold for arbitrary 
<!-- MATH
 $\delta\mathbf{c}$
 -->
<IMG
 WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img2.gif"
 ALT="$ \delta\mathbf{c}$">-vectors, it is clear that 
<!-- MATH
 $\mathcal{N}(A) = \mathcal{R}(B)$
 -->
<IMG
 WIDTH="104" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.gif"
 ALT="$ \mathcal{N}(A) = \mathcal{R}(B)$"> and that when <IMG
 WIDTH="68" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="$ y=B(x)$"> one has...
<BR>
...the <A NAME="tex2html3"
 HREF="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html">Pythagorians</A> knew infinitely many solutions in integers to 
<!-- MATH
 $a^2+b^2=c^2$
 -->
<IMG
 WIDTH="88" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.gif"
 ALT="$ a^2+b^2=c^2$">. 
That no non-trivial integer solutions exist for 
<!-- MATH
 $a^n+b^n=c^n$
 -->
<IMG
 WIDTH="93" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
 ALT="$ a^n+b^n=c^n$"> with integers <IMG
 WIDTH="43" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.gif"
 ALT="$ n&gt;2$"> has long 
been suspected (<A NAME="tex2html4"
 HREF="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html">Fermat, c.1637</A>). Only during the current decade has this been proved (<A NAME="tex2html5"
 HREF="http://www.pbs.org:80/wgbh/nova/proof">Wiles, 1995</A>).

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<P ALIGN="RIGHT"><A NAME="eqn:stress-sr"></A>
<!-- MATH
 \begin{eqnarray}
\htmlimage{thumbnail=.4} V \mathbf{\pi}^{sr} & = & \left<  \sum_i M_i \mathbf{V}_i \mathbf{V}_i
+ \sum_i \sum_{j>i} \mathbf{R}_{ij} \mathbf{F}_{ij}\right> \\\nonumber
                 & = & \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i
  + \sum_{i}\sum_{j>i}\sum_\alpha\sum_\beta \mathbf{r}_{i\alpha j\beta}\mathbf{f}_{i\alpha j\beta}
  - \sum_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha}   \right>
\end{eqnarray}
 -->
<A NAME="eqn:stress-sr"></A><A NAME="eqnarray52"
 HREF="img8.gif"><IMG
 WIDTH="140" HEIGHT="29" ALIGN="LEFT"
 SRC="Timg8.gif"
 ALT="\begin{eqnarray}V \mathbf{\pi}^{sr} &amp; = &amp; \left&lt; \sum_i M_i \mathbf{V}_i \mathbf...
...m_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right&gt;
\end{eqnarray}"></A>
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<BR CLEAR="ALL"></DIV><P></P>
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</DIV><A NAME="bgdefs"></A><A NAME="bdef"></A><A NAME="b0def"></A><A NAME="budef"></A><A NAME="gdef"></A>
<!-- MATH
 \begin{subequations}
\htmlimage{thumbnail=.3}\begin{align} B_{ij}^\alpha     & =
\left(B_{ij}^\alpha\right)_0 + \left(B_{ij}^\alpha\right)_a\\
  \left(B_{ij}^\alpha\right)_0  & = \frac{1}{2}\left(\frac{\d N_i^\alpha}{\d X_j}
        + \frac{\d N_j^\alpha} {\d X_i} \right)\\
  \left(B_{ij}^\alpha\right)_a  & = H_{ij}^{\alpha \beta} a^\beta\\
  H_{ij}^{\alpha \beta}         & =
    \frac{1}{2}\left( \frac{\d N_k^\alpha}{\d X_i} \frac{\d N_k^\beta}{\d X_j} 
    + \frac{\d N_k^\beta}{\d X_i} \frac{\d N_k^\alpha}{\d X_j} \right)\end{align}
\end{subequations}
 -->
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...} + \frac{\d N_k^\beta}{\d X_i} \frac{\d N_k^\alpha}{\d X_j} \right)\end{align}$"></A></TD></TR>
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<ADDRESS>
<I>Michel Goossens</I>
<BR><I>1999-03-30</I>
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