Introduction to Real Analysis

Description

This is a text that develops calculus "from scratch", with complete rigorous arguments. Its aim is to introduce the reader not only to the basic facts about calculus but, as importantly, to mathematical reasoning. It covers in great detail calculus of one variable and multivariable calculus. Additionally it offers a basic introduction to the topology of Euclidean space. It is intended to more advanced or highly motivated undergraduates. Introduction The Greek Alphabet Chapter 1. The basics of mathematical reasoning 1.1. Statements and predicates 1.2. Quantifiers 1.3. Sets 1.4. Functions 1.5. Exercises 1.6. Exercises for extra-credit Chapter 2. The Real Number System 2.1. The algebraic axioms of the real numbers 2.2. The order axiom of the real numbers 2.3. The completeness axiom 2.4. Visualizing the real numbers 2.5. Exercises Chapter 3. Special classes of real numbers 3.1. The natural numbers and the induction principle 3.2. Applications of the induction principle 3.3. Archimedes' Principle 3.4. Rational and irrational numbers 3.5. Exercises 3.6. Exercises for extra-credit Chapter 4. Limits of sequences 4.1. Sequences 4.2. Convergent sequences 4.3. The arithmetic of limits 4.4. Convergence of monotone sequences 4.5. Fundamental sequences and Cauchy's characterization of convergence 4.6. Series 4.7. Power series 4.8. Some fundamental sequences and series 4.9. Exercises 4.10. Exercises for extra-credit Chapter 5. Limits of functions 5.1. Definition and basic properties 5.2. Exponentials and logarithms 5.3. Limits involving infinities 5.4. One-sided limits 5.5. Some fundamental limits 5.6. Trigonometric functions: a less than completely rigorous definition 5.7. Useful trig identities. 5.8. Landau's notation 5.9. Exercises 5.10. Exercises for extra credit Chapter 6. Continuity 6.1. Definition and examples 6.2. Fundamental properties of continuous functions 6.3. Uniform continuity 6.4. Exercises 6.5. Exercises for extra-credit Chapter 7. Differential calculus 7.1. Linear approximation and derivative 7.2. Fundamental examples 7.3. The basic rules of differential calculus 7.4. Fundamental properties of differentiable functions 7.5. Table of derivatives 7.6. Exercises 7.7. Exercises for extra-credit Chapter 8. Applications of differential calculus 8.1. Taylor approximations 8.2. L'Hôpital's rule 8.3. Convexity 8.3.1. Basic facts about convex functions 8.3.2. Some classical applications of convexity 8.4. How to sketch the graph of a function 8.5. Antiderivatives 8.6. Exercises 8.7. Exercises for extra-credit Chapter 9. Integral calculus 9.1. The integral as area: a first look 9.2. The Riemann integral 9.3. Darboux sums and Riemann integrability 9.4. Examples of Riemann integrable functions 9.5. Basic properties of the Riemann integral 9.6. How to compute a Riemann integral 9.6.1. Integration by parts 9.6.2. Change of variables 9.7. Improper integrals 9.7.1. Euler's Gamma function 9.8. Length, area and volume 9.8.1. Length 9.8.2. Area 9.8.3. Solids of revolution 9.9. Exercises 9.10. Exercises for extra credit Chapter 10. Complex numbers and some of their applications 10.1. The field of complex numbers 10.1.1. The geometric interpretation of complex numbers 10.2. Analytic properties of complex numbers 10.3. Complex power series 10.4. Exercises Chapter 11. The geometry and topology of Euclidean spaces 11.1. Basic affine geometry 11.2. Basic Euclidean geometry 11.3. Basic Euclidean topology 11.4. Convergence 11.5. Normed vector spaces 11.6. Exercises 11.7. Exercises for extra credit Chapter 12. Continuity 12.1. Limits and continuity 12.2. Connectedness and compactness 12.2.1. Connectedness 12.2.2. Compactness 12.3. Topological properties of continuous maps 12.4. Continuous partitions of unity 12.5. Exercises 12.6. Exercises for extra credit Chapter 13. Multi-variable differential calculus 13.1. The differential of a map at a point 13.2. Partial derivatives and Fréchet differentials 13.3. The chain rule 13.4. Higher order partial derivatives 13.5. Exercises 13.6. Exercises for extra credit Chapter 14. Applications of multi-variable differential calculus 14.1. Taylor formula 14.2. Extrema of functions of several variables 14.3. Diffeomorphisms and the inverse function theorem 14.4. The implicit function theorem 14.5. Submanifolds of Rn 14.5.1. Definition and basic examples 14.5.2. Tangent spaces 14.5.3. Lagrange multipliers 14.6. Exercises 14.7. Exercises for extra credit Chapter 15. Multidimensional Riemann integration 15.1. Riemann integrable functions of several variables 15.1.1. The Riemann integral over a box 15.1.2. A conditional Fubini theorem 15.1.3. Functions Riemann integrable over Rn 15.1.4. Volume and Jordan measurability 15.1.5. The Riemann integral over arbitrary regions 15.2. Fubini theorem and iterated integrals 15.2.1. An unconditional Fubini theorem 15.2.2. Some applications 15.3. Change in variables formula 15.3.1. Formulation and some classical examples 15.3.2. Proof of the change of variables formula 15.4. Improper integrals 15.4.1. Locally integrable functions 15.4.2. Absolutely integrable functions 15.4.3. Examples 15.5. Exercises 15.6. Exercises for extra credit Chapter 16. Integration over submanifolds 16.1. Integration along curves 16.1.1. Integration of functions along curves 16.1.2. Integration of differential 1-forms over paths 16.1.3. Integration of 1-forms over oriented curves 16.1.4. The 2-dimensional Stokes' formula: a baby case 16.2. Integration over surfaces 16.2.1. The area of a parallelogram 16.2.2. Compact surfaces (with boundary) 16.2.3. Integrals over surfaces 16.2.4. Orientable surfaces in R3 16.2.5. The flux of a vector field through an oriented surface in R3 16.2.6. Stokes' Formulæ 16.3. Differential forms and their calculus 16.3.1. Differential forms on Euclidean spaces 16.3.2. Orientable submanifolds 16.3.3. Integration along oriented submanifolds 16.3.4. The general Stokes' formula 16.3.5. What are these differential forms anyway 16.4. Exercises 16.5. Exercises for extra credit Bibliography Index