This is for practice, the content is basicly from MathJax basic tutorial and quick reference

1. To see how any formula was written in any question or answer, including this one, right-click on the expression it and choose “Show Math As > TeX Commands”. (When you do this, the ‘$’ will not display. Make sure you add these. See the next point.)
2. For inline formulas, enclose the formula in $...$. For displayed formulas, use $$...$$.
   These render differently. For example, type $sum_{i=0}^n i^2 = frac{(n^2+n)(2n+1)}{6}$ to show $sum_{i=0}^n i^2 = frac{(n^2+n)(2n+1)}{6}$ (which is inline mode) or type $$sum_{i=0}^n i^2 = frac{(n^2+n)(2n+1)}{6}$$ to show:
   $sum_{i=0}^n i^2 = frac{(n^2+n)(2n+1)}{6}$
3. For Greek letters, use alpha, beta,…, omega: $alpha, beta, … omega$. For uppercase, use Gamma, Delta,…, Omega: $Gamma, Delta, …, Omega$.
4. For superscripts and subscripts, use ^ and _. For example, x_i^2: $x_i^2$, log_2 x: $log_2 x$.
5. Groups. Superscripts, subscripts, and other operations apply only to the next “group”. A “group” is either a single symbol, or any formula surrounded by curly braces {…}. If you do 10^10, you will get a surprise: $10^10$. But 10^{10} gives what you probably wanted: $10^{10}$.
   Use curly braces to delimit a formula to which a superscript or subscript applies: x^5^6 is an error; {x^y}^z is ${x^y}^z$, and x^{y^z} is $x^{y^z}$. Observe the difference between x_i^2: $x_i^2$ and x_{i^2}: $x_{i^2}$.
6. Parentheses Ordinary symbols ()[] make parentheses and brackets $(2+3)[4+4]$. Use { and } for curly braces ${}$.
   These do not scale with the formula in between, so if you write (frac{sqrt x}{y^3}) the parentheses will be too small: $(frac{sqrt x}{y^3})$. Using left(…right) will make the sizes adjust automatically to the formula they enclose: left(frac{sqrt x}{y^3}right) is $left(frac{sqrt x}{y^3}right)$.
   left and right apply to all the following sorts of parentheses: ( and ) $(x)$, [ and ] $[x]$, { and } ${x}$, | $|x|$, vert $vert x vert$, Vert $Vert xVert$, langle and rangle $langle x rangle$, lceil and rceil $lceil x rceil$, and lfloor and rfloor $lfloor x rfloor$. middle can be used to add additional dividers. There are also invisible parentheses, denoted by .: left.frac12rightrbrace is $left.frac12rightrbrace$.
   If manual size adjustments are required: Biggl(biggl(Bigl(bigl((x)bigr)Bigr)biggr)Biggr) gives $Biggl(biggl(Bigl(bigl((x)bigr)Bigr)biggr)Biggr)$.
7. Sums and integrals sum and int; the subscript is the lower limit and the superscript is the upper limit, so for example sum_1^n $sum_1^n$. Don’t forget {…} if the limits are more than a single symbol. For example, sum_{i=0}^infty i^2 is $sum_{i=0}^infty i^2$. Similarly, prod $prod$, int $int$, bigcup $bigcup$, bigcap $bigcap$, iint $iint$, iiint $iiint$, idotsint $idotsint$.
8. Fractions There are three ways to make these. frac ab applies to the next two groups, and produces $frac ab$; for more complicated numerators and denominators use {…}: frac{a+1}{b+1} is $frac {a+1} {b+1}$. If the numerator and denominator are complicated, you may prefer over, which splits up the group that it is in: {a+1over b+1} is ${a+1over b+1}$. Using `cfrac{a}{b}’ command is useful for continued fractions $cfrac{a}{b}$, more details for which are given in this sub-article.
9. Fonts

1. Radical signs Use sqrt, which adjusts to the size of its argument: 'sqrt{x^3} $sqrt{x^3}$; sqrt[3]{frac xy} $sqrt[3]{frac xy}$. For complicated expressions, consider using {...}^{1/2} instead.
2. Some special functions such as “lim”, “sin”, “max”, “ln”, and so on are normally set in roman font instead of italic font. Use lim, sin, etc. to make these: sin x: $sin x$, not sin x: $sinx$. Use subscripts to attach a notation to lim: $lim_{xto 0}$.
3. There are a very large number of special symbols and notations, too many to list here; see this shorter listing, or this exhaustive listing. Some of the most common include:
   • lt: $lt$, gt: $gt$, le: $le$, leq: $leq$, leqq: $leqq$, leqslant: $leqslant$, ge: $ge$, geq: $geq$, geqq: $geqq$, geqslant: $geqslant$, neq: $neq$.
     You can use not to put a slash through almost anything: notlt: $notlt$ but it often looks bad.
   • times: $times$, div: $div$, pm: $pm$, mp: $mp$, cdot is a centered dot: $x cdot y$.
   • cup: $cup$, cap: $cap$, setminus: $setminus$, subset: $subset$, subseteq: $subseteq$, subsetneq: $subsetneq$, supset: $supset$, in: $in$, notin: $notin$, emptyset: $emptyset$, varnothing: $varnothing$.
   • {n+1 choose 2k}: ${n+1 choose 2k}$ or binom{n+1}{2k}: $binom{n+1}{2k}$,
   • to: $to$, rightarrow: $rightarrow$, leftarrow: $leftarrow$, Rightarrow: $Rightarrow$, Leftarrow: $Leftarrow$, mapsto: $mapsto$
   • land: $land$, lor: $lor$, lnot: $lnot$, forall: $forall$, exists: $exists$, top: $top$, bot: $bot$, vdash: $vdash$, vDash: $vDash$.
   • star: $star$, ast: $ast$, oplus: $oplus$, circ: $circ$, bullet: $bullet$
   • approx: $approx$, sim: $sim$, simeq: $simeq$, cong: $cong$, equiv: $equiv$, prec: $prec$, lhd: $lhd$, therefore: $therefore$
   • infty: $infty$, aleph_0: $aleph_0$
   • nabla: $nabla$, partial: $partial$
   • Im: $Im$, Re: $Re$
   • For modular equivalence, use pmod like this: aequiv bpmod n: $aequiv bpmod n$.
   • ldots is the dots in $a_1, a_2, a_3, ldots, a_n$ a1,a2,…,an, cdots is the dots in $a_1 + a_2 + a_3 + cdots + a_n$
   • Some Greek letters have variant forms: epsilon: $epsilon$, varepsilon: $varepsilon$, phi: $phi$, varphi: $varphi$, and others.
   • Script lowercase l is ell: $ell$.
   • Detexify lets you draw a symbol on a web page and then lists the TEX symbols that seem to resemble it. These are not guaranteed to work in MathJax but are a good place to start. To check that a command is supported, note that MathJax.org maintains a list of currently supported LATEX commands, and one can also check Dr. Carol JVF Burns’s page of TEX Commands Available in MathJax.

Matrices

1. Use $$begin{matrix}…end{matrix}$$ In between the begin and end, put the matrix elements. End each matrix row with \, and separate matrix elements with &. For example:

   1 2 3 4 5 6 7

   $$ begin{matrix} 1 & x & x^2 \ 1 & y & y^2 \ 1 & z & z^2 \ end{matrix} $$

   produces:
   $% <![CDATA[ begin{matrix} 1 & x & x^2 \ 1 & y & y^2 \ 1 & z & z^2 \ end{matrix} %]]>$
   
   MathJax will adjust the sizes of the rows and columns so that everything fits.

2. To add brackets, either use left…right as in section 6 of the tutorial, or replace matrix with:
   pmatrix: $% <![CDATA[ begin{pmatrix} 1 & x & x^2 \ 1 & y & y^2 \ 1 & z & z^2 \ end{pmatrix} %]]>$
   bmatrix: $% <![CDATA[ begin{bmatrix} 1 & x & x^2 \ 1 & y & y^2 \ 1 & z & z^2 \ end{bmatrix} %]]>$
   Bmatrix: $% <![CDATA[ begin{Bmatrix} 1 & x & x^2 \ 1 & y & y^2 \ 1 & z & z^2 \ end{Bmatrix} %]]>$
   vmatrix: $% <![CDATA[ begin{vmatrix} 1 & x & x^2 \ 1 & y & y^2 \ 1 & z & z^2 \ end{vmatrix} %]]>$
   Vmatrix: $% <![CDATA[ begin{Vmatrix} 1 & x & x^2 \ 1 & y & y^2 \ 1 & z & z^2 \ end{Vmatrix} %]]>$
3. Use cdots $cdots$ ddots $ddots$ vdots $vdots$ when you want to omit some of the entries:
   $% <![CDATA[ begin{pmatrix} 1 & a_1 & a_1^2 & cdots & a_1^n \ 1 & a_2 & a_2^2 & cdots & a_2^n \ vdots & vdots& vdots & ddots & vdots \ 1 & a_m & a_m^2 & cdots & a_m^n end{pmatrix} %]]>$
4. For horizontally “augmented” matrices, put parentheses or brackets around a suitably-formatted table; see arrays below for details. Here is an example:
   $% <![CDATA[ left[ begin{array}{cc|c} 1&2&3 \ 4&5&6 end{array} right] %]]>$
   is produced by:

   1 2 3 4 5 6

   $$ left[ begin{array}{cc|c} 1&2&3 \ 4&5&6 end{array} right] $$

   The cc|c is the crucial part here; it says that there are three centered columns with a vertical bar between the second and third.

5. For vertically “augmented” matrices, use hline. For example:
   $% <![CDATA[ begin{pmatrix} a & b\ c & d\ hline 1 & 0\ 0 & 1 end{pmatrix} %]]>$
   is produced by:

   1 2 3 4 5 6 7 8 9

   $$ begin{pmatrix} a & b \ c & d \ hline 1 & 0 \ 0 & 1 end{pmatrix} $$

6. For small inline matrices use bigl(begin{smallmatrix} ... end{smallmatrix}bigr), e.g. $% <![CDATA[ bigl({begin{smallmatrix}a & b \ c & d end{smallmatrix}}bigr) %]]>$ is produced by: $$bigl({begin{smallmatrix}a & b \ c & d end{smallmatrix}}bigr)$$.

Aligned equations

Often people want a series of equations where the equals signs are aligned. To get this, use begin{align}…end{align}. Each line should end with \, and should contain an ampersand at the point to align at, typically immediately before the equals sign.
For example:

$% <![CDATA[ begin{align} sqrt{37} & = sqrt{frac{73^2-1}{12^2}} \ & = sqrt{frac{73^2}{12^2}cdotfrac{73^2-1}{73^2}} \ & = sqrt{frac{73^2}{12^2}}sqrt{frac{73^2-1}{73^2}} \ & = frac{73}{12}sqrt{1 - frac{1}{73^2}} \ & approx frac{73}{12}left(1 - frac{1}{2cdot73^2}right) end{align} %]]>$

The usual $$ marks that delimit the display may be omitted here.

Symbols

In general, you have to search in long tables about a specific symbol you’re looking for, things like Ψ, δ, ζ, ≥, ⊆ … And it turns out that this operation can be frustrating and time consuming, which can cause the buddy to abandon writing the complete $LaTeX$ sentence in his answer, or in some cases, the complete answer itself.

Here is the website: Detexify² No more frustration.

Definitions by cases (piecewise functions)

Use begin{cases}…end{cases}. End each case with a \, and use & before parts that should be aligned.

For example, you get this:

$$% <![CDATA[ f(n) = begin{cases} n/2, & text{if $n$ is even} \\ 3n+1, & text{if $n$ is odd} end{cases} %]]>$$

by write this:

1
2
3
4
5

    f(n) =
    begin{cases}
        n/2,  & text{if $n$ is even} 
        3n+1, & text{if $n$ is odd}
    end{cases}