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@@ -36,6 +36,7 @@
+ The Lorenz Equations
\[\begin{aligned}
\dot{x} & = \sigma(y-x) \\
\dot{y} & = \rho x - y - xz \\
@@ -43,6 +44,30 @@
\end{aligned} \]
+
+ The Cauchy-Schwarz Inequality
+
+ \[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]
+
+
+
+ A Cross Product Formula
+
+ \[\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
+ \mathbf{i} & \mathbf{j} & \mathbf{k} \\
+ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
+ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
+ \end{vmatrix} \]
+
+
+
+ An Identity of Ramanujan
+
+ \[ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
+ 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
+ {1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
+
+