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$\begin{array}{cc}\text{Bernoulli Trials}& P(E)\text{Probability of event E: Get exactly k heads in n coin flips.}=\left(\genfrac{}{}{0ex}{}{n}{k}\right)\text{Number of ways to get exactly k heads in n coin flips}{p\text{Probability of getting heads in one flip}}_{}^{k\text{Number of heads}}{(1-p)\text{Probability of getting tails in one flip}}_{}^{n-k\text{Number of tails}}\\ \text{Cauchy-Schwarz Inequality}& {\displaystyle {\left(\sum _{k=1}^n{a}_k^{}{b}_k^{}\right)}_{}^2\le \left(\sum _{k=1}^n{a}_k^2\right)\left(\sum _{k=1}^n{b}_k^2\right)}\\ \text{Cauchy Formula}& {\displaystyle f(z)·{Ind}_{\gamma }^{}(z)=\frac{1}{2\pi i}\underset{\gamma }{\overset{}{\oint }}\frac{f(\xi )}{\xi -z}d\xi }\\ \text{Cross Product}& V_1^{}\times V_2^{}=\left|\begin{array}{ccc}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}& {\displaystyle \left|\begin{array}{cccc}1& 1& \cdots & 1\\ v_1^{}& v_2^{}& \cdots & v_n^{}\\ v_1^2& v_2^2& \cdots & v_n^2\\ ⋮& ⋮& \ddots & ⋮\\ v_1^{n-1}& v_2^{n-1}& \cdots & v_n^{n-1}\end{array}\right|=\prod _{1\le i<j\le n}^{}(v_j^{}-v_i^{})}\\ \text{Lorenz Equations}& \begin{array}{rcl}\underset{}{\overset{\dot{}}{x}}& =& \sigma (y-x)\\ \underset{}{\overset{\dot{}}{y}}& =& \rho x-y-xz\\ \underset{}{\overset{\dot{}}{z}}& =& -\beta z+xy\end{array}\\ \text{Maxwell's Equations}& {\displaystyle \{\begin{array}{rcl}\nabla \text{}\times \underset{}{\overset{\leftharpoonup }{B}}-\frac{1}{c}\frac{\partial \text{}\underset{}{\overset{\leftharpoonup }{E}}}{\partial \text{}t}& =& \frac{4\pi }{c}\underset{}{\overset{\leftharpoonup }{j}}\\ \nabla \text{}·\underset{}{\overset{\leftharpoonup }{E}}& =& 4\pi \rho \\ \nabla \text{}\times \underset{}{\overset{\leftharpoonup }{E}}+\frac{1}{c}\frac{\partial \text{}\underset{}{\overset{\leftharpoonup }{B}}}{\partial \text{}t}& =& \underset{}{\overset{\leftharpoonup }{0}}\\ \nabla \text{}·\underset{}{\overset{\leftharpoonup }{B}}& =& 0\end{array}}\\ \text{Einstein Field Equations}& {\displaystyle 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+\dots }}}}\\ \text{Another Ramanujan identity}& {\displaystyle \sum _{k=1}^{\infty }\frac{1}{2_{}^{\lfloor k·\text{}\varphi \rfloor }}=\frac{1}{2_{}^0+\frac{1}{2_{}^1+\cdots \frac{1}{2_{}^1+\frac{1}{2_{}^2+\cdots \frac{1}{2_{}^3+\frac{1}{2_{}^5+\dots }}}}}}}\\ \text{Rogers-Ramanujan Identity}& {\displaystyle 1+\sum _{k=1}^{\infty }\frac{q_{}^{k_{}^2+k}}{(1-q)(1-q_{}^2)\cdots (1-q_{}^k)}\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})}\frac{1}{(1-q_{}^2)(1-q_{}^3)}\times \frac{1}{(1-q_{}^7)(1-q_{}^8)}\times \cdots ,for|q|<1.}\\ \text{Commutative Diagram}& \begin{array}{ccc}H& \leftarrow & K\\ \downarrow & & \uparrow \\ H& \to & K\end{array}\end{array}$

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