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\title{On teaching of Mathematics in technical universities}
\author[]{A.~P.~Buslaev (MADI(STU)), M.~I.~Goldman (RUDN),
G.~G.~Magaril-Il'yaev (MIREA(TU)), K.~Yu.~Osipenko (MATI--RSTU),
V.~M.~Tikhomirov (MSU), Yu.~A.~Farkov (MGRA), M.~V.~Yashina (MADI(STU))}
\address{The Moscow Automobile and road Institute (State Technical
University) (MADI--STU), Leningradskii prospekt, 64,125829, Moscow, Russia}
\email{busl@math.madi.ru}

\maketitle

A century of computers has come and an unbelievable range of opportunities
were brought to us. Thus almost $90\%$ of exercises, which formed the base
of mathematical education for more than a century (limits, differentiation,
integration, multi-variables functions, series etc) can be solved by
pushing only several keys on the keyboard \dots No one can ignore these
changes.

A group of professors of mathematics of different universities has decided
to join their efforts in creating a multifaceted manual of mathematics,
which would meet all the requirements of the new computer century in all
its aspects --- methods of teaching with the aim of rousing students in
studing mathematics and control of quality education.

An important characteristic of our work is its validity. We try to motivate
each step we make, such as the general planning, the planning of each
lecture, material selection and so on.

Each Lecture comprises the following components: 1) an epigraph, 2) a
summary, 3) a basic part, 4) discussions, 5) graphics, 6) an historical
excursus, 7) portraits, 8) a cycle of problems (a quiz, exercises, tests,
problems for personal solving), 9) a computer based comment, 10) a list of
literature. The main definitions are considered from three points of view
--- analytical, geometrical and from the point of view of natural sciences.
They are also followed by expressive computer-based illustrations.

The electronic version of our manual is using all the possibilities of
modern computer-based educational technologies, such as branched hyper-text
structure, sound and video fragments, interactive elements and controlled
animation. In the whole range electronic manuals for technical universities
we know we can see only limited of such possibilities. An interesting
variant of interactive course of mathematical analysis in English is being
working out by Bert G.~Wachsmuth (http://www.shu/edu/projects/reals/).

Original computer methods of problem solving (for example, applicated to
extremums-finding cases), supplementing traditional geometrical and
analytical methods will be presented in our manual. Such an approach is
extremely visual (the animation directly shows the answer and the process
of solving), and it introduces methods of mathematical modeling to the
students using materials that are well-known to them. Thus, future
engineers get acquainted not only with the basics of the university course
of mathematics, but they are also taught to use mathematics as a research
technique.

The multi-level structure of presenting the material provided with the
system of hyperlinks gives the student an opportunity to choose the level
that meets his interests and goals of studying. In the upper layer are
presented basic materials on each topic of ``High-mathematics course for
engineers" that almost all the authors of teaching programs of the course
consider necessary to go through.

The main directions of more specified consideration are the following:
\begin{itemize}
\item proofs of theorems and affirmations, supplementing the basic text;
\item answers for quizzes hints for problems and exercise from the main
text computer animations hyper-active diagrams and table, accompanied by
vocal comments problems and exercises of higher complexity discussions and
historical excursus.
\end{itemize}

Testing as a form of final control will be used at different stages of
educational process --- for interim control as well as for controlling the
results of the education.

We hope that our manual will be used in the system of distant education in
Russia.
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