$$\begin{array}{cc}\hfill \mathrm{\text{Bernoulli Trials}}\hfill & \hfill P(E)=\left(\frac{n}{k}\right)p_{}^k{(1-p)}_{}^{n-k}\hfill \\ \hfill \mathrm{\text{Cauchy-Schwarz Inequality}}\hfill & \hfill {\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)\hfill \\ \hfill \mathrm{\text{Cauchy Formula}}\hfill & \hfill f(z)·{\mathrm{Ind}}_{\gamma }^{}(z)=\frac{1}{2\pi i}\underset{\gamma }{\overset{}{\oint }}\frac{f(\xi )}{\xi -z}d\xi \hfill \\ \hfill \mathrm{\text{Cross Product}}\hfill & \hfill V_1^{}\times V_2^{}=\left|\begin{array}{ccc}\hfill i\hfill & \hfill j\hfill & \hfill k\hfill \\ \hfill \frac{\partial X}{\partial u}\hfill & \hfill \frac{\partial Y}{\partial u}\hfill & \hfill 0\hfill \\ \hfill \frac{\partial X}{\partial v}\hfill & \hfill \frac{\partial Y}{\partial v}\hfill & \hfill 0\hfill \end{array}\right|\hfill \\ \hfill \mathrm{\text{Vandermonde Determinant}}\hfill & \hfill \left|\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ \hfill v_1^{}\hfill & \hfill v_2^{}\hfill & \hfill \cdots \hfill & \hfill v_n^{}\hfill \\ \hfill v_1^2\hfill & \hfill v_2^2\hfill & \hfill \cdots \hfill & \hfill v_n^2\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill v_1^{n-1}\hfill & \hfill v_2^{n-1}\hfill & \hfill \cdots \hfill & \hfill v_n^{n-1}\hfill \end{array}\right|=\prod _{1\le i<j\le n}^{}(v_j^{}-v_i^{})\hfill \\ \hfill \mathrm{\text{Lorenz Equations}}\hfill & \hfill \begin{array}{ccc}\hfill \underset{}{\overset{\dot{}}{x}}\hfill & \hfill =\hfill & \hfill \sigma (y-x)\hfill \\ \hfill \underset{}{\overset{\dot{}}{y}}\hfill & \hfill =\hfill & \hfill \rho x-y-xz\hfill \\ \hfill \underset{}{\overset{\dot{}}{z}}\hfill & \hfill =\hfill & \hfill -\beta z+xy\hfill \end{array}\hfill \\ \hfill \mathrm{\text{Maxwell's Equations}}\hfill & \hfill \{\begin{array}{ccc}\hfill \nabla \times \underset{}{\overset{\leftharpoonup }{B}}-\frac{1}{c}\frac{\partial \underset{}{\overset{\leftharpoonup }{E}}}{\partial t}\hfill & \hfill =\hfill & \hfill \frac{4\pi }{c}\underset{}{\overset{\leftharpoonup }{j}}\hfill \\ \hfill \nabla ·\underset{}{\overset{\leftharpoonup }{E}}\hfill & \hfill =\hfill & \hfill 4\pi \rho \hfill \\ \hfill \nabla \times \underset{}{\overset{\leftharpoonup }{E}}+\frac{1}{c}\frac{\partial \underset{}{\overset{\leftharpoonup }{B}}}{\partial t}\hfill & \hfill =\hfill & \hfill \underset{}{\overset{\leftharpoonup }{0}}\hfill \\ \hfill \nabla ·\underset{}{\overset{\leftharpoonup }{B}}\hfill & \hfill =\hfill & \hfill 0\hfill \end{array}\hfill \\ \hfill \mathrm{\text{Einstein Field Equations}}\hfill & \hfill R_{\mu \nu }^{}-\frac{1}{2}{g}_{\mu \nu }^{}R=\frac{8\pi G}{c_{}^4}{T}_{\mu \nu }^{}\hfill \\ \hfill \mathrm{\text{Ramanujan Identity}}\hfill & \hfill \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 }}}}\hfill \\ \hfill \mathrm{\text{Another Ramanujan identity}}\hfill & \hfill \sum _{k=1}^{\infty }\frac{1}{2_{}^{\lfloor k·\phi \rfloor }}=\frac{1}{2_{}^0+\frac{1}{2_{}^1+\cdots }}\hfill \\ \hfill \mathrm{\text{Rogers-Ramanujan Identity}}\hfill & \hfill 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.\hfill \\ \hfill \mathrm{\text{Commutative Diagram}}\hfill & \hfill \begin{array}{ccc}\hfill H\hfill & \hfill \leftarrow \hfill & \hfill K\hfill \\ \hfill \downarrow \hfill & \hfill \hfill & \hfill \uparrow \hfill \\ \hfill H\hfill & \hfill \to \hfill & \hfill K\hfill \end{array}\hfill \end{array}$$