Sonnack.com

x

""

\displaystyle\sqrt{x^2+y^2}

_

\displaystyle{z}=(a+bi)=\eta(\cos\theta+{i}\sin\theta)=\eta{e}^{i\theta}

complex1

\displaystyle\eta=|z|=\sqrt{a^2+b^2},\;\;\;\theta=\arg(z)=\arctan\frac{a}{b}

complex2

\displaystyle\nabla=\hat{x}\frac{\partial}{\partial{x}}+\hat{y}\frac{\partial}{\partial{y}}+\hat{z}\frac{\partial}{\partial{z}}

nabla

\displaystyle\nabla^2=\hat{x}\frac{\partial^2}{\partial{x^2}}+\hat{y}\frac{\partial^2}{\partial{y^2}}+\hat{z}\frac{\partial^2}{\partial{z^2}}

nabla2

\displaystyle\Lambda=\eta{e}^{{i}2\pi\kappa{x}-\omega\sigma}

sine1

\displaystyle\Lambda=\eta_1{e}^{{i}2\pi\kappa_1{x}}+\eta_2{e}^{{i}2\pi\kappa_2{x}}+\eta_3{e}^{{i}2\pi\kappa_3{x}}+\ldots

sine2

\displaystyle{e}^{{i}\pi}-1=0

Euler

\displaystyle\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}

Lorentz

\displaystyle{E}={mc}^{2}

SR1

\displaystyle{E}=\sqrt{(mc^{2})^{2}+(pc)^{2}}

SR2

\displaystyle{G}_{\mu\nu}+\Lambda{g}_{\mu\nu}=\frac{8\pi{G}}{c^4}T_{\mu\nu}

GR

\displaystyle{T}={2}\pi\sqrt{\frac{{a}^{3}}{GM}}

orb.period

\displaystyle{v}\approx\sqrt{\frac{GM}{a}}\;\;\;\textsf{note:}\;{GM}=\mu

orb.speed1

\displaystyle{v}\approx\frac{{2}\pi{a}}{T}

orb.speed2

\displaystyle{a}=\sqrt[3]{\frac{GMT^2}{{4}\pi^{2}}}

orb.radius

\displaystyle{v}_{e}=\sqrt{\frac{{2}GM}{d}}

esc.speed

\displaystyle{g}=\frac{GM}{{r}^{2}}

surf.grav

\displaystyle{S}_\text{Boltzmann}={K}_{B}\log\Omega

entropyB

\displaystyle{S}_\text{Shannon}=\sum_{i}\,\text{P}\!\left({x}_{i}\right)\,\log_{2}\text{P}\!\left({x}_{i}\right)

entropyS

\displaystyle\frac{\partial}{\partial{t}}\,{T}(x,\!t)={D}\frac{\partial^{2}}{\partial{x}^{2}}\,{T}(x,\!t)

diffusion

\displaystyle{g}(x,y)={ae}^{-\left(\frac{x^2+y^2}{c^2}\right)}

Gauss2

\displaystyle{g}_x'(x,y)=\frac{\partial}{\partial{x}}{ae}^{-\left(\frac{x^2+y^2}{c^2}\right)}=\frac{-2ax}{c^2}\;{e}^{-\left(\frac{x^2+y^2}{c^2}\right)}

Gauss2dx

\displaystyle{g}_y'(x,y)=\frac{\partial}{\partial{y}}{ae}^{-\left(\frac{x^2+y^2}{c^2}\right)}=\frac{-2ay}{c^2}\;{e}^{-\left(\frac{x^2+y^2}{c^2}\right)}

Gauss2dy

\displaystyle{g}_x''(x,y)=\frac{\partial^2}{\partial{x^2}}{ae}^{-\left(\frac{x^2+y^2}{c^2}\right)}=\frac{4ax^2-2ac^2}{c^4}\;{e}^{-\left(\frac{x^2+y^2}{c^2}\right)}

Gauss2dx2

\displaystyle{g}_y''(x,y)=\frac{\partial^2}{\partial{y^2}}{ae}^{-\left(\frac{x^2+y^2}{c^2}\right)}=\frac{4ay^2-2ac^2}{c^4}\;{e}^{-\left(\frac{x^2+y^2}{c^2}\right)}

Gauss2dy2

\displaystyle|\Psi\rangle=|0\rangle+|1\rangle

Q1

\displaystyle|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|0\rangle+|1\rangle\right)

Q2

\displaystyle|\Psi\rangle=\frac{|0\rangle}{\sqrt{2}}+\frac{|1\rangle}{\sqrt{2}}

Q3

\displaystyle\hat{H}=\hat{T}\!+\!\hat{V}=\frac{\hbar^2}{2m}(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2})+V(\bold{r},t)

Hamilton

\displaystyle{i}\hbar\frac{d}{dt}\,\Psi(x,\!t)=\hat{H}\,\Psi(x,\!t)

Schrod1

\displaystyle{i}\hbar\frac{\partial}{\partial{t}}\,\Psi(x,\!t)=\!\left[\!-\frac{\hbar^2}{2m}\,\frac{\partial^2}{\partial{x}^2}+\!V(x,\!t)\right]\!\!\Psi(x,\!t)

Schrod2

\displaystyle\exp(x)=\sum_{n=0}^{\infty}\frac{\;\;x^n}{n!}

Expon1

\displaystyle\exp(x)=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}\cdots

Expon2

\displaystyle\exp(x)=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}\cdots

Expon3

\displaystyle\exp(x)=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}\cdots

Expon4

\displaystyle\frac{d}{dx}\left[f(x)\,g(x)\right]=f(x)\,\frac{d}{dx}\left[g(x)\right]+\frac{d}{dx}\left[f(x)\right]\,g(x)

product

\displaystyle\frac{d}{dx}\left[f(g(x))\right]=\frac{d}{du}\left[f(u)\right]\,\frac{d}{dx}\left[g(x)\right]

chain

\displaystyle{g}(x)={ae}^{-\left(\frac{x}{c}\right)^2}

Gauss1

\displaystyle{g}'\!(x)=\frac{d}{dx}\,{g}(x)=\frac{-2ax}{c^2}\;{ae}^{-\left(\frac{x}{c}\right)^2}

Gauss1dx

\displaystyle{g}''\!(x)=\frac{d^2}{dx^2}\,{g}(x)=\frac{-2ac^2+4ax^2}{c^4}\;{ae}^{-\left(\frac{x}{c}\right)^2}

Gauss1dx2

\displaystyle{g}'''\!(x)=\frac{d^3}{dx^3}\,{g}(x)=\frac{-8ax^{3}+12ac^{2}x}{c^6}\;{ae}^{-\left(\frac{x}{c}\right)^2}

Gauss1dx3

\displaystyle{r}=\sqrt{(x_0)^2+(y_0)^2}\\[6pt]\alpha=atan2(y_0,x_0)\\[6pt]x_1=r\times\cos(\alpha+\theta)\\y_1=r\times\sin(\alpha+\theta)

rot1

\displaystyle{P}_{R}=complex(\cos\theta,\sin\theta)\\[16pt]\\P_0 = complex(x_0, y_0)\\P_1 = P_R \times P_0

rot2

\displaystyle{V}^{2}=\begin{bmatrix}{1}\\{0}\end{bmatrix}=|{0}\rangle

V2C

\displaystyle{V}^{3}=\begin{bmatrix}{1}\\{0}\\{0}\end{bmatrix}=|{0}\rangle

V3C

\displaystyle{V}^{4}=\begin{bmatrix}{1}\\{0}\\{0}\\{0}\end{bmatrix}=|{0}\rangle

V4C

\displaystyle{V}^{2}=\begin{bmatrix}{1}&{0}\end{bmatrix}=\langle{0}|

V2R

\displaystyle{V}^{3}=\begin{bmatrix}{1}&{0}&{0}\end{bmatrix}=\langle{0}|

V3R

\displaystyle{V}^{4}=\begin{bmatrix}{1}&{0}&{0}&{0}\end{bmatrix}=\langle{0}|

V4R

T^2 = \displaystyle\begin{bmatrix}\hat{i}_{x}&\hat{j}_{x}\\\hat{i}_{y}&\hat{j}_{y}\end{bmatrix}

T2

T^3 = \displaystyle\begin{bmatrix}\hat{i}_{x}&\hat{j}_{x}&\hat{k}_{x}\\\hat{i}_{y}&\hat{j}_{y}&\hat{k}_{y}\\\hat{i}_{z}&\hat{j}_{z}&\hat{k}_{z}\end{bmatrix}

T3

I^2 = \displaystyle\begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}

I2

I^3 = \displaystyle\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\end{bmatrix}

I3

I^4 = \displaystyle\begin{bmatrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\end{bmatrix}

I4

\displaystyle{R}^{2}_{xy}(\theta)=\begin{bmatrix}{+\cos\theta}&{-\sin\theta}\\{+\sin\theta}&{+\cos\theta}\end{bmatrix}

R2xy

\displaystyle{R}^{2}_{xy}({0}^{\circ}) =\begin{bmatrix}{+1}&{0}\\{0}&{+1}\end{bmatrix}

R2xy0

\displaystyle{R}^{2}_{xy}({90}^{\circ}) =\begin{bmatrix}{0}&{-1}\\{+1}&{0}\end{bmatrix}

R2xy90

\displaystyle{R}^{2}_{xy}({180}^{\circ})=\begin{bmatrix}{-1}&{0}\\{0}&{-1}\end{bmatrix}

R2xy180

\displaystyle{R}^{2}_{xy}({270}^{\circ})=\begin{bmatrix}{0}&{+1}\\{-1}&{0}\end{bmatrix}

R2xy270

\displaystyle{R}^{2}_{xy}({15}^{\circ})=\begin{bmatrix}{+0.96592}&{-0.25881}\\{+0.25881}&{+0.96592}\end{bmatrix}

R2xy15

\displaystyle{R}^{2}_{xy}({30}^{\circ})=\begin{bmatrix}{+0.86602}&{-0.50000}\\{+0.50000}&{+0.86602}\end{bmatrix}

R2xy30

\displaystyle{R}^{2}_{xy}({45}^{\circ})=\begin{bmatrix}{+0.70711}&{-0.70711}\\{+0.70711}&{+0.70711}\end{bmatrix}

R2xy45

\displaystyle{R}^{2}_{xy}({60}^{\circ})=\begin{bmatrix}{+0.50000}&{-0.86602}\\{+0.86602}&{+0.50000}\end{bmatrix}

R2xy60

\displaystyle{R}^{2}_{xy}({120}^{\circ})=\begin{bmatrix}{-0.50000}&{-0.86602}\\{+0.86602}&{-0.50000}\end{bmatrix}

R2xy120

\displaystyle{R}^{2}_{xy}({240}^{\circ})=\begin{bmatrix}{-0.50000}&{+0.86602}\\{-0.86602}&{-0.50000}\end{bmatrix}

R2xy240

\displaystyle{R}^{3}_{xy}(\theta)=\begin{bmatrix}{+\cos\theta}&{-\sin\theta}&{0}\\{+\sin\theta}&{+\cos\theta}&{0}\\{0}&{0}&{+1}\end{bmatrix}

R3xy

\displaystyle{R}^{3}_{xy}(0^{\circ}) =\begin{bmatrix}{+1}&{0}&{0}\\{0}&{+1}&{0}\\{0}&{0}&{+1}\end{bmatrix}

R3xy0

\displaystyle{R}^{3}_{xy}(90^{\circ}) =\begin{bmatrix}{0}&{-1}&{0}\\{+1}&{0}&{0}\\{0}&{0}&{+1}\end{bmatrix}

R3xy90

\displaystyle{R}^{3}_{xy}(180^{\circ})=\begin{bmatrix}{-1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{+1}\end{bmatrix}

R3xy180

\displaystyle{R}^{3}_{xy}(270^{\circ})=\begin{bmatrix}{0}&{+1}&{0}\\{-1}&{0}&{0}\\{0}&{0}&{+1}\end{bmatrix}

R3xy270

\displaystyle{R}^{3}_{xz}(\theta)=\begin{bmatrix}{+\cos\theta}&{0}&{-\sin\theta}\\{0}&{+1}&{0}\\{+\sin\theta}&{0}&{+\cos\theta}\end{bmatrix}

R3xz

\displaystyle{R}^{3}_{xz}(0^{\circ}) =\begin{bmatrix}{+1}&{0}&{0}\\{0}&{+1}&{0}\\{0}&{0}&{+1}\end{bmatrix}

R3xz0

\displaystyle{R}^{3}_{xz}(90^{\circ}) =\begin{bmatrix}{0}&{0}&{-1}\\{0}&{+1}&{0}\\{+1}&{0}&{0}\end{bmatrix}

R3xz90

\displaystyle{R}^{3}_{xz}(180^{\circ})=\begin{bmatrix}{-1}&{0}&{0}\\{0}&{+1}&{0}\\{0}&{0}&{-1}\end{bmatrix}

R3xz180

\displaystyle{R}^{3}_{xz}(270^{\circ})=\begin{bmatrix}{0}&{0}&{+1}\\{0}&{+1}&{0}\\{-1}&{0}&{0}\end{bmatrix}

R3xz270

\displaystyle{R}^{3}_{yz}(\theta)=\begin{bmatrix}{+1}&{0}&{0}\\{0}&{+\cos\theta}&{-\sin\theta}\\{0}&{+\sin\theta}&{+\cos\theta}\end{bmatrix}

R3yz

\displaystyle{R}^{3}_{yz}(0^{\circ}) =\begin{bmatrix}{+1}&{0}&{0}\\{0}&{+1}&{0}\\{0}&{0}&{+1}\end{bmatrix}

R3yz0

\displaystyle{R}^{3}_{yz}(90^{\circ}) =\begin{bmatrix}{+1}&{0}&{0}\\{0}&{0}&{-1}\\{0}&{+1}&{0}\end{bmatrix}

R3yz90

\displaystyle{R}^{3}_{yz}(180^{\circ})=\begin{bmatrix}{+1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{-1}\end{bmatrix}

R3yz180

\displaystyle{R}^{3}_{yz}(270^{\circ})=\begin{bmatrix}{+1}&{0}&{0}\\{0}&{0}&{+1}\\{0}&{-1}&{0}\end{bmatrix}

R3yz270

\displaystyle{R}^{4}_{xy}(\theta)=\begin{bmatrix}{+\cos\theta}&{-\sin\theta}&{0}&{0}\\{+\sin\theta}&{+\cos\theta}&{0}&{0}\\{0}&{0}&{+1}&{0}\\{0}&{0}&{0}&{+1}\end{bmatrix}

R4xy

\displaystyle{R}^{4}_{xz}(\theta)=\begin{bmatrix}{+\cos\theta}&{0}&{-\sin\theta}&{0}\\{0}&{+1}&{0}&{0}\\{+\sin\theta}&{0}&{+\cos\theta}&{0}\\{0}&{0}&{0}&{+1}\end{bmatrix}

R4xz

\displaystyle{R}^{4}_{xw}(\theta)=\begin{bmatrix}{+\cos\theta}&{0}&{0}&{-\sin\theta}\\{0}&{+1}&{0}&{0}\\{0}&{0}&{+1}&{0}\\{+\sin\theta}&{0}&{0}&{+\cos\theta}\end{bmatrix}

R4xw

\displaystyle{R}^{4}_{yz}(\theta)=\begin{bmatrix}{+1}&{0}&{0}&{0}\\{0}&{+\cos\theta}&{-\sin\theta}&{0}\\{0}&{+\sin\theta}&{+\cos\theta}&{0}\\{0}&{0}&{0}&{+1}\end{bmatrix}

R4yz

\displaystyle{R}^{4}_{yw}(\theta)=\begin{bmatrix}{+1}&{0}&{0}&{0}\\{0}&{+\cos\theta}&{0}&{-\sin\theta}\\{0}&{0}&{+1}&{0}\\{0}&{+\sin\theta}&{0}&{+\cos\theta}\end{bmatrix}

R4yw

\displaystyle{R}^{4}_{zw}(\theta)=\begin{bmatrix}{+1}&{0}&{0}&{0}\\{0}&{+1}&{0}&{0}\\{0}&{0}&{+\cos\theta}&{-\sin\theta}\\{0}&{0}&{+\sin\theta}&{+\cos\theta}\end{bmatrix}

R4zw

\displaystyle\frac{{12}+{144}+{20}+({3}\times\sqrt{4})}{7}+({5}\times{11})={9}^{2}+{0}

limerick1

\displaystyle\int_{1}^{\sqrt[3]{3}}{z}^{2}\;{dx}\times\cos\!\left(\frac{{3}\times\pi}{9}\right)=\ln\sqrt[2]{e}

limerick2