+
+ The Lorenz Equations
+
+
+ \[\begin{aligned}
+ \dot{x} & = \sigma(y-x) \\
+ \dot{y} & = \rho x - y - xz \\
+ \dot{z} & = -\beta z + xy
+ \end{aligned} \]
+
+
+
+
+ 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
+
+
+ \[\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
+ \mathbf{i} & \mathbf{j} & \mathbf{k} \\
+ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
+ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
+ \end{vmatrix} \]
+
+
+
+
+ The probability of getting \(k\) heads when flipping \(n\) coins is
+
+
+ \[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]
+
+
+
+
+ An Identity of Ramanujan
+
+
+ \[ \frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} =
+ 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}}
+ {1+\frac{e^{-8\pi}} {1+\ldots} } } } \]
+
+
+
+
+ A Rogers-Ramanujan Identity
+
+
+ \[ 1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots =
+ \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}\]
+
+
+
+
+ Maxwell’s Equations
+
+
+ \[ \begin{aligned}
+ \nabla \times \vec{\mathbf{B}} -\, \frac1c\,
+ \frac{\partial\vec{\mathbf{E}}}{\partial t} & =
+ \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi
+ \rho \\
+ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\,
+ \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
+ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}
+ \]
+
+
+