Mathematical Analysis I / Edition 1

Mathematical Analysis I / Edition 1

by Vladimir A. Zorich

ISBN-10: 3540874518

ISBN-13: 9783540874515

Pub. Date: 11/28/2008

Publisher: Springer Berlin Heidelberg

This softcover edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, elliptic functions and distributions. Especially notable in this course is the clearly expressed orientation toward

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This softcover edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, elliptic functions and distributions. Especially notable in this course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books.

The first volume constitutes a complete course on one-variable calculus along with the multivariable differential calculus elucidated in an up-to-day, clear manner, with a pleasant geometric flavor.

Product Details

Springer Berlin Heidelberg
Publication date:
Universitext Series
Edition description:
1st ed. 2004. 2nd printing 2008
Sales rank:
Product dimensions:
6.10(w) x 9.20(h) x 1.30(d)

Table of Contents


Prefaces Preface to the English edition Prefaces to the fourth and third editions Preface to the second edition From the preface to the first edition

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. 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 Set-theoretic language for propositions
1.4.4 Exercises

2. The Real Numbers

2.1 Axioms and properties 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. Least upper bound

2.2 Classes of real numbers and computations
2.2.1 The natural numbers. 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 Corollaries
2.2.4 Geometric interpretation. Computational aspects 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 on completeness
2.3.1 The nested interval lemma
2.3.2 The finite covering lemma
2.3.3 The limit point lemma
2.3.4 Problems and exercises

2.4 Countable and uncountable sets
2.4.1 Countable sets Corollaries
2.4.2 The cardinality of the continuum Corollaries
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 a. General properties b. Passage to the limit and the arithmetic operations c. Passage to the limit and inequalities
3.1.3 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 Concluding remarks
3.1.4 Elementary facts about series a. The sum of a series and the CauchyCauchy, 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 Limits 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

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 Remarks to Theorem 2
4.2.3 Problems and exercises

5. Differential Calculus

5.1 Differentiable functions
5.1.1 Statement of the problem
5.1.2 Functions differentiable at a point
5.1.3 Tangents. Geometric meaning of the derivative
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 elementary functions
5.2.5 Differentiation of a very simple implicit function
5.2.6 Higher-order derivatives Examples
5.2.7 Problems and exercises

5.3 The basic theorems of differential calculus
5.3.1 Fermat's lemma and Rolle's theorem Remarks on Fermat's lemma
5.3.2 The theorems of Lagrange and Cauchy Remarks on Lagrange's theorem Corollaries of Lagrange's theorem Remarks on Cauchy's theorem
5.3.3 Taylor's formula
5.3.4 Problems and exercises

5.4 Differential calculus used to study functions
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'Hospital'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 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 Examples
5.5.3 Euler's formula and the elementary functions
5.5.4 Power series representation. Analyticity
5.5.5 Algebraic closedness of the field C
5.5.6 Problems and exercises

5.6 Examples of differential calculus in natural science
5.6.1 Motion of a body of variable mass
5.6.2 The barometric formula
5.6.3 Radioactive decay 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 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 int R(cos x, sin x) dx
5.7.5 Primitives of the form int R(x,y(x)) dx
5.7.6 Problems and exercises

6. Integration

6.1 Definition of the integral
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 interval function
6.2.3 Estimation, monotonicity, 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 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
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

7. Functions of Several Variables

7.1 The space Rm and 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
7.2.3 Problems and exercises

8. Differential Calculus in 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 at a point
8.2.2 Partial derivatives of a real-valued function
8.2.3 Coordinate representation. Jacobians
8.2.4 Partial derivatives and differentiability 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 composite mapping (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 Real-valued functions of several variables
8.4.1 The mean-value theorem
8.4.2 A sufficient condition for differentiability
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 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 Preliminary considerations
8.5.2 An elementary implicit function theorem
8.5.3 Transition to 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 to canonical form
8.6.3 Functional dependence
8.6.4 Local resolution of a diffeomorphism
8.6.5 Morse's lemma
8.6.6 Problems and exercises

8.7 Surfaces in Rn and constrained extrema
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. Introduction to several variables

4. Differential calculus of several variables

Examination Topics

1. First semester
1.1. Introduction and one-variable differential calculus

2. Second semester
2.1. Integration. Multivariable differential calculus


1. Classic works
1.1 Primary sources
1.2. Major comprehensive expository works

2. Textbooks

3. Classroom materials

4. Further reading

Subject Index

Name Index

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