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Vladimir.A.Zorich.Mathematical.Analysis.I.2nd.pdf
Mathematical Analysis I
Prefaces
Preface to the Second English Edition
Preface to the First English Edition
Preface to the Sixth Russian Edition
Preface to the Second Russian Edition
From the Preface to the First Russian Edition
Contents
Chapter 1: Some General Mathematical Concepts and Notation
1.1 Logical Symbolism
1.1.1 Connectives and Brackets
1.1.2 Remarks on Proofs
1.1.3 Some Special Notation
1.1.4 Concluding Remarks
1.1.5 Exercises
1.2 Sets and Elementary Operations on Them
1.2.1 The Concept of a Set
1.2.2 The Inclusion Relation
1.2.3 Elementary Operations on Sets
1.2.4 Exercises
1.3 Functions
1.3.1 The Concept of a Function (Mapping)
1.3.2 Elementary Classification of Mappings
1.3.3 Composition of Functions and Mutually Inverse Mappings
1.3.4 Functions as Relations. The Graph of a Function
a. Relations
b. Functions and Their Graphs
1.3.5 Exercises
1.4 Supplementary Material
1.4.1 The Cardinality of a Set (Cardinal Numbers)
1.4.2 Axioms for Set Theory
1.4.3 Remarks on the Structure of Mathematical Propositions and Their Expression in the Language of Set Theory
1.4.4 Exercises
Chapter 2: The Real Numbers
2.1 The Axiom System and Some General Properties of the Set of Real Numbers
2.1.1 Definition of the Set of Real Numbers
2.1.2 Some General Algebraic Properties of Real Numbers
a. Consequences of the Addition Axioms
b. Consequences of the Multiplication Axioms
c. Consequences of the Axiom Connecting Addition and Multiplication
d. Consequences of the Order Axioms
e. Consequences of the Axioms Connecting Order with Addition and Multiplication
2.1.3 The Completeness Axiom and the Existence of a Least Upper (or Greatest Lower) Bound of a Set of Numbers
2.2 The Most Important Classes of Real Numbers and Computational Aspects of Operations with Real Numbers
2.2.1 The Natural Numbers and the Principle of Mathematical Induction
a. Definition of the Set of Natural Numbers
b. The Principle of Mathematical Induction
2.2.2 Rational and Irrational Numbers
a. The Integers
b. The Rational Numbers
c. The Irrational Numbers
2.2.3 The Principle of Archimedes
2.2.4 The Geometric Interpretation of the Set of Real Numbers and Computational Aspects of Operations with Real Numbers
a. The Real Line
b. Defining a Number by Successive Approximations
c. The Positional Computation System
2.2.5 Problems and Exercises
2.3 Basic Lemmas Connected with the Completeness of the Real Numbers
2.3.1 The Nested Interval Lemma (Cauchy-Cantor Principle)
2.3.2 The Finite Covering Lemma (Borel-Lebesgue Principle, or Heine-Borel Theorem)
2.3.3 The Limit Point Lemma (Bolzano-Weierstrass Principle)
2.3.4 Problems and Exercises
2.4 Countable and Uncountable Sets
2.4.1 Countable Sets
2.4.2 The Cardinality of the Continuum
2.4.3 Problems and Exercises
Chapter 3: Limits
3.1 The Limit of a Sequence
3.1.1 Definitions and Examples
3.1.2 Properties of the Limit of a Sequence
a. General Properties
b. Passage to the Limit and the Arithmetic Operations
c. Passage to the Limit and Inequalities
3.1.3 Questions Involving the Existence of the Limit of a Sequence
a. The Cauchy Criterion
b. A Criterion for the Existence of the Limit of a Monotonic Sequence
c. The Number e
d. Subsequences and Partial Limits of a Sequence
3.1.4 Elementary Facts About Series
a. The Sum of a Series and the Cauchymat]Cauchy, A. Criterion for Convergence of a Series
b. Absolute Convergence. The Comparison Theorem and Its Consequences
c. The Number e as the Sum of a Series
3.1.5 Problems and Exercises
3.2 The Limit of a Function
3.2.1 Definitions and Examples
3.2.2 Properties of the Limit of a Function
a. General Properties of the Limit of a Function
b. Passage to the Limit and Arithmetic Operations
c. Passage to the Limit and Inequalities
d. Two Important Examples
3.2.3 The General Definition of the Limit of a Function (Limit over a Base)
a. Bases; Definition and Elementary Properties
b. The Limit of a Function over a Base
3.2.4 Existence of the Limit of a Function
a. The Cauchy Criterion
b. The Limit of a Composite Function
c. The Limit of a Monotonic Function
d. Comparison of the Asymptotic Behavior of Functions
3.2.5 Problems and Exercises
Chapter 4: Continuous Functions
4.1 Basic Definitions and Examples
4.1.1 Continuity of a Function at a Point
4.1.2 Points of Discontinuity
4.2 Properties of Continuous Functions
4.2.1 Local Properties
4.2.2 Global Properties of Continuous Functions
4.2.3 Problems and Exercises
Chapter 5: Differential Calculus
5.1 Differentiable Functions
5.1.1 Statement of the Problem and Introductory Considerations
5.1.2 Functions Differentiable at a Point
5.1.3 The Tangent Line; Geometric Meaning of the Derivative and Differential
5.1.4 The Role of the Coordinate System
5.1.5 Some Examples
5.1.6 Problems and Exercises
5.2 The Basic Rules of Differentiation
5.2.1 Differentiation and the Arithmetic Operations
5.2.2 Differentiation of a Composite Function (Chain Rule)
5.2.3 Differentiation of an Inverse Function
5.2.4 Table of Derivatives of the Basic Elementary Functions
5.2.5 Differentiation of a Very Simple Implicit Function
5.2.6 Higher-Order Derivatives
5.2.7 Problems and Exercises
5.3 The Basic Theorems of Differential Calculus
5.3.1 Fermat's Lemma and Rolle's Theorem
5.3.2 The Theorems of Lagrange and Cauchy on Finite Increments
Corollaries of Lagrange's Theorem
5.3.3 Taylor's Formula
5.3.4 Problems and Exercises
5.4 The Study of Functions Using the Methods of Differential Calculus
5.4.1 Conditions for a Function to be Monotonic
5.4.2 Conditions for an Interior Extremum of a Function
a. Young's Inequalities
b. Hölder's Inequalities
c. Minkowski's Inequalities
5.4.3 Conditions for a Function to be Convex
5.4.4 L'Hôpital's Rule
5.4.5 Constructing the Graph of a Function
a. Graphs of the Elementary Functions
b. Examples of Sketches of Graphs of Functions (Without Application of the Differential Calculus)
c. The Use of Differential Calculus in Constructing the Graph of a Function
5.4.6 Problems and Exercises
5.5 Complex Numbers and the Connections Among the Elementary Functions
5.5.1 Complex Numbers
a. Algebraic Extension of the Field R
b. Geometric Interpretation of the Field C
5.5.2 Convergence in C and Series with Complex Terms
5.5.3 Euler's Formula and the Connections Among the Elementary Functions
5.5.4 Power Series Representation of a Function. Analyticity
5.5.5 Algebraic Closedness of the Field C of Complex Numbers
5.5.6 Problems and Exercises
5.6 Some Examples of the Application of Differential Calculus in Problems of Natural Science
5.6.1 Motion of a Body of Variable Mass
5.6.2 The Barometric Formula
5.6.3 Radioactive Decay, Chain Reactions, and Nuclear Reactors
5.6.4 Falling Bodies in the Atmosphere
5.6.5 The Number e and the Function expx Revisited
5.6.6 Oscillations
5.6.7 Problems and Exercises
5.7 Primitives
5.7.1 The Primitive and the Indefinite Integral
5.7.2 The Basic General Methods of Finding a Primitive
a. Linearity of the Indefinite Integral
b. Integration by Parts
c. Change of Variable in an Indefinite Integral
5.7.3 Primitives of Rational Functions
5.7.4 Primitives of the Form R(cosx, sinx)dx
a.
b.
c.
5.7.5 Primitives of the Form R(x,y(x))dx
a.
b.
c. Elliptic Integrals
5.7.6 Problems and Exercises
Chapter 6: Integration
6.1 Definition of the Integral and Description of the Set of Integrable Functions
6.1.1 The Problem and Introductory Considerations
6.1.2 Definition of the Riemann Integral
a. Partitions
b. A Base in the Set of Partitions
c. Riemann Sums
d. The Riemann Integral
6.1.3 The Set of Integrable Functions
a. A Necessary Condition for Integrability
b. A Sufficient Condition for Integrability and the Most Important Classes of Integrable Functions
c. The Vector Space R[a,b]
d. Lebesgue's Criterion for Riemann Integrability of a Function
6.1.4 Problems and Exercises
6.2 Linearity, Additivity and Monotonicity of the Integral
6.2.1 The Integral as a Linear Function on the Space R[a,b]
6.2.2 The Integral as an Additive Function of the Interval of Integration
6.2.3 Estimation of the Integral, Monotonicity of the Integral, and the Mean-Value Theorem
a. A General Estimate of the Integral
b. Monotonicity of the Integral and the First Mean-Value Theorem
c. The Second Mean-Value Theorem for the Integral
6.2.4 Problems and Exercises
6.3 The Integral and the Derivative
6.3.1 The Integral and the Primitive
6.3.2 The Newton-Leibniz Formula
6.3.3 Integration by Parts in the Definite Integral and Taylor's Formula
6.3.4 Change of Variable in an Integral
6.3.5 Some Examples
6.3.6 Problems and Exercises
6.4 Some Applications of Integration
6.4.1 Additive Interval Functions and the Integral
6.4.2 Arc Length
6.4.3 The Area of a Curvilinear Trapezoid
6.4.4 Volume of a Solid of Revolution
6.4.5 Work and Energy
6.4.6 Problems and Exercises
6.5 Improper Integrals
6.5.1 Definition, Examples, and Basic Properties of Improper Integrals
6.5.2 Convergence of an Improper Integral
a. The Cauchy Criterion
b. Absolute Convergence of an Improper Integral
c. Conditional Convergence of an Improper Integral
6.5.3 Improper Integrals with More than One Singularity
6.5.4 Problems and Exercises
Chapter 7: Functions of Several Variables: Their Limits and Continuity
7.1 The Space Rm and the Most Important Classes of Its Subsets
7.1.1 The Set Rm and the Distance in It
7.1.2 Open and Closed Sets in Rm
7.1.3 Compact Sets in Rm
7.1.4 Problems and Exercises
7.2 Limits and Continuity of Functions of Several Variables
7.2.1 The Limit of a Function
7.2.2 Continuity of a Function of Several Variables and Properties of Continuous Functions
Local Properties of Continuous Functions
Global Properties of Continuous Functions
7.2.3 Problems and Exercises
Chapter 8: The Differential Calculus of Functions of Several Variables
8.1 The Linear Structure on Rm
8.1.1 Rm as a Vector Space
8.1.2 Linear Transformations L:Rm->Rn
8.1.3 The Norm in Rm
8.1.4 The Euclidean Structure on Rm
8.2 The Differential of a Function of Several Variables
8.2.1 Differentiability and the Differential of a Function at a Point
8.2.2 The Differential and Partial Derivatives of a Real-Valued Function
8.2.3 Coordinate Representation of the Differential of a Mapping. The Jacobi Matrix
8.2.4 Continuity, Partial Derivatives, and Differentiability of a Function at a Point
8.3 The Basic Laws of Differentiation
8.3.1 Linearity of the Operation of Differentiation
8.3.2 Differentiation of a Composition of Mappings (Chain Rule)
a. The Main Theorem
b. The Differential and Partial Derivatives of a Composite Real-Valued Function
c. The Derivative with Respect to a Vector and the Gradient of a Function at a Point
8.3.3 Differentiation of an Inverse Mapping
8.3.4 Problems and Exercises
8.4 The Basic Facts of Differential Calculus of Real-Valued Functions of Several Variables
8.4.1 The Mean-Value Theorem
8.4.2 A Sufficient Condition for Differentiability of a Function of Several Variables
8.4.3 Higher-Order Partial Derivatives
8.4.4 Taylor's Formula
8.4.5 Extrema of Functions of Several Variables
8.4.6 Some Geometric Images Connected with Functions of Several Variables
a. The Graph of a Function and Curvilinear Coordinates
b. The Tangent Plane to the Graph of a Function
c. The Normal Vector
d. Tangent Planes and Tangent Vectors
8.4.7 Problems and Exercises
8.5 The Implicit Function Theorem
8.5.1 Statement of the Problem and Preliminary Considerations
8.5.2 An Elementary Version of the Implicit Function Theorem
8.5.3 Transition to the Case of a Relation F(x1,…,xm, y)=0
8.5.4 The Implicit Function Theorem
8.5.5 Problems and Exercises
8.6 Some Corollaries of the Implicit Function Theorem
8.6.1 The Inverse Function Theorem
8.6.2 Local Reduction of a Smooth Mapping to Canonical Form
8.6.3 Functional Dependence
8.6.4 Local Resolution of a Diffeomorphism into a Composition of Elementary Ones
8.6.5 Morse's Lemma
8.6.6 Problems and Exercises
8.7 Surfaces in Rn and the Theory of Extrema with Constraint
8.7.1 k-Dimensional Surfaces in Rn
8.7.2 The Tangent Space
8.7.3 Extrema with Constraint
a. Statement of the Problem
b. A Necessary Condition for an Extremum with Constraint
c. A Sufficient Condition for a Constrained Extremum
8.7.4 Problems and Exercises
Some Problems from the Midterm Examinations
1 Introduction to Analysis (Numbers, Functions, Limits)
2 One-Variable Differential Calculus
3 Integration and Introduction to Several Variables
4 Differential Calculus of Several Variables
Examination Topics
1 First Semester
1.1 Introduction to Analysis and One-Variable Differential Calculus
2 Second Semester
2.1 Integration. Multivariable Differential Calculus
Appendix A: Mathematical Analysis (Introductory Lecture)
A.1 Two Words About Mathematics
A.2 Number, Function, Law
A.3 Mathematical Model of a Phenomenon (Differential Equations, or We Learn How to Write)
A.4 Velocity, Derivative, Differentiation
A.5 Higher Derivatives, What for?
A.5.1 Again Toward Numbers
A.5.2 And What to Do Next?
Appendix B: Numerical Methods for Solving Equations (An Introduction)
B.1 Roots of Equations and Fixed Points of Mappings
B.2 Contraction Mappings and Iterative Process
B.3 The Method of Tangents (Newton's Method)
Appendix C: The Legendre Transform (First Discussion)
C.1 Initial Definition of the Legendre Transform and the General Young Inequality
C.2 Specification of the Definition in the Case of Convex Functions
C.3 Involutivity of the Legendre Transform of a Function
C.4 Concluding Remarks and Comments
Appendix D: The Euler-MacLaurin Formula
D.1 Bernoulli Numbers
D.2 Bernoulli Polynomials
D.3 Some Known Operators and Series of Operators
D.4 Euler-MacLaurin Series and Formula
D.5 The General Euler-MacLaurin Formula
D.6 Applications
D.7 Again to the Actual Euler-MacLaurin Formula
Appendix E: Riemann-Stieltjes Integral, Delta Function, and the Concept of Generalized Functions
E.1 The Riemann-Stieltjes Integral
E.2 Case in Which the Riemann-Stieltjes Integral Reduces to the Riemann Integral
E.3 Heaviside Function and an Example of a Riemann-Stieltjes Integral Computation
E.4 Generalized Functions
E.4.1 Dirac's Delta Function. A Heuristic Description
E.5 The Correspondence Between Functions and Functionals
E.6 Functionals as Generalized Functions
E.7 Differentiation of Generalized Functions
E.8 Derivatives of the Heaviside Function and the Delta Function
Appendix F: The Implicit Function Theorem (An Alternative Presentation)
F.1 Formulation of the Problem
F.2 Some Reminders of Numerical Methods to Solve Equations
F.2.1 The Principle of the Fixed Point
F.3 The Implicit Function Theorem
F.3.1 Statement of the Theorem
F.3.2 Proof of the Existence of an Implicit Function
F.3.3 Continuity of an Implicit Function
F.3.4 Differentiability of an Implicit Function
F.3.5 Continuous Differentiability of an Implicit Function
F.3.6 Higher Derivatives of an Implicit Function
References
1 Classic Works
1.1 Primary Sources
1.2 Major Comprehensive Expository Works
1.3 Classical Courses of Analysis from the First Half of the Twentieth Century
2 Textbooks
3 Classroom Materials
4 Further Reading
Subject Index
Name Index
Universitext
Vladimir A. Zorich
Mathematical
Analysis I
Second Edition
Universitext
Universitext
Series Editors
Sheldon Axler
San Francisco State University, San Francisco, CA, USA
Vincenzo Capasso
Università degli Studi di Milano, Milano, Italy
Carles Casacuberta
Universitat de Barcelona, Barcelona, Spain
Angus MacIntyre
Queen Mary University of London, London, UK
Kenneth Ribet
University of California, Berkeley, CA, USA
Claude Sabbah
CNRS, École Polytechnique, Palaiseau, France
Endre Süli
University of Oxford, Oxford, UK
Wojbor A. Woyczy´nski
Case Western Reserve University, Cleveland, OH, USA
Universitext is a series of textbooks that presents material from a wide variety of
mathematical disciplines at master’s level and beyond. The books, often well class-
tested by their author, may have an informal, personal even experimental approach
to their subject matter. Some of the most successful and established books in the se-
ries have evolved through several editions, always following the evolution of teach-
ing curricula, to very polished texts.
Thus as research topics trickle down into graduate-level teaching, first textbooks
written for new, cutting-edge courses may make their way into Universitext.
For further volumes:
www.springer.com/series/223
Vladimir A. Zorich
Mathematical Analysis I
Second Edition
Vladimir A. Zorich
Department of Mathematics
Moscow State University
Moscow, Russia
Translators:
Roger Cooke (first English edition translated from the 4th Russian edition)
Burlington, Vermont, USA
and
Octavio Paniagua T. (Appendices A–F and new problems of the 6th Russian edition)
Berlin, Germany
Original Russian edition: Matematicheskij Analiz (Part I, 6th corrected edition, Moscow,
2012) MCCME (Moscow Center for Continuous Mathematical Education Publ.)
ISSN 0172-5939
Universitext
ISBN 978-3-662-48790-7
DOI 10.1007/978-3-662-48792-1
ISSN 2191-6675 (electronic)
ISBN 978-3-662-48792-1 (eBook)
Library of Congress Control Number: 2016930048
Mathematics Subject Classification (2010): 26-01, 26Axx, 26Bxx, 42-01
Springer Heidelberg New York Dordrecht London
© Springer-Verlag Berlin Heidelberg 2004, 2015
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
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Prefaces
Preface to the Second English Edition
Science has not stood still in the years since the first English edition of this book
was published. For example, Fermat’s last theorem has been proved, the Poincaré
conjecture is now a theorem, and the Higgs boson has been discovered. Other events
in science, while not directly related to the contents of a textbook in classical math-
ematical analysis, have indirectly led the author to learn something new, to think
over something familiar, or to extend his knowledge and understanding. All of this
additional knowledge and understanding end up being useful even when one speaks
about something apparently completely unrelated.1
In addition to the original Russian edition, the book has been published in En-
glish, German, and Chinese. Various attentive multilingual readers have detected
many errors in the text. Luckily, these are local errors, mostly misprints. They have
assuredly all been corrected in this new edition.
But the main difference between the second and first English editions is the addi-
tion of a series of appendices to each volume. There are six of them in the first and
five of them in the second. So as not to disturb the original text, they are placed at the
end of each volume. The subjects of the appendices are diverse. They are meant to be
useful to students (in mathematics and physics) as well as to teachers, who may be
motivated by different goals. Some of the appendices are surveys, both prospective
and retrospective. The final survey contains the most important conceptual achieve-
ments of the whole course, which establish connections between analysis and other
parts of mathematics as a whole.
1There is a story about Erd˝os, who, like Hadamard, lived a very long mathematical and human
life. When he was quite old, a journalist who was interviewing him asked him about his age. Erd˝os
replied, after deliberating a bit, “I remember that when I was very young, scientists established that
the Earth was two billion years old. Now scientists assert that the Earth is four and a half billion
years old. So, I am approximately two and a half billion years old.”
v
vi
Prefaces
I was happy to learn that this book has proven to be useful, to some extent, not
only to mathematicians, but also to physicists, and even to engineers from technical
schools that promote a deeper study of mathematics.
It is a real pleasure to see a new generation that thinks bigger, understands more
deeply, and is able to do more than the generation on whose shoulders it grew.
Moscow, Russia
2015
V. Zorich
Preface to the First English Edition
An entire generation of mathematicians has grown up during the time between the
appearance of the first edition of this textbook and the publication of the fourth
edition, a translation of which is before you. The book is familiar to many people,
who either attended the lectures on which it is based or studied out of it, and who
now teach others in universities all over the world. I am glad that it has become
accessible to English-speaking readers.
This textbook consists of two parts. It is aimed primarily at university students
and teachers specializing in mathematics and natural sciences, and at all those who
wish to see both the rigorous mathematical theory and examples of its effective use
in the solution of real problems of natural science.
Note that Archimedes, Newton, Leibniz, Euler, Gauss, Poincaré, who are held
in particularly high esteem by us, mathematicians, were more than mere math-
ematicians. They were scientists, natural philosophers. In mathematics resolving
of important specific questions and development of an abstract general theory are
processes as inseparable as inhaling and exhaling. Upsetting this balance leads to
problems that sometimes become significant both in mathematical education and in
science in general.
The textbook exposes classical analysis as it is today, as an integral part of the
unified Mathematics, in its interrelations with other modern mathematical courses
such as algebra, differential geometry, differential equations, complex and func-
tional analysis.
Rigor of discussion is combined with the development of the habit of working
with real problems from natural sciences. The course exhibits the power of con-
cepts and methods of modern mathematics in exploring specific problems. Various
examples and numerous carefully chosen problems, including applied ones, form
a considerable part of the textbook. Most of the fundamental mathematical notions
and results are introduced and discussed along with information, concerning their
history, modern state and creators. In accordance with the orientation toward natural
sciences, special attention is paid to informal exploration of the essence and roots of
the basic concepts and theorems of calculus, and to the demonstration of numerous,
sometimes fundamental, applications of the theory.
For instance, the reader will encounter here the Galilean and Lorentz transforms,
the formula for rocket motion and the work of nuclear reactor, Euler’s theorem
Prefaces
vii
on homogeneous functions and the dimensional analysis of physical quantities, the
Legendre transform and Hamiltonian equations of classical mechanics, elements of
hydrodynamics and the Carnot’s theorem from thermodynamics, Maxwell’s equa-
tions, the Dirac delta-function, distributions and the fundamental solutions, convo-
lution and mathematical models of linear devices, Fourier series and the formula
for discrete coding of a continuous signal, the Fourier transform and the Heisenberg
uncertainty principle, differential forms, de Rham cohomology and potential fields,
the theory of extrema and the optimization of a specific technological process, nu-
merical methods and processing the data of a biological experiment, the asymptotics
of the important special functions, and many other subjects.
Within each major topic the exposition is, as a rule, inductive, sometimes pro-
ceeding from the statement of a problem and suggestive heuristic considerations
concerning its solution, toward fundamental concepts and formalisms. Detailed at
first, the exposition becomes more and more compressed as the course progresses.
Beginning ab ovo the book leads to the most up-to-date state of the subject.
Note also that, at the end of each of the volumes, one can find the list of the main
theoretical topics together with the corresponding simple, but nonstandard problems
(taken from the midterm exams), which are intended to enable the reader both de-
termine his or her degree of mastery of the material and to apply it creatively in
concrete situations.
More complete information on the book and some recommendations for its use in
teaching can be found below in the prefaces to the first and second Russian editions.
Moscow, Russia
2003
V. Zorich
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