How can I draw a dot under a symbol in math mode?

You can use the \underaccent command from the accents package. Something along these lines:

\documentclass{article}
\usepackage{amsmath}
\usepackage{accents}

\newcommand*\underdot[1]{%
  \underaccent{\dot}{#1}}

\begin{document}
$\underdot{A}$
\end{document}

It's not in the TeX Book. If you don't mind cheating you could try the following but it's not nice:

\documentclass{minimal}
\usepackage{amsmath}

\def\myd#1{\text{\d{\ensuremath#1}}}
\begin{document}
$\myd{a}$
\end{document}