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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<H2 ALIGN="center"><A NAME="SECTION00000000000000000000">
Math examples</A>
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<DIV ALIGN="RIGHT">
<IMG
WIDTH="474" HEIGHT="160" ALIGN="BOTTOM" BORDER="0"
SRC="img1.gif"
ALT="\begin{eqnarray}
\phi(\lambda) & = & \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\in...
...\epsilon , \bar{\epsilon} & = & \mbox{actual/average energy loss}
\end{eqnarray}">
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Since (<A HREF="sampleMathImages.html#eqn:stress-sr">6</A>) or (<A HREF="sampleMathImages.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>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{} 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><IMG
WIDTH="555" HEIGHT="114" ALIGN="LEFT"
SRC="img8.gif"
ALT="\begin{eqnarray}V \mathbf{\pi}^{sr} & = & \left< \sum_i M_i \mathbf{V}_i \mathbf...
...m_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right>
\end{eqnarray}">
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<BR CLEAR="ALL"></DIV><P></P>
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<DIV ALIGN="RIGHT">
</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{}\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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<TD NOWRAP ALIGN="CENTER"><IMG
WIDTH="4" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img9.gif"
ALT="$\displaystyle \begin{align}B_{ij}^\alpha & = \left(B_{ij}^\alpha\right)_0 + \le...
...} + \frac{\d N_k^\beta}{\d X_i} \frac{\d N_k^\alpha}{\d X_j} \right)\end{align}$"></TD></TR>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<BR><HR>
<ADDRESS>
<I>Michel Goossens</I>
<BR><I>1999-03-30</I>
</ADDRESS>
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