Mathematical Analysis I (Universitext)

✍ Scribed by Vladimir A. Zorich

• Publisher: Springer
• Year: 2004
• Tongue: English
• Leaves: 598
• Edition: 2004
• Category: Library

✦ Synopsis

FIRST VOLUME OF THE AMAZING 2 VOLUME ANALYSIS TEXT BY THE INTERNATIONALLY REKNOWNED EXPERT,BASED ON COURSES TAUGHT AT MOSCOW STATE UNIVERSITY FOR DECADES TO GIFTED FRESHMAN,UNIQUE IN IT'S APPROACH IN THAT IT COMBINES THE HARD THEORY OF REAL AND COMPLEX ANALYSIS AND APPLICATIONS TO PHYSICS IN THE SAME PRESENTATION. MOST ANALYSIS TEXTS DON'T COVER BOTH AND THIS PRESENTATION IS VERY UNIQUE IN THIS MANNER. THIS IS A NEARLY PERFECT COPY OF THE 2004 1ST HARDCOVER EDITION,A GREAT COPY FOR CLASS!

✦ Table of Contents

Cover
Mathematical Analysis I
Copyright - ISBN 3540403868
Prefaces
Preface to the English Edition
Preface to the Fourth Russian Edition
Preface to the Third Russian edition
Preface to the Second Russian Edition
Prom the Preface to the First Russian Edition
Table of Contents
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
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
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
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
2.2.2 Rational and 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
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
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
3.1.3 Questions Involving the Existence of the Limit of a Sequence
3.1.4 Elementary Facts about 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
3.2.3 The General Definition of the Limit of a Function (Limit over a Base)
3.2.4 Existence of the Limit of a Function
3.2.5 Problems and Exercises
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
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
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
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
5.4.6 Problems and Exercises
5.5 Complex Numbers and the Connections Among the Elementary Functions
5.5.1 Complex Numbers
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 exp x 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
5.7.3 Primitives of Rational Functions
5.7.4 Primitives of the Form / R(cos x, sin x) dx
5.7.5 Primitives of the Form / R(x, y(x)) dx
5.7.6 Problems and Exercises
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
6.1.3 The Set of Integrable Functions
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
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
6.5.3 Improper Integrals with More than one Singularity
6.5.4 Problems and Exercises
7 Functions of Several Variables: their Limits and Continuity
7.1 The Space IR^m and the Most Important Classes of its Subsets
7.1.1 The Set IR^m and the Distance in it
7.1.2 Open and Closed Sets in IR^m
7.1.3 Compact Sets in IR^m
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
7.2.3 Problems and Exercises
8 The Differential Calculus of Functions of Several Variables
8.1 The Linear Structure on IR^m
8.1.1 IR^m as a Vector Space
8.1.2 Linear Transformations L : IR^m -> IR^n
8.1.3 The Norm in IR^m
8.1.4 The Euclidean Structure on IR^m
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)
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
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(x^1, ..., x^m, 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 IR^n and the Theory of Extrema with Constraint

8.7.2 The Tangent Space
8.7.3 Extrema with Constraint
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
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
A
B
C
D
E
F
G
H
I
J
L
M
N
O
P
Q
R
S
T
U
V
W
Z
Name Index