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 and Mutually Inverse Mappings 1.3.4 Functions as Relations. The Graph ofa 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 Remarks on the Structure of Mathematical Propositions and Their Expression in the Language of Set Theory 1.4.4 Exercises 2 The Real Numbers 2.1 The Axiom System and Some General Properties of the Set of Real Numbers 2.1.1 Definition of the Set of Real Numbers 2.1.2 Some General Algebraic Properties of Real Numbers 2.1.3 The Completeness Axiom and the Existence of a Least Upper (or Greatest Lower） Bound of a Set of Numbers 2.2 The Most Important Classes of Real Numbers and Computational Aspects of Operations with Real Numbers 2.2.1 The Natural Numbers and the Principle of Mathematical Induction 2.2.2 Rational and Irrational Numbers 2.2.3 The Principle of Archimedes 2.2.4 The Geometric Interpretation of the Set of Real Numbers and Computational Aspects of Operations with Real Numbers 2.2.5 Problems and Exercises 2.3 Basic Lemmas Connected with the Completeness of the Real Numbers 2.3.1 The Nested Interval Lemma (Cauchy-Cantor Principle） 2.3.2 The Finite Covering Lemma (Borel-Lebesgue Principle， or Heine-Borel Theorem） 2.3.3 The Limit Point Lemma (Bolzano-Weierstrass Principle） 2.3.4 Problems and Exercises 2.4 Countable and Uncountable Sets 2.4.1 Countable Sets 2.4.2 The Cardinality of the Continuum 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 3.1.3 Questions Involving the Existence of the Limit of a Sequence 3.1.4 Elementary Facts About 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 3.2.3 The General Definition of the Limit of a Function (Limit over a Base) 3.2.4 Existence of the Limit of a Function 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 …… 5 Differential Calculus 6 Integration 7 Functions of Several Variables: Their Limits and Continuity 8 The Differential Calculus of Functions of Several Variables Some Problems from the Midterm Examinations Examination Topics Appendix A Mathematical Analysis (Introductory Lecture) Appendix B Numerical Methods for Solving Equations (An Introduction) Appendix C The Legendre Transform (First Discussion) Appendix D The Euler-MacLaurin Formula Appendix E Riemann-Stieltjes Integral, Delta Function, and the Concept of Generalized Functions Appendix F The Implicit Function Theorem (An Alternative Presentation) References Subject Index Name Index
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