A Modern Introduction to Mathematical Analysis

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