This textbook presents all the basics for the first two years of a course in mathematical analysis, from the natural numbers to Stokes-Cartan Theorem.
The main novelty which distinguishes this book is the choice of introducing the Kurzweil-Henstock integral from the very beginning. Although this approach requires a small additional effort by the student, it will be compensated by a substantial advantage in the development of the theory, and later on when learning about more advanced topics.
The text guides the reader with clarity in the discovery of the many different subjects, providing all necessary tools – no preliminaries are needed. Both students and their instructors will benefit from this book and its novel approach, turning their course in mathematical analysis into a gratifying and successful experience.
Author(s): Alessandro Fonda
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 441
City: Cham
Preface
Contents
Preliminaries
The Symbols of Logic
Logical Propositions
The Language of Set Theory
First Symbols
Some Examples of Sets
Operations with Sets
The Concept of Function
Part I The Basics of Mathematical Analysis
1 Sets of Numbers and Metric Spaces
1.1 The Natural Numbers and the Induction Principle
1.1.1 Recursive Definitions
1.1.2 Proofs by Induction
1.1.3 The Binomial Formula
1.2 The Real Numbers
1.2.1 Supremum and Infimum
1.2.2 The Square Root
1.2.3 Intervals
1.2.4 Properties of Q and RQ
1.3 The Complex Numbers
1.3.1 Algebraic Equations in C
1.3.2 The Modulus of a Complex Number
1.4 The Space RN
1.4.1 Euclidean Norm and Distance
1.5 Metric Spaces
2 Continuity
2.1 Continuous Functions
2.2 Intervals and Continuity
2.3 Monotone Functions
2.4 The Exponential Function
2.5 The Trigonometric Functions
2.6 Other Examples of Continuous Functions
3 Limits
3.1 The Notion of Limit
3.2 Some Properties of Limits
3.3 Change of Variables in the Limit
3.4 On the Limit of Restrictions
3.5 The Extended Real Line
3.6 Some Operations with -∞ and +∞
3.7 Limits of Monotone Functions
3.8 Limits for Exponentials and Logarithms
3.9 Liminf and Limsup
4 Compactness and Completeness
4.1 Some Preliminaries on Sequences
4.2 Compact Sets
4.3 Compactness and Continuity
4.4 Complete Metric Spaces
4.5 Completeness and Continuity
4.6 Spaces of Continuous Functions
5 Exponential and Circular Functions
5.1 The Construction
5.1.1 Preliminaries for the Proof
5.1.2 Definition on a Dense Set
5.1.3 Extension to the Whole Real Line
5.2 Exponential and Circular Functions
5.3 Limits for Trigonometric Functions
Part II Differential and Integral Calculus in R
6 The Derivative
6.1 Some Differentiation Rules
6.2 The Derivative Function
6.3 Remarkable Properties of the Derivative
6.4 Inverses of Trigonometric and Hyperbolic Functions
6.5 Convexity and Concavity
6.6 L'Hôpital's Rules
6.7 Taylor Formula
6.8 Local Maxima and Minima
6.9 Analyticity of Some Elementary Functions
7 The Integral
7.1 Riemann Sums
7.2 δ-Fine Tagged Partitions
7.3 Integrable Functions on a Compact Interval
7.4 Elementary Properties of the Integral
7.5 The Fundamental Theorem
7.6 Primitivable Functions
7.7 Primitivation by Parts and by Substitution
7.8 The Taylor Formula with Integral Form Remainder
7.9 The Cauchy Criterion
7.10 Integrability on Subintervals
7.11 R-Integrable and Continuous Functions
7.12 Two Theorems Involving Limits
7.13 Integration on Noncompact Intervals
7.14 Functions with Vector Values
Part III Further Developments
8 Numerical Series and Series of Functions
8.1 Introduction and First Properties
8.2 Series of Real Numbers
8.3 Series of Complex Numbers
8.4 Series of Functions
8.4.1 Power Series
8.4.2 The Complex Exponential Function
8.4.3 Taylor Series
8.4.4 Fourier Series
8.5 Series and Integrals
9 More on the Integral
9.1 Saks–Henstock Theorem
9.2 L-Integrable Functions
9.3 Monotone Convergence Theorem
9.4 Dominated Convergence Theorem
9.5 Hake's Theorem
Part IV Differential and Integral Calculus in RN
10 The Differential
10.1 The Differential of a Scalar-Valued Function
10.2 Some Computational Rules
10.3 Twice Differentiable Functions
10.4 Taylor Formula
10.5 The Search for Maxima and Minima
10.6 Implicit Function Theorem: First Statement
10.7 The Differential of a Vector-Valued Function
10.8 The Chain Rule
10.9 Mean Value Theorem
10.10 Implicit Function Theorem: General Statement
10.11 Local Diffeomorphisms
10.12 M-Surfaces
10.13 Local Analysis of M-Surfaces
10.14 Lagrange Multipliers
10.15 Differentiable Manifolds
11 The Integral
11.1 Integrability on Rectangles
11.2 Integrability on a Bounded Set
11.3 The Measure
11.4 Negligible Sets
11.5 A Characterization of Measurable Bounded Sets
11.6 Continuous Functions and L-Integrable Functions
11.7 Limits and Derivatives under the Integration Sign
11.8 Reduction Formula
11.9 Change of Variables in the Integral
11.10 Change of Measure by Diffeomorphisms
11.11 The General Theorem on Change of Variables
11.12 Some Useful Transformations in R2
11.13 Cylindrical and Spherical Coordinates in R3
11.14 The Integral on Unbounded Sets
11.15 The Integral on M-Surfaces
11.16 M-Dimensional Measure
11.17 Length and Area
11.18 Approximation with Smooth M-Surfaces
11.19 The Integral on a Compact Manifold
12 Differential Forms
12.1 An Informal Definition
12.2 Algebraic Operations
12.3 The Exterior Differential
12.4 Differential Forms in R3
12.5 The Integral on an M-Surface
12.6 Pull-Back Transformation
12.7 Oriented Boundary of a Rectangle
12.8 Gauss Formula
12.9 Oriented Boundary of an M-Surface
12.10 Stokes–Cartan Formula
12.11 Physical Interpretation of Curl and Divergence
12.12 The Integral on an Oriented Compact Manifold
12.13 Closed and Exact Differential Forms
12.14 On the Precise Definition of a Differential Form
Bibliography
References Cited in the Book
Books on the Kurzweil–Henstock Integral
Some Textbooks on Exercises
Index
