Examples

Start from a classic

• $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

  Quadratic formula

  Algebra

  Open
• $$e^{i\pi }+1=0$$

  Euler's identity

  Algebra

  Open
• $${\int }_{-\infty }^{\infty }{e}^{-x^2}dx=\sqrt{\pi }$$

  Gaussian integral

  Calculus

  Open
• $$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

  Taylor series

  Calculus

  Open
• $$\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$$

  Basel problem

  Calculus

  Open
• $$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$$

  Normal distribution

  Statistics

  Open
• $$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$

  Bayes' theorem

  Statistics

  Open
• $$\begin{array}{rl}\nabla \cdot \mathbf{E}& =\frac{\rho }{{\epsilon }_0}\\ \nabla \times \mathbf{B}& ={\mu }_0\mathbf{J}+{\mu }_0{\epsilon }_0\frac{\partial \mathbf{E}}{\partial t}\end{array}$$

  Maxwell's equations

  Physics

  Open
• $$i\hslash \frac{\partial }{\partial t}\Psi =\hat{H}\Psi$$

  Schrödinger equation

  Physics

  Open
• $$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\Lambda g_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$$

  Einstein field equations

  Physics

  Open
• $$\sigma (\mathbf{z}{)}_i=\frac{e^{z_i}}{{\sum }_{j=1}^K{e}^{z_j}}$$

  Softmax

  Machine learning

  Open
• $$L=-\sum _i{y}_i\log {\hat{y}}_i$$

  Cross-entropy loss

  Machine learning

  Open
• $$\text{ce}CH4+2O2->CO2+2H2O$$

  Combustion of methane

  Chemistry

  Open
• $$\zeta (s)=\sum _{n=1}^{\infty }\frac{1}{n^s}=\prod _p\frac{1}{1-p^{-s}}$$

  Riemann zeta

  Number theory

  Open
• $$\det (A)=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$$

  Determinant

  Linear algebra

  Open
• $$\Delta =18abcd-4b^{3}d+b^2{c}^2-4a{c}^3-27a^2{d}^2$$

  Cubic formula discriminant

  Algebra

  Open
• $$(x+y{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{y}^k$$

  Binomial theorem

  Algebra

  Open
• $$a^2-b^2=(a-b)(a+b)$$

  Difference of squares

  Algebra

  Open
• $$\sum _{k=0}^{\infty }{r}^k=\frac{1}{1-r},|r|<1$$

  Geometric series

  Algebra

  Open
• $${\log }_{b}x=\frac{\ln x}{\ln b}$$

  Logarithm change of base

  Algebra

  Open
• $$a^2+b^2=c^2$$

  Pythagorean theorem

  Geometry

  Open
• $$c^2=a^2+b^2-2ab\cos \gamma$$

  Law of cosines

  Geometry

  Open
• $$A=\sqrt{s(s-a)(s-b)(s-c)},s=\frac{a+b+c}{2}$$

  Heron's formula

  Geometry

  Open
• $$V=\frac{4}{3}\pi r^3,S=4\pi r^2$$

  Sphere volume and surface

  Geometry

  Open
• $$\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$

  Angle addition

  Trigonometry

  Open
• $$e^{i\theta }=\cos \theta +i\sin \theta$$

  Euler's formula

  Trigonometry

  Open
• $$(\cos \theta +i\sin \theta {)}^n=\cos n\theta +i\sin n\theta$$

  De Moivre's theorem

  Trigonometry

  Open
• $$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

  Derivative definition

  Calculus

  Open
• $${\int }_a^b{f}^{\prime }(x)dx=f(b)-f(a)$$

  Fundamental theorem of calculus

  Calculus

  Open
• $$\int udv=uv-\int vdu$$

  Integration by parts

  Calculus

  Open
• $$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$$

  Chain rule

  Calculus

  Open
• $${\oint }_C(Ldx+Mdy)={\iint }_D\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dA$$

  Green's theorem

  Calculus

  Open
• $${\iiint }_V(\nabla \cdot \mathbf{F})dV=\iint \mathbf{F}\cdot d\mathbf{S}$$

  Divergence theorem

  Calculus

  Open
• $$n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$$

  Stirling's approximation

  Calculus

  Open
• $$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\infty }\left(a_n\cos nx+b_n\sin nx\right)$$

  Fourier series

  Calculus

  Open
• $$\hat{f}(\xi )={\int }_{-\infty }^{\infty }f(x)e^{-2\pi ix\xi }dx$$

  Fourier transform

  Calculus

  Open
• $$\mathcal{L}\{f\}(s)={\int }_0^{\infty }f(t)e^{-st}dt$$

  Laplace transform

  Calculus

  Open
• $$\frac{\partial u}{\partial t}=\alpha {\nabla }^{2}u$$

  Heat equation

  Differential equations

  Open
• $$\frac{{\partial }^{2}u}{\partial t^2}=c^2{\nabla }^{2}u$$

  Wave equation

  Differential equations

  Open
• $$\rho \left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot \nabla \mathbf{v}\right)=-\nabla p+\mu {\nabla }^2\mathbf{v}+\mathbf{f}$$

  Navier–Stokes

  Differential equations

  Open
• $$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$$

  Logistic growth

  Differential equations

  Open
• $$A\mathbf{v}=\lambda \mathbf{v},\det (A-\lambda I)=0$$

  Eigenvalue equation

  Linear algebra

  Open
• $$A=U\Sigma V^T$$

  Singular value decomposition

  Linear algebra

  Open
• $$R(\theta )=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

  Rotation matrix

  Linear algebra

  Open
• $$|⟨u,v⟩{|}^2\le ⟨u,u⟩\cdot ⟨v,v⟩$$

  Cauchy–Schwarz inequality

  Linear algebra

  Open
• $$\hat{\beta }=(X^{T}X{)}^{-1}{X}^{T}y$$

  Linear regression

  Statistics

  Open
• $$\sqrt{n}\left({\bar{X}}_n-\mu \right)\stackrel{d}{\to }\mathcal{N}(0,{\sigma }^2)$$

  Central limit theorem

  Statistics

  Open
• $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

  Binomial distribution

  Statistics

  Open
• $$P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}$$

  Poisson distribution

  Statistics

  Open
• $$t=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}}$$

  t-statistic

  Statistics

  Open
• $$\mathrm{Var}(X)=\mathbb{E}[X^2]-{\left(\mathbb{E}[X]\right)}^2$$

  Variance

  Statistics

  Open
• $$H(X)=-\sum _{x}p(x)\log p(x)$$

  Entropy

  Statistics

  Open
• $$D_{\mathrm{KL}}(P\parallel Q)=\sum _{x}P(x)\log \frac{P(x)}{Q(x)}$$

  Kullback–Leibler divergence

  Machine learning

  Open
• $${\theta }_{t+1}={\theta }_t-\eta {\nabla }_{\theta }\mathcal{L}({\theta }_t)$$

  Gradient descent

  Machine learning

  Open
• $$\mathrm{Attention}(Q,K,V)=\mathrm{softmax}\left(\frac{Q{K}^T}{\sqrt{d_k}}\right)V$$

  Attention

  Machine learning

  Open
• $$\sigma (x)=\frac{1}{1+e^{-x}}$$

  Sigmoid

  Machine learning

  Open
• $${\theta }_t={\theta }_{t-1}-\alpha \frac{{\hat{m}}_t}{\sqrt{{\hat{v}}_t}+ϵ}$$

  Adam update

  Machine learning

  Open
• $$V(s)=\underset{a}{\max }\left[R(s,a)+\gamma \sum _{s^{\prime }}P(s^{\prime }\mid s,a)V(s^{\prime })\right]$$

  Bellman equation

  Machine learning

  Open
• $$T(n)=aT\left(\frac{n}{b}\right)+f(n)$$

  Master theorem

  Computer science

  Open
• $$f(n)=O(g(n))⟺\exists c,n_0:\forall n\ge n_0,f(n)\le cg(n)$$

  Big-O definition

  Computer science

  Open
• $$E=m{c}^2$$

  Mass–energy equivalence

  Physics

  Open
• $$i\hslash \frac{\partial }{\partial t}|\psi ⟩=\hat{H}|\psi ⟩$$

  Time-dependent Schrödinger

  Physics

  Open
• $$\Delta x\Delta p\ge \frac{\hslash }{2}$$

  Heisenberg uncertainty

  Physics

  Open
• $$\mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times \mathbf{B}\right)$$

  Lorentz force

  Physics

  Open
• $$\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$$

  Lorentz factor

  Physics

  Open
• $$S=k_B\ln \Omega$$

  Boltzmann entropy

  Physics

  Open
• $$B_{\nu }(T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{h\nu /k_{B}T}-1}$$

  Planck's law

  Physics

  Open
• $$\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}}-\frac{\partial \mathcal{L}}{\partial q}=0$$

  Euler–Lagrange equation

  Physics

  Open
• $$\text{ce}6CO2+6H2O->[\text{light}]C6H12O6+6O2$$

  Photosynthesis

  Chemistry

  Open
• $$\text{ce}N2+3H2<=>2NH3$$

  Ammonia synthesis (Haber)

  Chemistry

  Open
• $$E=E^{\circ }-\frac{RT}{nF}\ln Q$$

  Nernst equation

  Chemistry

  Open
• $$\mathrm{pH}=\mathrm{p}{K}_a+\log \frac{[{\mathrm{A}}^{-}]}{[\mathrm{HA}]}$$

  Henderson–Hasselbalch

  Chemistry

  Open
• $$k=A{e}^{-E_a/RT}$$

  Arrhenius equation

  Chemistry

  Open
• $$\Delta G=\Delta H-T\Delta S$$

  Gibbs free energy

  Chemistry

  Open
• $$\pi (x)\sim \frac{x}{\ln x}$$

  Prime number theorem

  Number theory

  Open
• $$a^{p-1}\equiv 1(\mathrm{mod}p)$$

  Fermat's little theorem

  Number theory

  Open
• $$\phi (n)=n\prod _{p\mid n}\left(1-\frac{1}{p}\right)$$

  Euler's totient

  Number theory

  Open
• $$\phi =\frac{1+\sqrt{5}}{2}$$

  Golden ratio

  Number theory

  Open
• $$\pi =3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\cdots }}}$$

  Continued fraction

  Number theory

  Open
• $$\neg (p\wedge q)⟺\neg p\vee \neg q$$

  De Morgan's laws

  Logic & sets

  Open
• $$A\setminus B=A\cap B^c$$

  Set difference identity

  Logic & sets

  Open
• $$|A\cup B|=|A|+|B|-|A\cap B|$$

  Inclusion–exclusion

  Logic & sets

  Open
• $$|x|=\{\begin{array}{ll}x& x\ge 0\\ -x& x<0\end{array}$$

  Piecewise function

  Algebra

  Open
• $$\begin{array}{ccc}A& \stackrel{f}{\to }& B\\ g\downarrow & & \downarrow h\\ C& \underset{k}{\to }& D\end{array}$$

  Commutative diagram

  Category theory

  Open
• $$A=P{\left(1+\frac{r}{n}\right)}^{nt}$$

  Compound interest

  Finance

  Open
• $$\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0$$

  Black–Scholes

  Finance

  Open
• $$\mathrm{NPV}=\sum _{t=0}^T\frac{C_t}{(1+r{)}^t}$$

  Net present value

  Finance

  Open