ACTIVITIES | PERCENTAGES |
---|---|

Problem sets | 1/3 |

Two 1 hour exams | 1/3 |

Final exam | 1/3 |

When you click the Amazon logo to the left of any citation and purchase the book (or other media) from Amazon.com, MIT OpenCourseWare will receive up to 10% of this purchase and any other purchases you make during that visit. This will not increase the cost of your purchase. Links provided are to the US Amazon site, but you can also support OCW through Amazon sites in other regions. Learn more. |

Prerequisite

Multivariable Calculus (18.02); Differential Equations (18.03) or Honors Differential Equations (18.034)

These are formal prerequisites, meant to guarantee a certain mathematical maturity in the MIT students taking the course, and reflecting faculty opinion that most students wanting to study the proofs and abstract ideas of analysis are best served by learning first the standard techniques of calculus and differential equations and their applications to solving real-world problems.

In actuality, 18.100A requires only one-variable calculus; differential equations occur only in two Appendices in the textbook (usually omitted for lack of time), and multivariable calculus only for a single theorem (the Leibniz formula) near the end.

Description

This course is an introduction to devising mathematical proofs and learning to write them up. It is primarily for students with no prior experience with this. The class usually contains students from years 2, 3, 4, G in approximately equal numbers, and from a wide spectrum of majors: engineering, science, economics and business school. About 1/4 are math majors.

The subject matter for the first 2/3 of the syllabus (up to Exam 2) is the proofs of one-variable calculus theorems and arguments which use these theorems. The emphasis is on estimation and approximation, two basic tools of analysis.

The last third goes beyond calculus, getting into uniform convergence of series of functions and improper integrals, which involves several simultaneous limiting processes. The last theorem for example gives the justification for differentiating the Laplace transform under the integral sign, which involves interchanging the order in which three limits are taken.

In addition, toward the end there is a brief introduction to point-set topology in the plane: open and closed sets, compactness, continuous functions on compact sets. It is needed for most courses having analysis as a prerequisite, and here is used in working with integrals depending on a parameter.

Textbook

Mattuck, Arthur. *Introduction to Analysis*. Upper Saddle River, NJ: Prentice Hall, 1999. ISBN: 9780130811325.

Assignments

The accompanying fall schedule calls for three problem sets/week, with the homework collected at each class, and returned graded at the following class. This gives maximum feedback and is particularly useful at the beginning when students are learning to write proofs. A problem set usually has 3 -5 of the book's Exercises and Problems (from the end of each chapter), depending on their length or difficulty. Sometimes "Questions" from the book are included (these have model solutions given at the end of the chapter), as an aid in learning how to write up solutions.

The course is also given (usually in spring) using weekly or twice-weekly problem sets, which many students prefer; in any event, papers are returned at the next class, to give timely feedback.

Note

The textbook is by and large an adequate substitute for class attendance; students in the past have found it sufficiently clear. A few just read the book, get the assignments, and slip the homework under my door before or during class, retrieving the returned homework from a box outside my door.

Grading

ACTIVITIES | PERCENTAGES |
---|---|

Problem sets | 1/3 |

Two 1 hour exams | 1/3 |

Final exam | 1/3 |

Calendar

SES # | TOPICS |
---|---|

1 | Monotone sequences; completeness property |

2 | Estimations and approximations |

3 | Limit of a sequence |

4 | Error term; algebraic limit theorems |

5 | Limit theorems for sequences |

6 | Nested intervals; cluster points |

7 | Bolzano-Weierstrass theorem; Cauchy sequences |

8 | Completeness property for sets |

9 | Infinite series |

10 | Infinite series (cont.) |

11 | Power series |

12 | Functions; local and global properties |

Exam 1 covering Ses #1-12 | |

13 | Continuity |

14 | Continuity (cont.) |

15 | Intermediate-value theorem |

16 | Continuity theorems |

17 | Uniform continuity |

18 | Differentiation: local properties |

19 | Differentiation: global properties |

20 | Convexity; Taylor's theorem (skip proofs) |

21 | Integrability |

22 | Riemann integral |

23 | Fundamental theorems of calculus |

24 | Stirling's formula; improper integrals |

25 | Gamma function, convergence |

Exam 2 covering Ses# 13-25 | |

26 | Uniform convergence of series |

27 | Integration term-by-term |

28 | Differentiation term-by-term; analyticity |

29 | Quantifiers and negation |

30 | Continuous functions on the plane |

31 | Continuous functions on the plane (cont.); plane point-set topology |

32 | Compact sets and open sets |

33 | Differentiating finite integrals |

34 | Differentiating finite integrals (cont.); Fubini's theorem in rectangular regions |

35 | Uniform convergence of improper integrals |

36 | Differentiation and integration of improper integrals; applications |

37 | Comments; review |

Three-hour final exam during finals week |