\begin{array}{ll}
\text{Bernoulli\ Trials} & {{P(E)} = \left( \frac{n}{k} \right)p_{}^{k}{(1 - p)}_{}^{n - k}} \\
\text{Cauchy-Schwarz\ Inequality} & {\left( \sum\limits_{k = 1}^{n}a_{k}^{}b_{k}^{} \right)_{}^{2} \leq \left( \sum\limits_{k = 1}^{n}a_{k}^{2} \right)\left( \sum\limits_{k = 1}^{n}b_{k}^{2} \right)} \\
\text{Cauchy\ Formula} & {f(z)\, {Ind}_{\gamma}^{}(z) = \frac{1}{2\pi i}\oint\limits_{\gamma}^{}\frac{f(\xi)}{\xi - z}\, d\xi} \\
\text{Cross\ Product} & {V_{1}^{} \times V_{2}^{} = \left| \begin{array}{lll}
i & j & k \\
\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\
\end{array} \right|} \\
\text{Vandermonde\ Determinant} & {\left| \begin{array}{llll}
1 & 1 & \cdots & 1 \\
v_{1}^{} & v_{2}^{} & \cdots & v_{n}^{} \\
v_{1}^{2} & v_{2}^{2} & \cdots & v_{n}^{2} \\
 \vdots & \vdots & \ddots & \vdots \\
v_{1}^{n - 1} & v_{2}^{n - 1} & \cdots & v_{n}^{n - 1} \\
\end{array} \right| = \prod\limits_{1 \leq i < j \leq n}^{}(v_{j}^{} - v_{i}^{})} \\
\text{Lorenz\ Equations} & \begin{array}{lll}
\overset{}{\underset{}{x}} & = & {\sigma(y - x)} \\
\overset{}{\underset{}{y}} & = & {\rho x - y - xz} \\
\overset{}{\underset{}{z}} & = & {- \beta z + xy} \\
\end{array} \\
\text{Maxwell's\ Equations} & \left\{ \begin{array}{lll}
{\nabla \times \overset{\leftharpoonup}{\underset{}{B}} - \,\frac{1}{c}\,\frac{\partial\overset{\leftharpoonup}{\underset{}{E}}}{\partial t}} & = & {\frac{4\pi}{c}\,\overset{\leftharpoonup}{\underset{}{j}}} \\
{\nabla \overset{\leftharpoonup}{\underset{}{E}}} & = & {4\pi\rho} \\
{\nabla \times \overset{\leftharpoonup}{\underset{}{E}}\, + \,\frac{1}{c}\,\frac{\partial\overset{\leftharpoonup}{\underset{}{B}}}{\partial t}} & = & \overset{\leftharpoonup}{\underset{}{0}} \\
{\nabla \overset{\leftharpoonup}{\underset{}{B}}} & = & 0 \\
\end{array} \right. \\
\text{Einstein\ Field\ Equations} & {R_{\mu\nu}^{} - \frac{1}{2}\, g_{\mu\nu}^{}\, R = \frac{8\pi G}{c_{}^{4}}\, T_{\mu\nu}^{}} \\
\text{Ramanujan\ Identity} & {\frac{1}{(\sqrt{\varphi\sqrt{5}} - \varphi)e_{}^{\frac{25}{\pi}}} = 1 + \frac{e_{}^{- 2\pi}}{1 + \frac{e_{}^{- 4\pi}}{1 + \frac{e_{}^{- 6\pi}}{1 + \frac{e_{}^{- 8\pi}}{1 + \ldots}}}}} \\
\text{Another\ Ramanujan\ identity} & {\sum\limits_{k = 1}^{\infty}\frac{1}{2_{}^{\lfloor k \varphi\rfloor}} = \frac{1}{2_{}^{0} + \frac{1}{2_{}^{1} + \cdots}}} \\
\text{Rogers-Ramanujan\ Identity} & {1 + {\sum\limits_{k = 1}^{\infty}\frac{q_{}^{k_{}^{2} + k}}{(1 - q)(1 - q_{}^{2})\cdots(1 - q_{}^{k})}} = {\prod\limits_{j = 0}^{\infty}\frac{1}{(1 - q_{}^{5j + 2})(1 - q_{}^{5j + 3})}},\text{ \,}\text{ \,}{for}\ |q| < 1.} \\
\text{Commutative\ Diagram} & \begin{array}{lll}
H & \leftarrow & K \\
 \downarrow & & \uparrow \\
H & \rightarrow & K \\
\end{array} \\
\end{array}
