$\int_1^a {1\over x}\,dx$
$r\,dr\,d\theta$
$x\,dy - y\,dx$
$n!\,(n+1)!$
$51!\over 12!\, 39!$
$\sqrt2\,x$
$\sqrt{\,\log x}$
$O\bigl(1/\sqrt n\,\bigr)$
$[\,0,1)$
$\{\,x\mid x\ge 0\,\}$
$x^2\!/2$
$n/\!\log n$
$\Gamma_{\!2}+\Delta^{\!2}$
$\left(1\over1+x^2\right)^{\!2}$
$R_i{}_j{}_{\!kl}$
$\int_0^x\!\int_0^y dF(u,v)$
$\int\!\!\!\int_D dx\,dy$
$x_1+\cdots+x_n$
$x_1=\cdots=x_n=0$
$A_1\times\cdots\times A_n$
$f(x_1,\ldits,x_n)$
$x_1x_2\ldots\x_n$
$(1-x)(1-x^2)\ldots(1-x^n)$
$\left$ not $$
$\left|-x\right| = \left|+x\right|$ not $|-x|=|+x|$
$\sin 2\theta$ not $\sin(2\theta)$
$\log n$ not $\log(n)$
\[F_n=F_{n-1}+F_{n-2} \qquad n \ge 2.\]
\[F_n=F_{n-1}+F_{n-2} \qquad \mbox{for all $n \ge
2$}.\]
Mixing text and math:
$\{\,p\mid\mbox{$p$ and $p+2$ are prime}\,\}$
``Dr.~Miner'' not ``Dr. Miner''
(the spacing will be wrong).