This textbook provides a comprehensive introduction to mathematical calculus. Written for advanced undergraduate and graduate students, it teaches the fundamental mathematical concepts, methods and tools required for various areas of economics and the social sciences, such as optimization and measure theory. The reader will be introduced to topological, metric and normed spaces, learning about numerical sequences, series, and differential and integral calculus. These concepts are introduced using the axiomatic approach as a tool for logical reasoning, consistency, and formalization of ideas. The book follows a theorem-proving approach, stressing the limitations of applying the different theorems, while providing thought-provoking counter-examples. Each chapter features exercises that facilitate learning and allow students to apply and test important concepts and tools.
Author(s): Giulio Bottazzi
Series: Classroom Companion: Economics
Publisher: Springer
Year: 2023
Language: English
Pages: 319
City: Cham
Preface
Contents
1 Preliminaries
1.1 Sets, Equivalence Relations, and Functions
1.2 Order Relations, Supremum, and Infimum
1.3 Countable Sets
1.4 Operations and Fields
1.5 Real Numbers
1.6 Convexity and Concavity
Exercises
2 Topology
2.1 Definition and Basic Properties
2.2 Base of a Topology
2.2.1 Countability
2.2.2 Euclidean Topology
2.3 Cover and Compactness
2.3.1 Hausdorff Spaces
2.3.2 Compactness in Euclidean Topology
2.3.3 The Extended Real Number System
2.4 Connectedness
2.5 Limit of Functions and Continuity
2.5.1 Continuity in Euclidean Topology
Exercises
3 Metric Spaces
3.1 Definition and Basic Properties
3.2 Metric and Topology
3.3 Uniform Continuity
3.4 Lipschitz Continuity
Exercises
4 Normed Spaces
4.1 Definition and Basic Properties
4.2 Example of Normed Spaces
4.2.1 Euclidean Norm in mathbbRn
4.2.2 p-Norm in mathbbRn
4.2.3 Operator Norm
4.3 Finite-Dimensional Normed Spaces
4.3.1 Equivalence of Norms in mathbbRn
Exercises
5 Sequences and Series
5.1 Sequences in Topological Spaces
5.1.1 Subsequences
5.1.2 Sequences and Functions
5.1.3 Uniqueness of Limit
5.2 Sequences in Metric Spaces
5.2.1 Cauchy Sequences and Complete Spaces
5.3 Sequences in mathbbR
5.3.1 Upper and Lower Limits
5.3.2 Infinity and Infinitesimals
5.4 Sequences in Normed Spaces
5.5 Series in mathbbR
5.5.1 Series with Decreasing Terms
5.5.2 Tests Based on the Asymptotic Behaviour of Terms
5.6 Sequences and Series of Functions
5.6.1 Uniform Convergence
Exercises
6 Differential Calculus of Functions of One Variable
6.1 Limit of Real Functions
6.2 Continuity of Real Functions
6.3 Differential Analysis
6.3.1 Higher-Order Derivatives
6.3.2 Derivatives and Function Behaviour
6.3.3 Derivatives and Limits
6.4 Taylor Polynomial and Power Series Expansion
Exercises
7 Differential Calculus of Functions of Several Variables
7.1 Limits and Continuity in mathbbRn
7.2 Differential Analysis in mathbbRn
7.3 Mean Value Theorems
7.4 Higher-Order Derivatives and Taylor Polynomial
7.4.1 Local Maxima and Minima
7.5 Inverse Function Theorem
7.6 Implicit Function Theorem
7.6.1 Real Functions of Two Variables
7.6.2 Real Functions of Several Variables
7.6.3 Vector Functions of Several Variables
7.6.4 Dependent and Independent Functions
7.7 Constrained Optimisation
7.7.1 One Dimensional Problems
7.7.2 Two Dimensional Problems
7.7.3 Theorems of the Alternatives
7.7.4 First-Order Conditions
7.7.5 Second-Order Conditions
7.7.6 Envelope Theorem
7.7.7 Minimisation Problems
Exercises
8 Integral Calculus
8.1 Definite Integrals
8.1.1 Properties of the Definite Integral
8.1.2 Riemann Integrable Functions
8.1.3 Improper Integrals
8.1.4 Integral of Vector-Valued Functions
8.2 The Fundamental Theorem of Calculus
8.3 Riemann–Stieltjes Integral
8.3.1 Stieltjes Integrable Functions
8.3.2 Properties of the Stieltjes Integral
Exercises
9 Measure Theory
9.1 Algebras, Measurable Spaces, and Measures
9.1.1 Complete Measure Space
9.1.2 Borel σ-Algebra
9.1.3 Lebesgue Measure
9.2 Measurable Functions
9.2.1 Measurable Real-Valued Functions
9.3 Lebesgue Integral
9.3.1 Lebesgue and Riemann Integral on mathbbR
9.4 Product Measure Space
9.4.1 Product σ-Algebra
9.4.2 Product Measure
9.4.3 Multiple Integrals
9.5 Probability Measure
9.5.1 Multiple Random Variables
9.5.2 Banach Space of Square Summable Random Variables
Exercises
Appendix A Cauchy Initial Value Problem
Appendix B Brouwer Fixed Point Theorem
Appendix Index
