|
@@ -36,6 +36,7 @@
|
|
|
</section>
|
|
|
|
|
|
<section>
|
|
|
+ <h2>The Lorenz Equations</h2>
|
|
|
\[\begin{aligned}
|
|
|
\dot{x} & = \sigma(y-x) \\
|
|
|
\dot{y} & = \rho x - y - xz \\
|
|
@@ -43,6 +44,30 @@
|
|
|
\end{aligned} \]
|
|
|
</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>
|
|
|
+
|
|
|
+ \[\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
|
|
|
+ \mathbf{i} & \mathbf{j} & \mathbf{k} \\
|
|
|
+ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
|
|
|
+ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
|
|
|
+ \end{vmatrix} \]
|
|
|
+ </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>
|