这是我第一次来这里，而且我对乳胶也还很陌生，但我已经设法（借助我在网上找到的模板）制作了这张漂亮的三列公式表。我的问题是我需要在第一页的中间列上方留出一个间隙，只是一个空白处。我试过使用，\vspace但没有成功。想知道我是否可以在这里找到帮助。
（抱歉，格式化我尽力了）
\documentclass[12pt,landscape]{article}
\usepackage{multicol}
\usepackage{calc}
\usepackage{ifthen}
\usepackage[portrait]{geometry}
\usepackage{hyperref}
\usepackage{amsmath}
% To Do:
% \listoffigures \listoftables
% \setcounter{secnumdepth}{0}
% This sets page margins to .5 inch if using letter paper, and to 1cm
% if using A4 paper. (This probably isn't strictly necessary.)
% If using another size paper, use default 1cm margins.
\ifthenelse{\lengthtest { \paperwidth = 11in}}
{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} }
{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}}
{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
}
% Turn off header and footer
\pagestyle{empty}
% Redefine section commands to use less space
\makeatletter
\renewcommand{\section}{\@startsection{section}{1}{0mm}%
{-1ex plus -.5ex minus -.2ex}%
{0.5ex plus .2ex}%x
{\normalfont\large\bfseries}}
\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}%
{-1explus -.5ex minus -.2ex}%
{0.5ex plus .2ex}%
{\normalfont\normalsize\bfseries}}
\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}%
{-1ex plus -.5ex minus -.2ex}%
{1ex plus .2ex}%
{\normalfont\small\bfseries}}
\makeatother
% Define BibTeX command
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
% Don't print section numbers
\setcounter{secnumdepth}{0}
\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt plus 0.5ex}
% -----------------------------------------------------------------------
\begin{document}
\raggedright
\footnotesize
\begin{multicols}{3}
% multicol parameters
% These lengths are set only within the two main columns
%\setlength{\columnseprule}{0.25pt}
\setlength{\premulticols}{1pt}
\setlength{\postmulticols}{1pt}
\setlength{\multicolsep}{1pt}
\setlength{\columnsep}{2pt}
\section{Mechanics}
$v=u+at$ \\
\vspace{1mm}
$s=\frac{v+u}{2}t$. \\
\vspace{1mm}
\vspace{1mm}
$v^2=u^2+2as$ \\
\vspace{1mm}
$s=ut+\frac{1}{2}at^2$ \\
\vspace{1mm}
$p=mv$ \\
\vspace{1mm}
$\sum F= \frac{dp}{dt}$ \\
\vspace{1mm}
$\int_{t_1}^{t_2}Fdt=p_2-p_1$ - Impulse \\
\vspace{1mm}
$P=\frac{F}{A}$ \\
\vspace{1mm}
$F=ma$\\
\vspace{1mm}
$E_k=1/2mv^2$ \\
\vspace{1mm}
$\mu=Fd_\perp$ - moment \\
\vspace{1mm}
$E_p=mgh$\\
\vspace{1mm}
$W=\int F \cdot dx=F\Delta x$\\
\vspace{1mm}
$P_{avg}=\frac{\Delta W}{\Delta t}$\\
\vspace{1mm}
$P_{inst}=\frac{dW}{dT}=F\cdot v$\\
\vspace{1mm}
\section{Torque}
$\tau=Fd_{\perp}$ - About midpoint\\
\vspace{1mm}
$\tau=r \times F$ - vector torque\\
\vspace{1mm}
$W=\tau(\theta_2-\theta_1)=\tau \Delta \theta$\\
\vspace{1mm}
$P=\tau\omega$\\
\vspace{1mm}
$L=r\times p=m r \times v $ - AM particle\\
\vspace{1mm}
$L=I\omega$ - AM rigid body\\
\vspace{1mm}
$\sum \tau=\frac{dL}{dt}$
\vspace{1mm}
\section{Rotational motion}
$\omega=\frac{d\theta}{dt}$\\
\vspace{1mm}
$\alpha=\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$\\
\vspace{1mm}
Equations for constant $\alpha$:
$\theta=\theta_o+\omega_ot=\frac{1}{2}\alpha t^2$\\
\vspace{1mm}
$\omega=\omega_o+\alpha t$\\
\vspace{1mm}
$\omega^2=\omega_o^2+2\alpha(\theta-\theta_o)$\\
\vspace{1mm}
$E_k=\frac{1}{2}I\omega^2$\\
\vspace{1mm}
$I=\int r^2 dm$
\vspace{1mm}
\section{SHM}
$\omega=2\pi f=\frac{2\pi}{T}$\\
\vspace{1mm}
$F=-kx$\\
\vspace{1mm}
$E_s=\frac{1}{2}kx^2$\\
\vspace{1mm}
$E=\frac{1}{2}mv^2+\frac{1}{2}kx^2=const$\\
\vspace{1mm}
$T_s=2\pi \sqrt{\frac{m}{k}}$\\
\vspace{1mm}
$T_{sp}=2\pi \sqrt{\frac{L}{g}}$\\
\vspace{1mm}
$T_{physP}=2\pi \sqrt{\frac{I}{mgd}}$
\section{Waves}
$v=f\lambda$\\
\vspace{1mm}
$k=\frac{2\pi}{\lambda}$\\
\vspace{1mm}
$\omega=2\pi f$\\
\vspace{1mm}
General Wavefunction for free wave:\\
$y(x,t)=Asin(\omega t-kx)$\\
\vspace{1mm}
Wave Equation:\\
$\frac{\partial^2y}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2y}{\partial t^2}$\\
\vspace{1mm}
$E=hf=\hbar\omega=\frac{hc}{\lambda}$\\
\vspace{1mm}
$eV_o=hf-\phi$ - photoelectric effect\\
\vspace{1mm}
$f_L=\frac{v\pm v_L}{v \pm v_s}f_s$\\
\vspace{1mm}
$c=1/\sqrt{\mu_o \varepsilon_o}$\\
\vspace{1mm}
$n=c/v$\\
\vspace{1mm}
$n_asin(\theta_a)=n_bsin(\theta_b)$\\
\vspace{1mm}
$sin(\theta_c)=\frac{n_b}{n_a}$\\
\vspace{1mm}
$dsin(\theta)=m\lambda$ - constructive\\
\vspace{1mm}
$dsin(\theta)=(m+\frac{1}{2})\lambda$ - destructive\\
\vspace{1mm}
$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$ - object,image distance\\
\vspace{1mm}
\section{Elasticity}
\newlength{\MyLen}
\settowidth{\MyLen}{\texttt{letterpaper}/\texttt{a4paper} \ }
$ Stess=F/A$\\
\vspace{1mm}
$Strain=\delta l / l_o$\\
\vspace{1mm}
\textit{Youngs Mod}$=Stess/Strain$\\
\section{Thermodynamics}
\settowidth{\MyLen}{\texttt{multicol} }
$pV=nRT$\\
\vspace{1mm}
$moles=m/A$\\
\vspace{1mm}
Kinetic energy per molecule\\
$E_k=n/2k_bT$- n=Degrees of freedom\\
\vspace{1mm}
$v_{rms}=\sqrt{3k_bT/m}=\sqrt{3RT/A}$\\
\vspace{1mm}
$\gamma=C_p/C_v$\\
\vspace{1mm}
$\gamma=5/3$ - Monatomic\\
\vspace{1mm}
$\gamma=7/5$ - Diatomic \\
\vspace{1mm}
$v=\sqrt{\frac{\gamma P}{\rho}}$\\
\vspace{1mm}
\section{Entropy-Heat}
$dS=\frac{dQ}{T}$\\
\vspace{1mm}
$Q=mc\Delta T$\\
\vspace{1mm}
$Q=mL$\\
\vspace{1mm}
$dQ=L dm$ - if $m$ changes\\
\vspace{1mm}
Consider what changes,
i.e sign on \textit{Q}.
Also for a large bath,
$\Delta S=\frac{\Delta Q}{T}$\\
\vspace{1mm}
$\frac{dQ}{dt}=k \frac{\Delta T}{ x}$\\
\section{Relativity}
$\beta=v/c$\\
\vspace{1mm}
$\gamma=1/\sqrt{1-\beta^2}$\\
\vspace{1mm}
$p=\gamma mv$\\
\vspace{1mm}
$E=\gamma mc^2$\\
\vspace{1mm}
$E^2=(pc)^2+(mc^2)^2$\\
\vspace{1mm}
$\Delta t=\gamma \Delta t_o$\\
\vspace{1mm}
$\Delta L= \frac{\Delta L_o}{\gamma}$\\
\vspace{1mm}
$E_k=(1-\gamma)mc^2$
\section{Electromagnetism}
$F=\frac{Q_1 Q_2}{4\pi \varepsilon_o r^2} $\\
\vspace{1mm}
$E=\frac{Q}{4\pi \varepsilon_o r^2}\hat{r}$\\
\vspace{1mm}
$p=q\cdot l$ - Dipole moment\\
\vspace{1mm}
$\tau=p \times E$ - Torque\\
\vspace{1mm}
$u=-p\cdot E $ - Potential Energy\\
\vspace{1mm}
$E=-\nabla V$\\
\vspace{1mm}
$V=-\int E \cdot dl $\\
\vspace{1mm}
$\int E \cdot dA=\frac{Q_{encl}}{\varepsilon_o}=\frac{\sum_i q_i}{\varepsilon_o}$\\
\vspace{1mm}
$C=Q/V$\\
\vspace{1mm}
$C=\varepsilon_o \frac{A}{d} $ - parallel plate\\
\vspace{1mm}
$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+..+\frac{1}{C_n}$ - Series\\
\vspace{1mm}
$C=C_1+C_2+..+C_n$ - parallel\\
\vspace{1mm}
$U=\frac{Q^2}{2C}=\frac{1}{2}CV^2=\frac{1}{2}QV$ - stored E\\
\vspace{1mm}
$u_e=\frac{1}{2}\varepsilon_o \varepsilon_r E^2$ - $E_E$ density\\
\vspace{1mm}
$u_b=\frac{1}{2}\frac{B^2}{\mu_o \mu_r}$ - $E_B$ density\\
\vspace{1mm}
$J=\sum_i n_iq_iv_{d_i}$\\
\vspace{1mm}
$\rho=\frac{E}{J}$ - resistivity\\
\vspace{1mm}
$\sigma=\frac{1}{\rho}$ - conductivity\\
\vspace{1mm}
$R=\frac{\rho L}{A}$\\
\vspace{1mm}
$V=\varepsilon-Ir$ - across a Battery\\
\vspace{1mm}
$P=VI=I^2R=\frac{V^2}{R}$\\
\vspace{1mm}
$I=\frac{dQ}{dt}=nqAv_d$\\
\vspace{1mm}
$R=R_1+R_2+..+R_n$ - series\\
\vspace{1mm}
$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+..+\frac{1}{R_n}$ - parallel\\
\vspace{1mm}
$F=qE$\\
\vspace{1mm}
$F=q \ v\times B$\\
\vspace{1mm}
$F=Il\times B$ \\
\vspace{1mm}
$B=\frac{\mu_o I}{2\pi r}$ - current carrying wire\\
\vspace{1mm}
$B=\frac{\mu_o I}{2r}$ - center of a loop\\
\vspace{1mm}
Multiply by N for N loops.
\vspace{1mm}
$B=\frac{\mu_o I}{2(x^2+r^2)^{3/2}}$ - $x$ from center of loop\\
\vspace{1mm}
$\mu_e=\frac{e\hbar}{2m_e}$ \\
\vspace{1mm}
$\mu_N=\frac{e\hbar}{2m_N}$\\
\vspace{1mm}
$\varepsilon=-\frac{d\Phi_b}{dt}=-\frac{B\cdot dA}{dt}$\\
\vspace{1mm}
$E_b=-\mu_{N / e} \cdot B$\\
\vspace{1mm}
$\int B \cdot dl=\mu_o I_{encl}$\\
\vspace{1mm}
\section{Capacitors}
Charging:\\
$Q=Q_o(1-e^{-t/RC})$\\
\vspace{1mm}
$I=I_o e^{-t/RC}$\\
\vspace{1mm}
Discharging:\\
$Q=Q_oe^{-t/RC}$\\
\vspace{1mm}
$I=I e^{-t/RC}$\\
\vspace{1mm}
Where $RC=\tau$ is the time constant\\
\vspace{1mm}
\section{Gravity}
$F=G\frac{Mm}{r^2}$\\
\vspace{1mm}
$g=G\frac{M}{r^2}$\\
\vspace{1mm}
\section{Quantum mechanics}
$\left(-\frac{\hbar}{2m}\nabla+V\right)\psi=E\psi $\\
\vspace{1mm}
$\Delta p \Delta x \geq \frac{\hbar}{2}$\\
\vspace{1mm}
$f(E)=\frac{1}{e^{(E-E_f)/k_bT}+1}$\\
\vspace{1mm}
$\lambda_db=\frac{h}{p}$\\
\vspace{1mm}
\end{multicols}
\end{document}
答案1
我认为您指的是分栏符，而不是？这可以（应该？）通过将内容移至下一列来General Wavefunction实现。\columnbreak
我认为应该\columnbreak在\section{Electromagnetism}和处添加一个\section{Gravity}。我更正了一些拼写错误，但可能还有其他我目前找不到的...
对于其他一些问题，我不会用这个答案来讨论。
\documentclass[12pt,landscape]{article}
\usepackage{multicol}
\usepackage{calc}
\usepackage{ifthen}
\usepackage[portrait]{geometry}
\usepackage{hyperref}
\usepackage{amsmath}
% To Do:
% \listoffigures \listoftables
% \setcounter{secnumdepth}{0}
% This sets page margins to .5 inch if using letter paper, and to 1cm
% if using A4 paper. (This probably isn't strictly necessary.)
% If using another size paper, use default 1cm margins.
\ifthenelse{\lengthtest { \paperwidth = 11in}}
{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} }
{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}}
{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
}
% Turn off header and footer
\pagestyle{empty}
% Redefine section commands to use less space
\makeatletter
\renewcommand{\section}{\@startsection{section}{1}{0mm}%
{-1ex plus -.5ex minus -.2ex}%
{0.5ex plus .2ex}%x
{\normalfont\large\bfseries}}
\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}%
{-1explus -.5ex minus -.2ex}%
{0.5ex plus .2ex}%
{\normalfont\normalsize\bfseries}}
\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}%
{-1ex plus -.5ex minus -.2ex}%
{1ex plus .2ex}%
{\normalfont\small\bfseries}}
\makeatother
% Define BibTeX command
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
% Don't print section numbers
\setcounter{secnumdepth}{0}
\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt plus 0.5ex}
% -----------------------------------------------------------------------
\begin{document}
\raggedright
\footnotesize
\begin{multicols}{3}
% multicol parameters
% These lengths are set only within the two main columns
%\setlength{\columnseprule}{0.25pt}
\setlength{\premulticols}{1pt}
\setlength{\postmulticols}{1pt}
\setlength{\multicolsep}{1pt}
\setlength{\columnsep}{2pt}
\section{Mechanics}
$v=u+at$ \\
\vspace{1mm}
$s=\frac{v+u}{2}t$. \\
\vspace{1mm}
\vspace{1mm}
$v^2=u^2+2as$ \\
\vspace{1mm}
$s=ut+\frac{1}{2}at^2$ \\
\vspace{1mm}
$p=mv$ \\
\vspace{1mm}
$\sum F= \frac{dp}{dt}$ \\
\vspace{1mm}
$\int_{t_1}^{t_2}Fdt=p_2-p_1$ - Impulse \\
\vspace{1mm}
$P=\frac{F}{A}$ \\
\vspace{1mm}
$F=ma$\\
\vspace{1mm}
$E_k=1/2mv^2$ \\
\vspace{1mm}
$\mu=Fd_\perp$ - moment \\
\vspace{1mm}
$E_p=mgh$\\
\vspace{1mm}
$W=\int F \cdot dx=F\Delta x$\\
\vspace{1mm}
$P_{avg}=\frac{\Delta W}{\Delta t}$\\
\vspace{1mm}
$P_{inst}=\frac{dW}{dT}=F\cdot v$\\
\vspace{1mm}
\section{Torque}
$\tau=Fd_{\perp}$ - About midpoint\\
\vspace{1mm}
$\tau=r \times F$ - vector torque\\
\vspace{1mm}
$W=\tau(\theta_2-\theta_1)=\tau \Delta \theta$\\
\vspace{1mm}
$P=\tau\omega$\\
\vspace{1mm}
$L=r\times p=m r \times v $ - AM particle\\
\vspace{1mm}
$L=I\omega$ - AM rigid body\\
\vspace{1mm}
$\sum \tau=\frac{dL}{dt}$
\vspace{1mm}
\section{Rotational motion}
$\omega=\frac{d\theta}{dt}$\\
\vspace{1mm}
$\alpha=\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$\\
\vspace{1mm}
Equations for constant $\alpha$:
$\theta=\theta_o+\omega_ot=\frac{1}{2}\alpha t^2$\\
\vspace{1mm}
$\omega=\omega_o+\alpha t$\\
\vspace{1mm}
$\omega^2=\omega_o^2+2\alpha(\theta-\theta_o)$\\
\vspace{1mm}
$E_k=\frac{1}{2}I\omega^2$\\
\vspace{1mm}
$I=\int r^2 dm$
\vspace{1mm}
\section{SHM}
$\omega=2\pi f=\frac{2\pi}{T}$\\
\vspace{1mm}
$F=-kx$\\
\vspace{1mm}
$E_s=\frac{1}{2}kx^2$\\
\vspace{1mm}
$E=\frac{1}{2}mv^2+\frac{1}{2}kx^2=const$\\
\vspace{1mm}
$T_s=2\pi \sqrt{\frac{m}{k}}$\\
\vspace{1mm}
$T_{sp}=2\pi \sqrt{\frac{L}{g}}$\\
\vspace{1mm}
$T_{physP}=2\pi \sqrt{\frac{I}{mgd}}$
\section{Waves}
$v=f\lambda$\\
\vspace{1mm}
$k=\frac{2\pi}{\lambda}$\\
\vspace{1mm}
$\omega=2\pi f$\\
\vspace{1mm}
\columnbreak
General Wavefunction for free wave:\\
$y(x,t)=Asin(\omega t-kx)$\\
\vspace{1mm}
Wave Equation:\\
$\frac{\partial^2y}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2y}{\partial t^2}$\\
\vspace{1mm}
$E=hf=\hbar\omega=\frac{hc}{\lambda}$\\
\vspace{1mm}
$eV_o=hf-\phi$ - photoelectric effect\\
\vspace{1mm}
$f_L=\frac{v\pm v_L}{v \pm v_s}f_s$\\
\vspace{1mm}
$c=1/\sqrt{\mu_o \varepsilon_o}$\\
\vspace{1mm}
$n=c/v$\\
\vspace{1mm}
$n_asin(\theta_a)=n_bsin(\theta_b)$\\
\vspace{1mm}
$sin(\theta_c)=\frac{n_b}{n_a}$\\
\vspace{1mm}
$dsin(\theta)=m\lambda$ - constructive\\
\vspace{1mm}
$dsin(\theta)=(m+\frac{1}{2})\lambda$ - destructive\\
\vspace{1mm}
$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$ - object,image distance\\
\vspace{1mm}
\section{Elasticity}
\newlength{\MyLen}
\settowidth{\MyLen}{\texttt{letterpaper}/\texttt{a4paper} \ }
$ Stress=F/A$\\
\vspace{1mm}
$Strain=\delta l / l_o$\\
\vspace{1mm}
\textit{Youngs Mod}$=Stress/Strain$\\
\section{Thermodynamics}
\settowidth{\MyLen}{\texttt{multicol} }
$pV=nRT$\\
\vspace{1mm}
$moles=m/A$\\
\vspace{1mm}
Kinetic energy per molecule\\
$E_k=n/2k_bT$- n=Degrees of freedom\\
\vspace{1mm}
$v_{rms}=\sqrt{3k_bT/m}=\sqrt{3RT/A}$\\
\vspace{1mm}
$\gamma=C_p/C_v$\\
\vspace{1mm}
$\gamma=5/3$ - Monatomic\\
\vspace{1mm}
$\gamma=7/5$ - Diatomic \\
\vspace{1mm}
$v=\sqrt{\frac{\gamma P}{\rho}}$\\
\vspace{1mm}
\section{Entropy-Heat}
$dS=\frac{dQ}{T}$\\
\vspace{1mm}
$Q=mc\Delta T$\\
\vspace{1mm}
$Q=mL$\\
\vspace{1mm}
$dQ=L dm$ - if $m$ changes\\
\vspace{1mm}
Consider what changes,
i.e sign on \textit{Q}.
Also for a large bath,
$\Delta S=\frac{\Delta Q}{T}$\\
\vspace{1mm}
$\frac{dQ}{dt}=k \frac{\Delta T}{ x}$\\
\section{Relativity}
$\beta=v/c$\\
\vspace{1mm}
$\gamma=1/\sqrt{1-\beta^2}$\\
\vspace{1mm}
$p=\gamma mv$\\
\vspace{1mm}
$E=\gamma mc^2$\\
\vspace{1mm}
$E^2=(pc)^2+(mc^2)^2$\\
\vspace{1mm}
$\Delta t=\gamma \Delta t_o$\\
\vspace{1mm}
$\Delta L= \frac{\Delta L_o}{\gamma}$\\
\vspace{1mm}
$E_k=(1-\gamma)mc^2$
\columnbreak
\section{Electromagnetism}
$F=\frac{Q_1 Q_2}{4\pi \varepsilon_o r^2} $\\
\vspace{1mm}
$E=\frac{Q}{4\pi \varepsilon_o r^2}\hat{r}$\\
\vspace{1mm}
$p=q\cdot l$ - Dipole moment\\
\vspace{1mm}
$\tau=p \times E$ - Torque\\
\vspace{1mm}
$u=-p\cdot E $ - Potential Energy\\
\vspace{1mm}
$E=-\nabla V$\\
\vspace{1mm}
$V=-\int E \cdot dl $\\
\vspace{1mm}
$\int E \cdot dA=\frac{Q_{encl}}{\varepsilon_o}=\frac{\sum_i q_i}{\varepsilon_o}$\\
\vspace{1mm}
$C=Q/V$\\
\vspace{1mm}
$C=\varepsilon_o \frac{A}{d} $ - parallel plate\\
\vspace{1mm}
$\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+..+\frac{1}{C_n}$ - Series\\
\vspace{1mm}
$C=C_1+C_2+..+C_n$ - parallel\\
\vspace{1mm}
$U=\frac{Q^2}{2C}=\frac{1}{2}CV^2=\frac{1}{2}QV$ - stored E\\
\vspace{1mm}
$u_e=\frac{1}{2}\varepsilon_o \varepsilon_r E^2$ - $E_E$ density\\
\vspace{1mm}
$u_b=\frac{1}{2}\frac{B^2}{\mu_o \mu_r}$ - $E_B$ density\\
\vspace{1mm}
$J=\sum_i n_iq_iv_{d_i}$\\
\vspace{1mm}
$\rho=\frac{E}{J}$ - resistivity\\
\vspace{1mm}
$\sigma=\frac{1}{\rho}$ - conductivity\\
\vspace{1mm}
$R=\frac{\rho L}{A}$\\
\vspace{1mm}
$V=\varepsilon-Ir$ - across a Battery\\
\vspace{1mm}
$P=VI=I^2R=\frac{V^2}{R}$\\
\vspace{1mm}
$I=\frac{dQ}{dt}=nqAv_d$\\
\vspace{1mm}
$R=R_1+R_2+..+R_n$ - series\\
\vspace{1mm}
$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+..+\frac{1}{R_n}$ - parallel\\
\vspace{1mm}
$F=qE$\\
\vspace{1mm}
$F=q \ v\times B$\\
\vspace{1mm}
$F=Il\times B$ \\
\vspace{1mm}
$B=\frac{\mu_o I}{2\pi r}$ - current carrying wire\\
\vspace{1mm}
$B=\frac{\mu_o I}{2r}$ - center of a loop\\
\vspace{1mm}
Multiply by N for N loops.
\vspace{1mm}
$B=\frac{\mu_o I}{2(x^2+r^2)^{3/2}}$ - $x$ from center of loop\\
\vspace{1mm}
$\mu_e=\frac{e\hbar}{2m_e}$ \\
\vspace{1mm}
$\mu_N=\frac{e\hbar}{2m_N}$\\
\vspace{1mm}
$\varepsilon=-\frac{d\Phi_b}{dt}=-\frac{B\cdot dA}{dt}$\\
\vspace{1mm}
$E_b=-\mu_{N / e} \cdot B$\\
\vspace{1mm}
$\int B \cdot dl=\mu_o I_{encl}$\\
\vspace{1mm}
\section{Capacitors}
Charging:\\
$Q=Q_o(1-e^{-t/RC})$\\
\vspace{1mm}
$I=I_o e^{-t/RC}$\\
\vspace{1mm}
Discharging:\\
$Q=Q_oe^{-t/RC}$\\
\vspace{1mm}
$I=I e^{-t/RC}$\\
\vspace{1mm}
Where $RC=\tau$ is the time constant\\
\vspace{1mm}
\columnbreak
\section{Gravity}
$F=G\frac{Mm}{r^2}$\\
\vspace{1mm}
$g=G\frac{M}{r^2}$\\
\vspace{1mm}
\section{Quantum mechanics}
$\left(-\frac{\hbar}{2m}\nabla+V\right)\psi=E\psi $\\
\vspace{1mm}
$\Delta p \Delta x \geq \frac{\hbar}{2}$\\
\vspace{1mm}
$f(E)=\frac{1}{e^{(E-E_f)/k_bT}+1}$\\
\vspace{1mm}
$\lambda_db=\frac{h}{p}$\\
\vspace{1mm}
\end{multicols}
\end{document}
