additional math examples #531

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Hakim El Hattab 2013-08-12 09:24:29 -04:00
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</section> </section>
<section> <section>
<h2>The Lorenz Equations</h2>
\[\begin{aligned} \[\begin{aligned}
\dot{x} &amp; = \sigma(y-x) \\ \dot{x} &amp; = \sigma(y-x) \\
\dot{y} &amp; = \rho x - y - xz \\ \dot{y} &amp; = \rho x - y - xz \\
@ -43,6 +44,30 @@
\end{aligned} \] \end{aligned} \]
</section> </section>
<section>
<h2>The Cauchy-Schwarz Inequality</h2>
\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]
</section>
<section>
<h2>A Cross Product Formula</h2>
\[\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
\mathbf{i} &amp; \mathbf{j} &amp; \mathbf{k} \\
\frac{\partial X}{\partial u} &amp; \frac{\partial Y}{\partial u} &amp; 0 \\
\frac{\partial X}{\partial v} &amp; \frac{\partial Y}{\partial v} &amp; 0
\end{vmatrix} \]
</section>
<section>
<h2>An Identity of Ramanujan</h2>
\[ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
{1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
</section>
</div> </div>
</div> </div>