Make LaTeX output a dot when typing an asterisk

Simply change the character mapping for math mode:

\DeclareMathSymbol{*}{\mathbin}{symbols}{"01}

Have bold italic math letters

The standard TeX Computer Modern font set has a bold italic math font (cmmib10), LaTeX added some smaller sizes (cmmib5 etc.). But when \mathbf is used, the font set switches to cmbx, resulting in roman boldface. To get around this, simply define \mathitbf:

\DeclareMathAlphabet{\mathitbf}{OML}{cmm}{b}{it}

You may notice that lowercase greek is not affected by this change, although cmmib includes these glyphs as well. This is because lowercase greek is defined to be resistant against font changes (it is assigned category \mathord). So you may want to redefine the lowercase greek letters to be of category \mathalpha, which follows the font changes:

\DeclareMathSymbol{\alpha}{\mathalpha}{letters}{"0B}
\DeclareMathSymbol{\beta}{\mathalpha}{letters}{"0C}
\DeclareMathSymbol{\gamma}{\mathalpha}{letters}{"0D}
\DeclareMathSymbol{\delta}{\mathalpha}{letters}{"0E}
\DeclareMathSymbol{\epsilon}{\mathalpha}{letters}{"0F}
\DeclareMathSymbol{\zeta}{\mathalpha}{letters}{"10}
\DeclareMathSymbol{\eta}{\mathalpha}{letters}{"11}
\DeclareMathSymbol{\theta}{\mathalpha}{letters}{"12}
\DeclareMathSymbol{\iota}{\mathalpha}{letters}{"13}
\DeclareMathSymbol{\kappa}{\mathalpha}{letters}{"14}
\DeclareMathSymbol{\lambda}{\mathalpha}{letters}{"15}
\DeclareMathSymbol{\mu}{\mathalpha}{letters}{"16}
\DeclareMathSymbol{\nu}{\mathalpha}{letters}{"17}
\DeclareMathSymbol{\xi}{\mathalpha}{letters}{"18}
\DeclareMathSymbol{\pi}{\mathalpha}{letters}{"19}
\DeclareMathSymbol{\rho}{\mathalpha}{letters}{"1A}
\DeclareMathSymbol{\sigma}{\mathalpha}{letters}{"1B}
\DeclareMathSymbol{\tau}{\mathalpha}{letters}{"1C}
\DeclareMathSymbol{\upsilon}{\mathalpha}{letters}{"1D}
\DeclareMathSymbol{\phi}{\mathalpha}{letters}{"1E}
\DeclareMathSymbol{\chi}{\mathalpha}{letters}{"1F}
\DeclareMathSymbol{\psi}{\mathalpha}{letters}{"20}
\DeclareMathSymbol{\omega}{\mathalpha}{letters}{"21}
\DeclareMathSymbol{\varepsilon}{\mathalpha}{letters}{"22}
\DeclareMathSymbol{\vartheta}{\mathalpha}{letters}{"23}
\DeclareMathSymbol{\varpi}{\mathalpha}{letters}{"24}
\DeclareMathSymbol{\varrho}{\mathalpha}{letters}{"25}
\DeclareMathSymbol{\varsigma}{\mathalpha}{letters}{"26}
\DeclareMathSymbol{\varphi}{\mathalpha}{letters}{"27}

You may want to have a look at the package fixmath, which incorporates these changes.

There are some other glyphs that may be useful, too:

\DeclareMathSymbol{\imath}{\mathalpha}{letters}{"7B}
\DeclareMathSymbol{\jmath}{\mathalpha}{letters}{"7C}

Get italic uppercase greek per default

(La)TeX defines uppercase greek letters to be roman, what, for example, may lead to confusion between the steradian Ω and the unit of electrical resistance Ω (Ohm). That's why I prefer to have uppercase greek variables in italic but I need to preserve the ability to change to roman with \mathrm or to bold italic with \mathitbf. The solution is simple since one just has to switch uppercase greek from math symbol font ‘operators’ (which normally is connected to cmr) to ‘letters’ (normally cmmi).

\DeclareMathSymbol{\Gamma}{\mathalpha}{letters}{"00}
\DeclareMathSymbol{\Delta}{\mathalpha}{letters}{"01}
\DeclareMathSymbol{\Theta}{\mathalpha}{letters}{"02}
\DeclareMathSymbol{\Lambda}{\mathalpha}{letters}{"03}
\DeclareMathSymbol{\Xi}{\mathalpha}{letters}{"04}
\DeclareMathSymbol{\Pi}{\mathalpha}{letters}{"05}
\DeclareMathSymbol{\Sigma}{\mathalpha}{letters}{"06}
\DeclareMathSymbol{\Upsilon}{\mathalpha}{letters}{"07}
\DeclareMathSymbol{\Phi}{\mathalpha}{letters}{"08}
\DeclareMathSymbol{\Psi}{\mathalpha}{letters}{"09}
\DeclareMathSymbol{\Omega}{\mathalpha}{letters}{"0A}

The package fixmath includes these definitions, too.

Quantities

The code

\let\unitsep=\,     % Inserted at spaces in second arg, separates units (A s)
\let\numunitsep=\;  % Separator between number and unit (12.3 V)
\makeatletter
\def\quan#1{\ifmmode \let\@shiftmath=\relax \else \let\@shiftmath=$\fi
\@shiftmath \uppercase{\def\@quanarg{#1}}%
\expandafter\@quannum\@quanarg E\@noexp\@end \futurelet\@nextchar\@quanunit}
\def\@quannum#1E#2\@end{\@quanmant#1\@end \ifx\@noexp#2 \else \@quanexp#2\fi}
\def\@quanmant#1{\ifx#1\@end \let\@next=\relax \else
\if,#1\mathord,\else #1\fi \let\@next=\@quanmant \fi \@next}
\def\@quanexp#1E\@noexp{\times 10^{#1}}
\def\@quanunit{\if[\@nextchar
\def\@next[##1]{\numunitsep\mathrm{##1}\endgroup\@shiftmath}%
\begingroup\@quanspace \else \let\@next=\@shiftmath \fi \@next}
{\catcode`\ =\active\gdef\@quanspace{\catcode`\ =\active\let =\unitsep}}
\makeatother

Example:

... evaluates to $x = \quan{1.54e3}[kg m s^2]$, which is ...

… evaluates to x = 1.54 × 10³ kg m s², which is …

Upright μ

Sometimes one needs an upright μ to correctly typeset the SI-conform abbreviation of 10^-6 (“micro”), for example. Upright lowercase greek unfortunately is not included in the standard Computer Modern font set, so \mathrm{\mu} does not work (see also the remark about \mathitbf). One way out of this is to use the text companion fonts (TC-fonts) that are coming with the EC-fonts, which are standard in every LaTeX distribution now. They provide a command \textmu, which (issued in horizontal mode!) produces the desired upright μ. So what you have to do is add

\usepackage{textcomp}

\hbox{\textmu}

Defining additional operators

(La)TeX has a built-in set of operators like sin, cos, log, …. To add a new definition, you can either use AMSLaTeX's DeclareMathOperator, or simply do it like the LaTeX kernel does it:

\newcommand{\grad}{\mathop{\mathrm{grad}}\nolimits}
\renewcommand{\Re}{\mathop{\mathrm{Re}}\nolimits}
\renewcommand{\Im}{\mathop{\mathrm{Im}}\nolimits}

Smash the width of boxes

The most tricky formula layouts can be achieved with phantom, vphantom and smash. The latter prints its argument in a box of zero height and is useful when setting roots for example.

If one has sums or integrals with boundaries (sub- or superscripts) which boxes exceed a certain width, too much white space will surround the sum/integral sign. In this case it would be desireable to have a command to smash the width of a box, centering its content:

\def\clap#1{\hbox to0pt{\hss#1\hss}}
\def\hsmash{\mathpalette\dohsmash}
\def\dohsmash#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

Multiline equation environments

LaTeX's standard environment eqnarray produces particularly ugly layouts, which is at large due to the fact that space is added between the three columns. Another annoying thing is that the middle column is set in textstyle, whereas the left and right columns use displaystyle. Furthermore, the three column layout is often not appropriate, because two columns are sufficient to get proper alignment — if lines do not have to be aligned against each other, even a simple one-centered-column approach would make it. But eqnarray would display its middle column in the wrong math style! AMSLaTeX therefore provides (among others) the align and multiline environments. But if that is all you want from AMSLaTeX, you do not have to load the whole package. The following lines are borrowed from the eqnarray definition inside the LaTeX kernel and mimic their AMSLaTeX counterparts:

\makeatletter

\def\eqnalign{%
   \stepcounter{equation}%
   \def\@currentlabel{\p@equation\theequation}%
   \global\@eqnswtrue
   \m@th
   \global\@eqcnt\z@
   \tabskip\@centering
   \let\\\@eqncr
   $$\everycr{}\halign to\displaywidth\bgroup
       \global\@eqcnt\@ne
         \hskip\@centering$\displaystyle\tabskip\z@skip{##}$
      &\global\@eqcnt\tw@  
         $\displaystyle{##}$\hfil\tabskip\@centering
      &\global\@eqcnt\thr@@ \hb@xt@\z@\bgroup\hss##\egroup
         \tabskip\z@skip
      \cr
}
\def\endeqnalign{%
      \@@eqncr
      \egroup
      \global\advance\c@equation\m@ne
   $$\@ignoretrue
}
\@namedef{eqnalign*}{\def\@eqncr{\nonumber\@seqncr}\eqnalign}
\@namedef{endeqnalign*}{\nonumber\endeqnalign}

\def\eqnlines{%
   \stepcounter{equation}%
   \def\@currentlabel{\p@equation\theequation}%
   \global\@eqnswtrue
   \m@th
   \global\@eqcnt\z@
   \tabskip\@centering
   \let\\\@eqncr
   $$\everycr{}\halign to\displaywidth\bgroup
       \global\@eqcnt\tw@
         \hfil$\displaystyle{##}$\hfil\tabskip\@centering
      &\global\@eqcnt\thr@@ \hb@xt@\z@\bgroup\hss##\egroup
         \tabskip\z@skip
      \cr
}
\def\endeqnlines{%
      \@@eqncr
      \egroup
      \global\advance\c@equation\m@ne
   $$\@ignoretrue
}
\@namedef{eqnlines*}{\def\@eqncr{\nonumber\@seqncr}\eqnlines}
\@namedef{endeqnlines*}{\nonumber\endeqnlines}

\makeatother