The mathematical analysis. Volume 1

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Mathematical Analysis ◽  
Differential Calculus ◽  
Real Variable ◽  
Hyperbolic Functions ◽  
Integral Calculus ◽  
Postgraduate Students ◽  
First Semester ◽  
Improper Integrals ◽  
Basic Concepts ◽  
Sets Theory

The aim of the tutorial is to help students to master the basic concepts and methods of the study of calculus. Volume 1 explores the following topics: theory of sets, theory of limits; differential calculus of functions of one variable; investigation of the properties of functions and graphing; integral calculus of functions of one real variable (indefinite, definite and improper integrals), the technique of integration; hyperbolic functions; applications to the analysis and solution of practical problems. These topics are studied in universities, as a rule, in the first semester in the framework of self-discipline "Mathematical analysis" or the course "Higher mathematics", "Mathematics". Great attention is paid to comparison of these methods, the proper choice of study design tasks, analyze complex situations that arise in the study of these branches of mathematical analysis. For teachers, students and postgraduate students studying mathematical analysis.

Mathematical analysis in examples and tasks. Part 1

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Mathematical Analysis ◽  
Continuous Functions ◽  
Advanced Mathematics ◽  
Hyperbolic Functions ◽  
Integral Calculus ◽  
Postgraduate Students ◽  
Background Material ◽  
Improper Integrals ◽  
Sets Theory ◽  
Independent Work

The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 1 of the proposed cycle of workshops on the following topics: theory of sets, theory of limits, theory of continuous functions; differential calculus of functions of one variable, its application to the study of the properties of functions and graph; integral calculus of functions of one variable: indefinite, definite, improper integrals; hyperbolic functions; applications of integral calculus to the analysis and solution of practical problems. For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the obtained knowledge provides exercises and problems with answers and guidance. For teachers, students and postgraduate students studying advanced mathematics.

The mathematical analysis. Volume 2

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Field Theory ◽  
Quality Control ◽  
Fourier Series ◽  
Mathematical Analysis ◽  
Differential Calculus ◽  
Analytic Geometry ◽  
Postgraduate Students ◽  
Several Variables ◽  
Basic Concepts ◽  
Control Knowledge

The aim of the tutorial is to help students to master the basic concepts and methods of the study of calculus. In volume 2 we study analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Taylor series and Maclaurin, and Fourier series; applications to the analysis and solution of applied problems. Great attention is paid to comparison of these methods, the proper choice of study design tasks, analyze complex situations that arise in the study of these branches of mathematical analysis. For self-training and quality control knowledge given test questions. For teachers, students and postgraduate students studying mathematical analysis.

Mathematical analysis in examples and tasks. Part 2

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Field Theory ◽  
Graduate Students ◽  
Mathematical Analysis ◽  
Differential Calculus ◽  
Advanced Mathematics ◽  
Analytic Geometry ◽  
Several Variables ◽  
Background Material ◽  
Basic Concepts ◽  
Independent Work

The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 2 of the proposed cycle of workshops on the following topics: analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Fourier series; applications to the analysis and solution of applied problems. These topics are studied in universities, usually in the second semester in the discipline "Mathematical analysis" or the course "Higher mathematics", "Mathematics". For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the acquired knowledge in each section designed exercises and tasks with answers and guidance. It is recommended that teachers, students and graduate students studying advanced mathematics.

scholarly journals Integral calculus on quantum exterior algebras

2014 ◽  
Vol 11 (04) ◽  
pp. 1450026 ◽  
Author(s):  
Serkan Karaçuha ◽  
Christian Lomp
Keyword(s):  
Differential Calculus ◽  
Polynomial Algebra ◽  
Exterior Algebra ◽  
Integral Calculus ◽  
De Rham Complex ◽  
Integral Forms ◽  
Rham Complex ◽  
De Rham ◽  
Quantum Polynomial

Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.

Introduction to the axiomatic theory of elementary functions

2021 ◽  
Author(s):  
Andrey Shishkin
Keyword(s):  
Complex Variable ◽  
Real Variable ◽  
Complex Variables ◽  
State University ◽  
Axiomatic Theory ◽  
Elementary Functions ◽  
State Pedagogical ◽  
Pedagogical Institute ◽  
State Pedagogical Institute ◽  
Basic Concepts

Contains an exposition of the basic concepts and theorems of the axiomatic theory of the basic elementary functions of real and complex variables. The textbook is written on the basis of lectures given by the author for a number of years at the Armavir State Pedagogical University, at the Slavyansk-on-Kuban State Pedagogical Institute and at the branch of the Kuban State University in Slavyansk-on-Kuban. It is intended for students of natural-mathematical profiles of preparation of the direction "Pedagogical education". It can be used in the study of mathematical analysis, the theory of functions of a real variable, the theory of functions of a complex variable, etc.

Optimizing length of planar curves

2020 ◽  
Vol 1 (1) ◽  
pp. 31-38
Author(s):  
Vojtech Kloud ◽  
Dusan Bednarik
Keyword(s):  
Calculus Of Variations ◽  
Shortest Path ◽  
Mathematical Analysis ◽  
Existence Of A Solution ◽  
Planar Curves ◽  
Basic Concepts

This article focuses on the problem of finding a shortest path in plane with obstacles. Problems of such nature occur for instance in robotics or transport and are of great importance. The problem is analyzed using the methods of mathematical analysis and calculus of variations. Definitions of basic concepts of the problem are given. From these definitions, useful properties, such as convexity of the length functional, are proven. These properties are used to show the existence of a solution in one of the considered cases of the problem. Other case of the problem was considered, where it is established under which conditions does a shortest path attain its general form and what this form looks like.

scholarly journals The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 42
Author(s):  
Yuri N. Lovyagin ◽  
Nikita Y. Lovyagin
Keyword(s):  
Line Segment ◽  
Cauchy Sequence ◽  
Research Method ◽  
Axiomatic Theory ◽  
Real Numbers ◽  
Integral Calculus ◽  
Standard Analysis ◽  
Monotonic Sequence ◽  
Basic Concepts ◽  
Non Standard Analysis

This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions.

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