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The idea of designing this three-part problem book belongs to Professor B. Demidovich, but his early death prevented him from putting it into practice. The book was prepared by a group of us some of whom have had long pedagogical experience lecturing higher mathematics to engineering students. We have tried to do our best to develop and realize the ideas and plans of the late Professor Demidovich. The problems contained in the book cover all the branches of higher mathematics studied by engineering students. In addition, our problem book includes a number of problems which will enable the reader to repeat the main topics of mathe­ matical analysis and vector algebra from his school course, studying them more carefully. The present book (Part 1) entitled “Linear Algebra and Fundamentals of Mathematical Analysis” consists of seven chapters (1 to 7) and includes the topics which, as a rule, are studied during the first two semesters of the course. Among them are: vector algebra with elements of analytical geometry, linear algebra, and also differential calculus of functions of one and several variables, and integral cal­ culus of functions of one variable. Each chapter is supplied with answers to all the compu­ tational problems. The answers are given at the end of each chapter. Those problems marked with an * are provided with hints and the ones with **, with solutions to be found with answers. Each section in all chapters is supplied with a brief in­ troduction containing both the relevant theoretical material (definitions, formulas, and theorems) and a plenty of thor­ oughly worked examples. Contributing Authors: V.A. Bolgov, B.P. Demidovich, V.A. Efimenko, A.V. Efimov, A.F. Karakulin, S.M. Kogan, G.L. Lunts, E.F. Porshneva, A.S. Pospelov, S.V. Frolov, R.Ya. Shostak, A.R. Yanpolskii Translated from the Russian by Leonid Levant

The idea of designing this three-part problem book belongs to Professor B. Demidovich, but his early death prevented him from putting it into practice. The book was prepared by a group of us some of whom have had long pedagogical experience lecturing higher mathematics to engineering students. We have tried to do our best to develop and realize the ideas and plans of the late Professor Demidovich. The problems contained in the book cover all the branches of higher mathematics studied by engineering students. In addition, our problem book includes a number of problems which will enable the reader to repeat the main topics of mathe­ matical analysis and vector algebra from his school course, studying them more carefully. The present book (Part 1) entitled “Linear Algebra and Fundamentals of Mathematical Analysis” consists of seven chapters (1 to 7) and includes the topics which, as a rule, are studied during the first two semesters of the course. Among them are: vector algebra with elements of analytical geometry, linear algebra, and also differential calculus of functions of one and several variables, and integral cal­ culus of functions of one variable. Each chapter is supplied with answers to all the compu­ tational problems. The answers are given at the end of each chapter. Those problems marked with an * are provided with hints and the ones with **, with solutions to be found with answers. Each section in all chapters is supplied with a brief in­ troduction containing both the relevant theoretical material (definitions, formulas, and theorems) and a plenty of thor­ oughly worked examples. Contributing Authors: V.A. Bolgov, B.P. Demidovich, V.A. Efimenko, A.V. Efimov, A.F. Karakulin, S.M. Kogan, G.L. Lunts, E.F. Porshneva, A.S. Pospelov, S.V. Frolov, R.Ya. Shostak, A.R. Yanpolskii Translated from the Russian by Leonid Levant

 THIS book is planned as a textbook of analysis for first and second year mathematics students at Russian universities and consequently is divided into two volumes. In compiling the book I have made extensive use of my three-volume Course of Difl'erential and Integral Calculus, revising and abridging it in order to adapt it to the official mathematical analysis programme and to make it meet the requirements of a lecture course.   The tasks I set myself and the points by which I was guided are as follows:     1. First and foremost to provide a systematic and, as far as possible, rigorous treatment of the fundamentals of mathematical analysis. I consider it obligatory for the contents of a textbook to be presented in a logical sequence, in order to achieve a clearly defined and systematic presentation of the facts. This does not, however, prevent the lecturer from deviating from a strict systematic approach, but, perhaps, even helps him in this respect. In my own lecture courses, for example, I usually put aside for a while such difficult tasks for beginners as the theory of real numbers, the principle of convergence or the properties of continuous functions.   2. To uphold my own opinion that a course of mathematical analysis should not appear to students to be merely a long chain of “definitions” and “theorems”, but that it should also serve as a guide to action. Students must be taught to apply the theorems in practice in order to assist them in mastering the computational apparatus of analysis. Although this can be achieved largely with the help of exercises, I have also included some examples in my treatment of the theoretical material. The total number of these examples is, out of necessity, small, but they have been selected in such a way as to prepare students for conscientious work on the exercises.   3. It is well known that mathematical analysis has diverse and remarkable applications both in mathematics itself and in related scientific fields. Whilst students will realiZe this more and more as time passes, it is essential that they should learn and get used to the relationship of mathematical analysis with other mathematical sciences and with the requirements of practical work whilst studying the fundamentals of analysis. For this very reason I have provided, wherever possible, examples of the application of analysis not only to geometry, but also to mechanics, physics and engineering.   4. The problem of completing analytic work up to numerical results is of both theoretical and practical importance. Since an “exact” or “closed form” solution of a problem in analysis is possible in the simplest cases only, it is important to acquaint students with the use of approximate methods. Some attention has been given to this within the pages of this book.   5. By way of a brief explanation of my treatment of the subject matter, I have first of all considered the concept of a limit which plays the principal role among the fundamental concepts of analysis and which crops up in diverse forms literally throughout the entire course. Hence arises the problem of establishing a unified form of all variations of the limit. This is not only important from the viewpoint of principles but also vital from a practical standpoint, to obviate the necessity of having to construct the theory of limits anew each time it arises. There are two ways of achieving this aim: we can either immediately give the general definition of the limit of “directed variable” (following, for example, Shatunovskii and Moore, or Smith), or we can reduce every limit to the simplest case of the limit of a variable ranging over an enumerated sequence of values. The first alternative is difficult for beginners, and I have, therefore, chosen the second method of approaching the problem. The definition  of each new limit is given first by means of the limit of a sequence and only later on “in 3—5 language”.   6. To indicate a second feature of my treatment of the subject matter I have in Volume II, when speaking of curvilinear and surface integrals, emphasized the difference between the curvilinear and surface “integrals of first kind” (the exact counterparts of the ordinary and double integral over unoriented domains) and similar “integrals of second kind” (where the analogy partly vanishes). Experience has convinced me that this distinction not only leads to a better understanding of the material, but is also convenient in applications.   7. As a short appendix to the book I have included a brief account of elliptic integrals and in several cases I have presented problems with solutions involving elliptic integrals. This may help to destroy the harmful illusion, acquired by merely solving simple problems, that the results of analytic calculations must necessarily be “elementary”.   8. In various places throughout the book the reader will come across remarks of an historical nature. Moreover, Volume I ends with a chapter entitled, “Historical survey of the development of the fundamental concepts of mathematical analysis” and Volume II concludes with “An outline of further developments in mathematical analysis”. However, neither of these two “surveys” has been introduced to serve as a substitute for a complete history of mathematical analysis, which students meet with later in general courses on “the history of mathematics”. The first survey touches upon the origin of the concepts, whilst the final chapter in Volume II aims at providing the reader with at least a general idea of the chronology of the most important events in the history of analysis.  At this point, and in connection with the preceding paragraph,   A warning to potential readers of this book : The sequence in which I have treated various topics is closely connected with modern demands for strict mathematical rigour—demands which have become more and more acute over the years. 

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 THIS book is planned as a textbook of analysis for first and second year mathematics students at Russian universities and consequently is divided into two volumes. In compiling the book I have made extensive use of my three-volume Course of Difl'erential and Integral Calculus, revising and abridging it in order to adapt it to the official mathematical analysis programme and to make it meet the requirements of a lecture course.   The tasks I set myself and the points by which I was guided are as follows:     1. First and foremost to provide a systematic and, as far as possible, rigorous treatment of the fundamentals of mathematical analysis. I consider it obligatory for the contents of a textbook to be presented in a logical sequence, in order to achieve a clearly defined and systematic presentation of the facts. This does not, however, prevent the lecturer from deviating from a strict systematic approach, but, perhaps, even helps him in this respect. In my own lecture courses, for example, I usually put aside for a while such difficult tasks for beginners as the theory of real numbers, the principle of convergence or the properties of continuous functions.   2. To uphold my own opinion that a course of mathematical analysis should not appear to students to be merely a long chain of “definitions” and “theorems”, but that it should also serve as a guide to action. Students must be taught to apply the theorems in practice in order to assist them in mastering the computational apparatus of analysis. Although this can be achieved largely with the help of exercises, I have also included some examples in my treatment of the theoretical material. The total number of these examples is, out of necessity, small, but they have been selected in such a way as to prepare students for conscientious work on the exercises.   3. It is well known that mathematical analysis has diverse and remarkable applications both in mathematics itself and in related scientific fields. Whilst students will realiZe this more and more as time passes, it is essential that they should learn and get used to the relationship of mathematical analysis with other mathematical sciences and with the requirements of practical work whilst studying the fundamentals of analysis. For this very reason I have provided, wherever possible, examples of the application of analysis not only to geometry, but also to mechanics, physics and engineering.   4. The problem of completing analytic work up to numerical results is of both theoretical and practical importance. Since an “exact” or “closed form” solution of a problem in analysis is possible in the simplest cases only, it is important to acquaint students with the use of approximate methods. Some attention has been given to this within the pages of this book.   5. By way of a brief explanation of my treatment of the subject matter, I have first of all considered the concept of a limit which plays the principal role among the fundamental concepts of analysis and which crops up in diverse forms literally throughout the entire course. Hence arises the problem of establishing a unified form of all variations of the limit. This is not only important from the viewpoint of principles but also vital from a practical standpoint, to obviate the necessity of having to construct the theory of limits anew each time it arises. There are two ways of achieving this aim: we can either immediately give the general definition of the limit of “directed variable” (following, for example, Shatunovskii and Moore, or Smith), or we can reduce every limit to the simplest case of the limit of a variable ranging over an enumerated sequence of values. The first alternative is difficult for beginners, and I have, therefore, chosen the second method of approaching the problem. The definition  of each new limit is given first by means of the limit of a sequence and only later on “in 3—5 language”.   6. To indicate a second feature of my treatment of the subject matter I have in Volume II, when speaking of curvilinear and surface integrals, emphasized the difference between the curvilinear and surface “integrals of first kind” (the exact counterparts of the ordinary and double integral over unoriented domains) and similar “integrals of second kind” (where the analogy partly vanishes). Experience has convinced me that this distinction not only leads to a better understanding of the material, but is also convenient in applications.   7. As a short appendix to the book I have included a brief account of elliptic integrals and in several cases I have presented problems with solutions involving elliptic integrals. This may help to destroy the harmful illusion, acquired by merely solving simple problems, that the results of analytic calculations must necessarily be “elementary”.   8. In various places throughout the book the reader will come across remarks of an historical nature. Moreover, Volume I ends with a chapter entitled, “Historical survey of the development of the fundamental concepts of mathematical analysis” and Volume II concludes with “An outline of further developments in mathematical analysis”. However, neither of these two “surveys” has been introduced to serve as a substitute for a complete history of mathematical analysis, which students meet with later in general courses on “the history of mathematics”. The first survey touches upon the origin of the concepts, whilst the final chapter in Volume II aims at providing the reader with at least a general idea of the chronology of the most important events in the history of analysis.  At this point, and in connection with the preceding paragraph,   A warning to potential readers of this book : The sequence in which I have treated various topics is closely connected with modern demands for strict mathematical rigour—demands which have become more and more acute over the years. Historically speaking, therefore, the development of mathematical analysis has not been followed as closely as it might have been. Thus, Chapter 1 is devoted to “real numbers”, Chapter 3 to the “theory of limits”, and it is not until Chapter 5 that I have commenced to give a systematic account of the differential and integral calculus. The historical sequence of events was, of course, the complete reverse. The differential and integral calculus were founded in the seventeenth century and developed in the eighteenth century, being applied to numerous important problems; the theory of limits became the foundation-stone of mathematical analysis at the beginning of the nineteenth century and only in the second half of the nineteenth century did a clearly defined concept of real numbers come into being, which justified the most refined propositions of the theory of limits. 

This book is based on the lectures read by authors at Moscow State University for a number of years. As in Part 1, the authors strived to make presentation systematic and to set off the most important notions and theorems. Besides the basic curriculum material, this book contains some additional questions that play an important part in various branches of modern mathematics and physics (the theory of measure and Lebesgue integrals, the theory of Hilbert spaces and of self-adjoint linear operators in these spaces, questions of regularization of Fourier series, the theory of differential forms in Euclidean spaces, etc.}. Some of the topics, such as the conditions for termwise differentiation and termwise integration of functional sequences and functional series, the theorem on the change of variables in a multiple integral, Green’s and Stokes’s formulas, necessary conditions for a bounded functiomto be integrable in the sense of Riemann and in the sense of Lebesgue, are treated more generally and under weaker assumptions than usual. As in Part 1, we discuss in this book some questions related to computational mathematics, including first of all approximate cal­ culation of multiple integrals in the supplement to Chapter 2 and calculation of the values of functions from the approximate values of Fourier coefficients (A.N. Tichonoff’s regularization method) in the Appendix. The material of this book, together with that of Part 1 published earlier, constitutes an entire university course in mathematical analysis.

Abstract .   The law of impossibility is one of the fundamental concepts in mathematics that has a crucial role in probability theory and statistical analysis . This essay examines in depth the basic principles of the law of improbability , from its mathematical definition to its application in various fields of science . Through an analytical approach , this study explores how the concept of mathematical impossibility differs from impossibility in an empirical context , as well as explaining the paradoxes that arise in the space of continuous probability . The results of the analysis show that a proper understanding of the law of impossibility is not only important for the development of mathematical theories , but also has significant practical implications in decision -making, risk analysis , and statistical inference .  

-introduction to analysis; vector algebra and analytical geometry; determinants and matrices; systems of linear equations; elements of linear algebra; differential calculus-functions of one variable; integral calculus-functions of one variable; differential calculus-functions of several variables-

The book is based on the lectures delivered by the authors at the Physics Department of Moscow University. In the course of systematic exposition of the material, the authors gave prominence to the most important concepts and theorems, the most significant of the latter being called fundamental theorems. The authors tried to give the formulations of new concepts and theorems not long before their direct application. The sequence of presentation of the material corresponds to that accepted at the Physics Department of Moscow University. The chapter “Preliminary Information on the Main Concepts of Mathematical Analysis,” in particular, comes before the exposition of the systematic course because this chapter deals with some very important physical problems and discusses mathematical means for their solution. Thus, it becomes clear from the very beginning what problems and concepts constitute the subject matter of mathematical analysis. Our lecturing experience shows that such a preliminary elucidation of the range of problems treated in the course of the analysis makes it appreciably easier for the students to understand the basic mathematical concepts. The authors paid due attention to the ever-increasing role of computations and methods of approximation; that is why they tried to use algorithmic form in proving theorems and carrying out calculations wherever it was possible. In Chap. 12, in particular, the main emphasis is laid on the algorithmic aspect of the methods of approximation, and only then are the methods substantiated. In addition to the main part of the material, the authors consider it possible to include some sections given in small type. In writing the book, the authors borrowed some methodical techniques from the course of lectures by N. V. Efimov, as well as from the well-known books by E. Goursat, De La Valle Poussin, and F. Franklin. The authors express their deep gratitude to A. N. Tikhonov and A. G. Sveshnikov for their valuable ideas and instructions, and also for their enormous help at all stages of writing this book. Thanks are also due to I. A. Shishmarev, whose editing of the book was very beneficial. The authors also want to express their gratitude to N. V. Efimov and L. D. Kudryavtsev for a large number of methodical instructions, B. M. Budak and S. Y. Fomin for reviewing separate chapters and making their remarks, and B. Khimchenko, P. Zaykin, and A. Zolotarev for their help in preparing the manuscript for print. The authors consider it necessary to point out that to a large extent their pedagogical views were influenced by their talks with I. M. Vinogradov and A. A. Samarsky, to whom they are also very thankful. Translated from the Russian by Irene Aleksanova.

About the Book This book is based on lectures delivered by the authors at Moscow State University over several years. As in Part 1, the authors aim to present the material systematically, highlighting the most important notations and theorems. In addition to the core curriculum, the book explores several important questions that play a significant role in various branches of modern mathematics and physics, such as the theory of measure and Lebesgue integrals, Hilbert spaces and self-adjoint linear operators, Fourier series regularisation, and the theory of differential forms in Euclidean spaces. Some topics, such as the conditions for term-wise differentiation and integration of functional sequences and series, variable changes in multiple integrals, and Green’s and Stokes’ formulas, are treated more generally and under weaker assumptions than usual. The book also addresses computational mathematics, including approximate methods for evaluating multiple integrals and calculating function values from Fourier coefficients. The material, along with Part 1 (Volume 1), constitutes a comprehensive university course in mathematical analysis. The text allows for some flexibility, as chapters on advanced topics such as Lebesgue integrals, Hilbert spaces, and related supplements can be skipped without affecting comprehension of the main material. The authors acknowledge their indebtedness to various contributors for valuable advice and criticisms, particularly A. N. Tikhonov, A. G. Svesshnikov, Sh. A. Alimov, L. D. Kudryavtsev, S. A. Lomov, P. S. Modenov, and Ya. M. Zhileikin for their assistance in field theory and approximate methods for evaluating multiple integrals. Translated from the Russian by Vladimir Shokurov

About the Book This book is based on lectures delivered by the authors at Moscow State University over several years. As in Part 1, the authors aim to present the material systematically, highlighting the most important notations and theorems. In addition to the core curriculum, the book explores several important questions that play a significant role in various branches of modern mathematics and physics, such as the theory of measure and Lebesgue integrals, Hilbert spaces and self-adjoint linear operators, Fourier series regularisation, and the theory of differential forms in Euclidean spaces. Some topics, such as the conditions for term-wise differentiation and integration of functional sequences and series, variable changes in multiple integrals, and Green’s and Stokes’ formulas, are treated more generally and under weaker assumptions than usual. The book also addresses computational mathematics, including approximate methods for evaluating multiple integrals and calculating function values from Fourier coefficients. The material, along with Part 1 (Volume 1), constitutes a comprehensive university course in mathematical analysis. The text allows for some flexibility, as chapters on advanced topics such as Lebesgue integrals, Hilbert spaces, and related supplements can be skipped without affecting comprehension of the main material. The authors acknowledge their indebtedness to various contributors for valuable advice and criticisms, particularly A. N. Tikhonov, A. G. Svesshnikov, Sh. A. Alimov, L. D. Kudryavtsev, S. A. Lomov, P. S. Modenov, and Ya. M. Zhileikin for their assistance in field theory and approximate methods for evaluating multiple integrals. Translated from the Russian by Vladimir Shokurov

About the Book This book is based on lectures delivered by the authors at Moscow State University over several years. As in Part 1, the authors aim to present the material systematically, highlighting the most important notations and theorems. In addition to the core curriculum, the book explores several important questions that play a significant role in various branches of modern mathematics and physics, such as the theory of measure and Lebesgue integrals, Hilbert spaces and self-adjoint linear operators, Fourier series regularisation, and the theory of differential forms in Euclidean spaces. Some topics, such as the conditions for term-wise differentiation and integration of functional sequences and series, variable changes in multiple integrals, and Green’s and Stokes’ formulas, are treated more generally and under weaker assumptions than usual. The book also addresses computational mathematics, including approximate methods for evaluating multiple integrals and calculating function values from Fourier coefficients. The material, along with Part 1 (Volume 1), constitutes a comprehensive university course in mathematical analysis. The text allows for some flexibility, as chapters on advanced topics such as Lebesgue integrals, Hilbert spaces, and related supplements can be skipped without affecting comprehension of the main material. The authors acknowledge their indebtedness to various contributors for valuable advice and criticisms, particularly A. N. Tikhonov, A. G. Svesshnikov, Sh. A. Alimov, L. D. Kudryavtsev, S. A. Lomov, P. S. Modenov, and Ya. M. Zhileikin for their assistance in field theory and approximate methods for evaluating multiple integrals. Translated from the Russian by Vladimir Shokurov