LaTeX Practice

Topics

• LaTeX Art

Resources

• Cheatsheet
• LaTeX Shortcuts
• MathJax Basics
• LaTeX Suite Github

$${\int }_{-2\pi }^{2\pi }f(x)dx$$ $$\text{Injective (one-to-one):}\forall x_1,x_2\in X,f(x_1)=f(x_2)\to x_1=x_2$$

$\text{Injective (one-to-one): }\forall x_1,x_2\in X,f(x_1)=f(x_2)\to x_1=x_2$
$\omega =2\pi f=\frac{2\pi }{T}$
$k=2\pi \xi =\frac{2\pi }{\lambda }$
$v_{\text{phase}}=\frac{\omega }{k}=f\lambda =\frac{\lambda }{\tau }=\frac{2\pi f}{k}=\frac{\omega \lambda }{2\pi }$
$v_{\text{group}}=\frac{\partial E}{\partial p}=\frac{\partial \omega }{\partial k}$
$v_{\text{phase}}{v}_{\text{group}}=c^2$
$v=\frac{\omega }{|k|\cos \theta }=\frac{\lambda }{T\cos \theta }$
$E=\hslash \omega =\frac{h}{T}=hf$
$p=\hslash k=\frac{h}{\lambda }=h\xi$
$n\lambda =d\sin \theta$
$\lambda =\frac{{\lambda }_C}{\sqrt{\frac{\kappa }{mc}+{\left(\frac{K}{m{c}^2}\right)}^2}},{\lambda }_c=\frac{h}{mc}=\frac{2\pi }{k_C}$
$m_e{c}^2\approx 511\text{ keV}$
$\frac{1}{{\lambda }_{t^2}}-\frac{1}{{\lambda }^2}=\frac{1}{{\lambda }_C^2}$
$R=\frac{\lambda }{2\text{NA}}$
$\frac{e^2}{4\pi \epsilon }=1.44\text{ eV}\cdot \text{nm}$
$\hslash c=197\text{ eV }\cdot \text{nm}$
$\alpha =\frac{e^2}{4\pi \epsilon \hslash c}\approx \frac{1}{137}$
$E\lambda \approx 400\pi \text{ eV}\cdot \text{nm}$
$\Delta t\Delta \omega \ge \frac{1}{2},\Delta x\Delta k\ge \frac{1}{2}$
$\Delta t\Delta E\ge \frac{\hslash }{2},\Delta x\Delta p\ge \frac{\hslash }{2}$
$f(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }a(k)e^{ikx}dk$
$a(k)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f(x)e^{-ikx}dx$
$\text{To time: }x\to t,k\to \omega ,i\to -i$
$f(x)=\frac{1}{2\pi }\iint f(x^{\prime })e^{}$
$\text{Commutative: }A\cup B=B\cup A,A\cap B=B\cap A$
$\text{Associative: }(A\cup B)\cup C=A\cup (B\cup C),(A\cap B)\cap C=A\cap (B\cap C)$
$\text{Distributive: }A\cap (B\cup C)=(A\cap B)\cup (A\cap C),A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
$\text{De Morgan’s Laws: }(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime },(A\cap B{)}^{\prime }UBKCK=A^{\prime }\cup B^{\prime }$
$\text{Relation: }aRb=(a,b)\in R$
$\Delta t\Delta \omega$
$\Delta t\Delta \omega$
$E=K$
$\frac{1}{2}{m}_e{v}^2=\frac{hc}{\lambda }-\varphi$
$\frac{1}{2}m_{e}(0.1c)^{2}>\frac{hc}{\lambda}-\phi$$

$$\sum _{i=0}^{\infty }f(x)$$ $$\begin{array}{r}1000\\ \underline{\times 11}\\ 1000\\ \underline{1000}\\ 11000\end{array}$$ $$\Vert \begin{array}{ccc}2& 2& 2\\ 3& & \end{array}\Vert$$

$\sum 2$
${\mathbf{A}}^{-1}={\mathbf{A}}^{\top }\to \mathbf{A}\text{ is orthonormal}$

$\frac{kg{m}^2}{s^2}$
$\frac{1}{\pi \sqrt{2a}}{\int }_{-\infty }^{\infty }\frac{\sin (ka)}{k}{e}^{i\left(kx-\frac{h{k}^{2}t}{2m}\right)}\mathrm{d}k$
KK 5.12: Ascent of Sap in Trees
Find the maxt height that water will rise in a tree
$\tau \ln \left(\frac{n(h)}{n_Q}+Mgh=\tau \ln \left(\frac{n(0)}{n_Q}\right)\right)$
$Mgh=\tau \left(\ln \left(\frac{n(0)}{n_Q}\right)-\ln \left(\frac{rn(0)}{n_Q}\right)\right)$
$Mgh=\tau \ln \left(\frac{1}{r}\right)\Rightarrow h=\frac{\tau }{Mg}\ln \left(\frac{1}{r}\right)$
$R_{nl}(r)=\sqrt{{\left(\frac{2}{n{a}_0}\right)}^3\frac{(n-l-1)!}{2n(n+l)!}}{e}^{-r/n{a}_0}{\left(\frac{2r}{n{a}_0}\right)}^l\left[L_{n-l-1}^{2l+1}\left(\frac{2r}{n{a}_0}\right)\right]$