\documentclass{article}

% this is a comment - LaTeX does not read from % to the end of the line
\title{Simple example}
\author{Helper}

\date{\today}

\begin{document}

\section{Text}

Write as usual. You can make \textbf{bold} or \textit{italic}.

You can write 
\large larger \Large larger \LARGE and larger and then \normalsize back  normal. Latex ignores whitespace: several spaces         is the same as single. You can make space using \ \ \ \ . 

A line of whitespaces gives paragraph. Break line using \\ without new paragraph.  

\section{Math}

Inline math is using $dollars$: $y=\sin x$. Display math (outside text) is using
\[
y=\sum_i \sin x_i
\]
without numbering or 
\begin{equation}
\int_a^b \frac{df}{dx}dx=f(b)-f(a)
\end{equation}
with numbers. 

Subscripts make like this $x_i$, superscripts  like this $a^n$. More complicated should be enclosed in \{ and \}: $X_{ias}, Y^{\alpha\beta\gamma}$. 

Elementary functions: $\sin, \cos,\exp,\tan,\cot,\sinh,\cosh,\ln$. Warning: you should put space between a keyword like $\sin$ and argument: $\sin x$. 

Fraction: 

\[
\frac{x}{y}
\]

Square root:

\[
\sqrt{xuy}
\]

Other symbols: 

sum

\[
\sum_{n=1}^\infty x_n
\]

integral

\[
\int_a^b\frac{dx}{x}=\ln\frac{|b|}{|a|}=\ln\left|\frac{b}{a}\right|
\] 

Vector make using math bold: $\mathbf{E}$ (preferred) or $\vec{E}$. Better way is to use 
\verb+\usepackage{bm}+ before \verb+\begin{document}+ and then \verb+\bm{E}+.

This is enough to start. 



\end{document} 