  The aim of this Tutorial is to present the basic mathematical techniques
required for an accurate description of cold trapped atoms, both Bose and
Fermi. The term {\it cold} implies that considered temperatures are low, such
that quantum theory is necessary, even if temperatures are finite. And the term
{\it atoms} means that the considered particles are structureless, being
defined by their masses and mutual interactions. Atoms are {\it trapped} in the
sense that they form a finite quantum system, though their number can be very
large allowing for the use of the methods of statistical mechanics. This
Tutorial is the first part of several tutorials, giving general mathematical
techniques for both types of particle statistics. The following tutorials will
be devoted separately to Bose atoms and Fermi atoms. The necessity of carefully
explaining basic techniques is important for avoiding numerous misconceptions
often propagating in literature.
