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ZorichPreface to the Second Russian EditionIn this second edition of the book, along with an attempt to remove the misprintsthat occurred in the first edition,2 certain alterations in the exposition have beenmade (mainly in connection with the proofs of individual theorems), and some newproblems have been added, of an informal nature as a rule.The preface to the first edition of this course of analysis (see below) contains ageneral description of the course. The basic principles and the aim of the expositionare also indicated there. Here I would like to make a few remarks of a practicalnature connected with the use of this book in the classroom.Usually both the student and the teacher make use of a text, each for his ownpurposes.At the beginning, both of them want most of all a book that contains, along withthe necessary theory, as wide a variety of substantial examples of its applicationsas possible, and, in addition, explanations, historical and scientific commentary, anddescriptions of interconnections and perspectives for further development.
But whenpreparing for an examination, the student mainly hopes to see the material that willbe on the examination. The teacher likewise, when preparing a course, selects onlythe material that can and must be covered in the time alloted for the course.In this connection, it should be kept in mind that the text of the present bookis noticeably more extensive than the lectures on which it is based. What caused2 No need to worry: in place of the misprints that were corrected in the plates of the first edition(which were not preserved), one may be sure that a host of new misprints will appear, which soenliven, as Euler believed, the reading of a mathematical text.Prefacesixthis difference? First of all, the lectures have been supplemented by essentially anentire problem book, made up not so much of exercises as substantive problems ofscience or mathematics proper having a connection with the corresponding partsof the theory and in some cases significantly extending them.
Second, the booknaturally contains a much larger set of examples illustrating the theory in action thanone can incorporate in lectures. Third and finally, a number of chapters, sections, orsubsections were consciously written as a supplement to the traditional material.This is explained in the sections “On the introduction” and “On the supplementarymaterial” in the preface to the first edition.I would also like to recall that in the preface to the first edition I tried to warnboth the student and the beginning teacher against an excessively long study ofthe introductory formal chapters. Such a study would noticeably delay the analysisproper and cause a great shift in emphasis.To show what in fact can be retained of these formal introductory chapters ina realistic lecture course, and to explain in condensed form the syllabus for sucha course as a whole while pointing out possible variants depending on the studentaudience, at the end of the book I give a list of problems from the midterm exam,along with some recent examination topics for the first two semesters, to which thisfirst part of the book relates.
From this list the professional will of course discern theorder of exposition, the degree of development of the basic concepts and methods,and the occasional invocation of material from the second part of the textbook whenthe topic under consideration is already accessible for the audience in a more generalform.In conclusion I would like to thank colleagues and students, both known and unknown to me, for reviews and constructive remarks on the first edition of the course.It was particularly interesting for me to read the reviews of A.N. Kolmogorov andV.I.
Arnol’d. Very different in size, form, and style, these two have, on the professional level, so many inspiring things in common.Moscow, Russia1997V. ZorichFrom the Preface to the First Russian EditionThe creation of the foundations of the differential and integral calculus by Newtonand Leibniz three centuries ago appears even by modern standards to be one of thegreatest events in the history of science in general and mathematics in particular.Mathematical analysis (in the broad sense of the word) and algebra have intertwined to form the root system on which the ramified tree of modern mathematicsis supported and through which it makes its vital contact with the nonmathematicalsphere.
It is for this reason that the foundations of analysis are included as a necessary element of even modest descriptions of so-called higher mathematics; and it isprobably for that reason that so many books aimed at different groups of readers aredevoted to the exposition of the fundamentals of analysis.xPrefacesThis book has been aimed primarily at mathematicians desiring (as is proper) toobtain thorough proofs of the fundamental theorems, but who are at the same timeinterested in the life of these theorems outside of mathematics itself.The characteristics of the present course connected with these circumstances reduce basically to the following:In the Exposition Within each major topic the exposition is as a rule inductive,sometimes proceeding from the statement of a problem and suggestive heuristicconsiderations toward its solution to fundamental concepts and formalisms.Detailed at first, the exposition becomes more and more compressed as the courseprogresses.An emphasis is placed on the efficient machinery of smooth analysis.
In the exposition of the theory I have tried (to the extent of my knowledge) to point out themost essential methods and facts and avoid the temptation of a minor strengtheningof a theorem at the price of a major complication of its proof.The exposition is geometric throughout wherever this seemed worthwhile in order to reveal the essence of the matter.The main text is supplemented with a rather large collection of examples, andnearly every section ends with a set of problems that I hope will significantly complement even the theoretical part of the main text.
Following the wonderful precedent of Pólya and Szegő, I have often tried to present a beautiful mathematical resultor an important application as a series of problems accessible to the reader.The arrangement of the material was dictated not only by the architectureof mathematics in the sense of Bourbaki, but also by the position of analysisas a component of a unified mathematical or, one should rather say, naturalscience/mathematical education.In Content This course is being published in two books (Part 1 and Part 2).The present Part 1 contains the differential and integral calculus of functions ofone variable and the differential calculus of functions of several variables.In differential calculus we emphasize the role of the differential as a linear standard for describing the local behavior of the variation of a variable.
In addition tonumerous examples of the use of differential calculus to study functional relations(monotonicity, extrema) we exhibit the role of the language of analysis in writingsimple differential equations – mathematical models of real-world phenomena andthe substantive problems connected with them.We study a number of such problems (for example, the motion of a body of variable mass, a nuclear reactor, atmospheric pressure, motion in a resisting medium)whose solution leads to important elementary functions. Full use is made of the language of complex variables; in particular, Euler’s formula is derived and the unityof the fundamental elementary functions is shown.The integral calculus has consciously been explained as far as possible using intuitive material in the framework of the Riemann integral. For the majority of applica-Prefacesxitions, this is completely adequate.3 Various applications of the integral are pointedout, including those that lead to an improper integral (for example, the work involved in escaping from a gravitational field, and the escape velocity for the Earth’sgravitational field) or to elliptic functions (motion in a gravitational field in the presence of constraints, pendulum motion).The differential calculus of functions of several variables is very geometric.
Inthis topic, for example, one studies such important and useful consequences of theimplicit function theorem as curvilinear coordinates and local reduction to canonicalform for smooth mappings (the rank theorem) and functions (Morse’s lemma), andalso the theory of extrema with constraint.Results from the theory of continuous functions and differential calculus are summarized and explained in a general invariant form in two chapters that link up naturally with the differential calculus of real-valued functions of several variables.These two chapters open the second part of the course.
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