as a section of higher
mathematics is its classical part and therefore changes little in terms of its
teaching.
Generally, it contains all the basic concepts of higher mathematics, which are then found in subsequent more specialized courses. Since there are quite a large number of such concepts, the presentation of mathematical analysis is reduced to the definition of all these concepts and, accordingly, related theorems, among which there are also very complex ones. In this case, the proofs of complex theorems are omitted.
The course of mathematical analysis proposed below is based on a somewhat different principle. Briefly, it can be characterized as follows: first, a list of basic theorems is selected, and then only those concepts that are necessary to prove the theorems of the selected list are introduced.
As a result of this approach, with a noticeable reduction in the volume of the book, all theorems are proved completely.
To whom do we offer this course? First of all, not for professional mathematicians, but for those who use mathematics not at all occasionally. For example, physicists, applied mathematicians, programmers and specialists in other, very different areas at the junction with mathematics. Both students and those who already have a degree. Secondly, students who, for various reasons, missed classes. And finally, schoolchildren who are bored with solving monotonous problems, and especially those, actually I have met few of them, who feel the beauty of mathematics and for whom the beauty of theory inspires much more than even very elegant solutions to Olympiad problems.
What grade should you be at to read it?
Read the first paragraph. If everything is clear, then it is already possible.And for professionals. More specifically characterizing the suggested course, I note, for example, that in it the compactness property of a segment is used only as a property of a nested system of segments to have a common point. Neither the classical definition of compactness and, accordingly, the Borel-Lebesgue lemma, nor sequential compactness is present in it, although the existence of a convergent subsequence in a bounded one is proved. In addition, the most difficult theorem in differential calculus, Taylor's theorem, is proved by trivial integration by parts in integral calculus.
I also omit proofs in little-o notation based on the classical definition, replacing o(f(x)) by the product of function f(x) by an infinitesimal function.
Finally, I think that power series should be teach right away for complex numbers, having previously explained that practically everything that is true for analysis with real variables remains true for complex numbers.
to be continued
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