?
Курс математического анализа для студентов-бакалавров инженерных факультетов
The great mathematical formalism
of the presentation of classical
Tutorialss is not suitable for the perceptions of a modern student.
I tried to achieve a more simplified and compressed material. This was achieved with the help of a more visual, often geometric presentation of both the basic concepts and proofs. Moreover, in the geometric definition of the basic concepts, they were always reduced to those accepted in the standard statement.. In particular, they are given as the main geometric definitions of the derivative, definite integrals. In the course, double integrals were not considered in detail, but for familiarization, a formula was introduced for reducing the double integral to a iterated one in the simplest region. .. The consideration of improper integrals of the first kind is necessary to draw an analogy with number series. The classical definition was given . The improper integrals of the second kind were not considered, since they are not used further.
The concept of differentiability here reduces to the existence of a tangent line or a tangential plane for a function graph..
In addition, I tried to reduce the number of basic concepts and theorems. For example, instead of 15 definitions of the functijns limits of one variable, I propose one from which all the others are obtained by elementary substitutions. In conclusion second wundtrful limit for sequences, a simplified Bernoulli inequality was used instead of the Newton binomial theorem.
For functions of several variables, the Taylor formula is not derived. A sufficient condition for a local extremum is strictly obtained without this formula. (Note that in this case the Taylor formula for functions of one variable is given in all forms with rigorous inference, since it is necessary to obtain non-standard equivalence relations).
We compare numerical series with previously studied improper integrals of the first kind.
If I saved on definitions, I tried to bring as much proofs as possible, simplifying them as much as possible, sometimes just replacing them with explanations
