$$\begin{array}{cc}\mathrm{\text{Bernoulli Trials}}& P(E)=\left(\frac{n}{k}\right)p_{}^k{(1-p)}_{}^{n-k}\\ \mathrm{\text{Cauchy-Schwarz Inequality}}& {\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)\\ \mathrm{\text{Cauchy Formula}}& f(z)·{\mathrm{Ind}}_{\gamma }^{}(z)=\frac{1}{2\pi i}\underset{\gamma }{\overset{}{\oint }}\frac{f(\xi )}{\xi -z}d\xi \\ \mathrm{\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|\\ \mathrm{\text{Vandermonde Determinant}}& \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^{})\\ \mathrm{\text{Lorenz Equations}}& \begin{array}{ccc}\underset{}{\overset{\dot{}}{x}}& =& \sigma (y-x)\\ \underset{}{\overset{\dot{}}{y}}& =& \rho x-y-xz\\ \underset{}{\overset{\dot{}}{z}}& =& -\beta z+xy\end{array}\\ \mathrm{\text{Maxwell's Equations}}& \{\begin{array}{ccc}\nabla \mathrm{\text{}}\times \underset{}{\overset{\leftharpoonup }{B}}-\frac{1}{c}\frac{\partial \mathrm{\text{}}\underset{}{\overset{\leftharpoonup }{E}}}{\partial \mathrm{\text{}}t}& =& \frac{4\pi }{c}\underset{}{\overset{\leftharpoonup }{j}}\\ \nabla \mathrm{\text{}}·\underset{}{\overset{\leftharpoonup }{E}}& =& 4\pi \rho \\ \nabla \mathrm{\text{}}\times \underset{}{\overset{\leftharpoonup }{E}}+\frac{1}{c}\frac{\partial \mathrm{\text{}}\underset{}{\overset{\leftharpoonup }{B}}}{\partial \mathrm{\text{}}t}& =& \underset{}{\overset{\leftharpoonup }{0}}\\ \nabla \mathrm{\text{}}·\underset{}{\overset{\leftharpoonup }{B}}& =& 0\end{array}\\ \mathrm{\text{Einstein Field Equations}}& R_{\mu \nu }^{}-\frac{1}{2}{g}_{\mu \nu }^{}R=\frac{8\pi G}{c_{}^4}{T}_{\mu \nu }^{}\\ \mathrm{\text{Ramanujan Identity}}& \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi )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 }}}}\\ \mathrm{\text{Another Ramanujan identity}}& \sum _{k=1}^{\infty }\frac{1}{2_{}^{\lfloor k·\mathrm{\text{}}\phi \rfloor }}=\frac{1}{2_{}^0+\frac{1}{2_{}^1+\cdots }}\\ \mathrm{\text{Rogers-Ramanujan Identity}}& 1+\sum _{k=1}^{\infty }\frac{q_{}^{k_{}^2+k}}{(1-q)(1-q_{}^2)\cdots (1-q_{}^k)}=\prod _{j=0}^{\infty }\frac{1}{(1-q_{}^{5j+2})(1-q_{}^{5j+3})},for\left|q\right|<1.\\ \mathrm{\text{Commutative Diagram}}& \begin{array}{ccc}H& \leftarrow & K\\ \downarrow & & \uparrow \\ H& \to & K\end{array}\end{array}$$