Foundations of higher mathematics. Part 1. Linear algebra and analytical geometry
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Дата
2019
Науковий керівник
Укладач
Редактор
Назва журналу
ISSN
E-ISSN
Назва тому
Видавець
Одеський національний університет імені І. І. Мечникова
Анотація
The modern level of development of a science results in that more and more
specialties which had before the applied (technical) character are included in sphere
of university education. First of all to such specialties we should refer specialties in
the field of computer sciences. Features of training of students on these specialties at
university generate a need of the accelerated studying of a course of higher mathematics, which has the volume coming nearer to the university course. Such challenge
is issued by this text-book on higher mathematics issues which is intended for students of the universities specializing in the field of computer sciences. Here the reader will find many perfectly developed pages as the course of the general mathematics
cannot be original work. The reason of it that a course carries out the first contact to
new knowledge and it is intended for the persons finished the school education and
having only principles of elementary mathematics knowledge. Feature of the given
text-book is also the uniform methodical approach to a statement of the entire higher
mathematics course, consisting that the basic mathematical concepts follow from the
general concepts and from logic concepts with the following distribution of a material.
The course is divided into five books.
The book 1 contains some logic concepts, the elementary concepts concerning
to sets and operations on them (union, intersection, difference, product), and also the
basic mathematical concepts, namely: concept of function or mapping; concept of n
– dimensional arithmetic space.
The book 2 is dedicated to the linear algebra. From fundamental concept of mapping,
concepts of internal and external laws of a composition are introduced. Conditions at
which operations of these laws on a set transform them into groups, rings, fields and
vector spaces are considered. It is investigated: a field of complex numbers; a ring of
multinomials; vector space of multinomials; vector space of free vectors in geometrical space; vectors in n -dimension arithmetic space. Concepts of matrixes, determinants and system of the linear equations result from concepts of vector space and
linear mapping of one vector space to another one. In the separate chapter it is considered reduction of matrixes by changing of basis to more simple form. Rather in tail, it is shown for reduction of the square matrix to the diagonal type, and the
square-law form to the canonic type.
The book 3 contains a number of concepts of analytical geometry required by
the program: the equations of a straight line on the plane and in the space; the equations of a plane; curves and surfaces of the second order, the equation of curves and
surfaces of the second order are reduced to the canonical type with use of square-law
forms. These geometrical concepts act as the direct appendix of the book 2 or as
transferring of results of this book on language of geometry as it is made in it for free
vectors in geometrical space.
The book 4 is dedicated to the mathematical analysis. Numerical functions of
one and many real variables are considered. Concepts of limit and continuity are introduced for these functions. The book comes to an end with the statement of differential and integral calculus.
In the book 5 the chapters are collected which are concerning to the concepts, having
technical character at a level of the general mathematics course, these are differential
equations and lines.
The statement of a theoretical material is accompanied by the illustrative examples and the solutions of typical problems. With the purpose of reinforcement of educational material, here the exercises for independent work are offered.
Опис
Ключові слова
general concepts, function, mapping, arithmetic space Rn, linear algebra, complex numbers, multinomials, vector spaces, matrixes, linear equation systems, matrix reduction, analytical geometry, straight line on the plane, straight line, plane, three-dimensional, space, lines and surfaces of the second order
Бібліографічний опис
Foundations of higher mathematics. Part 1. Linear algebra and analytical geometry: Textbook / A. V. Tyurin, A.Yu. Akhmerov – Odessa : Odessa National University after I. I. Mechnikov, 2019. – 262 p.