Introduction to Analysis

Last update: December 6, 2002

*Check regularly the end of this page for announcements and postings!*

**Lecturer:** Detlef Gromoll, Math 5-116

Phone: 632-8290,
email: *detlef@math.sunysb.edu*

Classes: TuTh 2:20 - 3:40, Chemistry 128

Office Hours: M 2-3 (5-116) & Th 11-12 (UG Office P-143), and by appointment

**Teaching Assistant:** Wonkuk Kim, Math 3-118

email: *wkim@math.sunysb.edu*

Recitation: MF 12:40 - 1:35, Light Engineering 152

Office Hours: W 3-4 & F 2-3, and by appointment

**Course Description:** MAT 320 and MAT 319 are introductory courses in analysis, required for math majors. The courses meet together for the
first six weeks, then split into separate lectures and recitations of October 14.

Beyond being a more thorough introduction than regular calculus to one of the pillars of modern mathematics, these courses serve the important purpose of teaching you how to *rigorously prove* and *apply* results in analysis, including results related to the notions of limit, continuity, derivative, integral, and infinite series. Inherent in all of these notions is the concept of *approximation*. As we shall see, a good grasp of this latter concept is essential not only in proving "pure" results in analysis, but is also crucial in "applied" problems requiring estimations. In any approximation a key question is *how do you estimate the error*? In the first part of this course, we look at some types of algebraic manipulations that can be used in error estimation; we will also look at more powerful methods involving the mean value theorem for derivatives.

**Difference between the courses:** MAT 320 will be more comprehensive and will provide a firm grounding for further study. MAT 319 will have more of an emphasis on topics which arise in high-school calculus. Students planning to go on to graduate school in mathematics or other areas with a heavy math component are advised to take MAT 322 and MAT 324 as well. Students wanting to take MAT 322 or MAT 324 (or the seminars
MAT 401 or MAT 402) will need to take MAT 320, not MAT 319.

**Prerequisites:** C or higher in MAT 200 or permission of instructor; plus one of the following: MAT 203, 205, 211, AMS 261, or A- or higher in MAT 127, 132, 142 or AMS 161. Anyone lacking these prerequisites risks deregistration.

**Text:** Daryl Geller, *A Bridge to Analysis*, available only at Kinko's Copies on campus. There may be additional handouts.

**Grading:** The first test, on the first three chapters of the book (up to page 117) will be given in class on **Thursday, October 10**. It will be graded by Monday, October 14, at which time the classes will split. You will be allowed to switch your registration (from MAT 319 to MAT 320, or vice versa) at that time. Your results on the test will help you to make an informed decision. After October 14, Professor Geller will lecture in MAT 319 and Professor Gromoll will lecture in MAT 320. The second test will be given in class on **Tuesday, November 19**. The final examination will be held on **Tuesday, December 17, 2 - 4:30**. Students are expected to ensure when they register for the courses that they will be available for the final examination, and that they do not have too many final exams on that date.

The final course grades in MAT 320 will be determined as follows:

Homework 25%, two midterms 20% each, final exam 35%.
We will give up to 20% extra credit for independent study type projects to be assigned not later than mid November. They can also be used toward the writing requirement in Mathematics and thus should be in the form of a careful and very detailed presentation, with all extra references cited. The grades of A- and A will be reserved for students who demonstrate a substantial ability to apply the concepts of these courses independently and somewhat creatively. Please note that there will be no curve in these courses when determining grades. Incompletes will be granted only if documented circumstances beyond your control prevent you from completing the course work, according to strict University rules.

**Homework:** The only way to learn the material is to work problems for yourself. Each week, you should attempt to do all of the problems from the sections which are covered in class. We will ask you to hand some of them in. Your homework will be graded meticulously and will give you *vital feedback* on where you are making mistakes. Homework is a means to an end, the "end" being for you to learn the material. I *encourage* you to work on homework together with friends. However, since homework earns credit, all participants of any submitted homework should put their names on the paper. It is understood that by doing they confirm that they have contributed equally to the work. It has been our experience after many years of teaching that those who consistently do the homework are those that learn the most and do the best in the course. Trying to learn the material in any math course just before the exams rarely produces good results. Never be shy about asking how to do a homework problem, we are always happy to help you.

**Approximate Course Schedule:**

Chapter 1: 9/3 -- 9/13

Chapter 2: 9/17 -- 10/1

Chapter 3: 10/2 -- 10/7

Sections 4.1 and 4.2: 10/8

First Test (on Chapters 1-3): 10/10

After the split, MAT 320 continues:

Rest of Chapter 4: 10/14 -- 10/25

Chapter 5: 10/28 -- 11/12

Second Test (on Chapters 4-5): 11/19

Chapter 6: 11/14 -- 12/13

Final Examination (cumulative): 12/17

**Special Needs:** If you have a physical, psychiatric,
medical, or learning disability that could adversely affect
your ability to carry out assigned course work, we urge you to contact
the staff in the Disabled Student Services office (DSS),
Room 133 Humanities, 632-6748/TDD. DSS will review your
situation and determine, with you, what accomodations are
necessary and appropriate. All information and documentation
of disability is confidential.

**Week by Week Details:**

10/14-18 | Review Sections 4.1, 4.2 | Study 4.2.1-4 | Homework (due 10/21):

3&4 p125, 5 p133, 7&9 p136, 11&14 p154

*Notes:* 1. Makeup Test 1 will be given on Thursday, November 7, 7-9:30pm.

2. I you have noy done it yet, you need to switch registration to MAT 320 right now (using the form, *not* SOLAR).

10/21-25 | Review 4.2.3-4 | Study 4.3 in detail | Homework (due 10/28):

16 p162, 18&19 p163, 22 p168

10/28-11/01 | Review 4.3 | Study 4.3.1 and 4.4 in detail | Homework:

26 p170, 27&28 p183

11/04-08 | Review 4.4 | Study 5.1 and 5.2 in detail; begin reading 5.3 | Homework (due 11/11):

1 p199, 3&4 p202, 6 p203, 8 p205, 9 p206

11/11-15 | Review 5.2 | Study 5.3-5 in detail; begin reading 5.6 | Homework (due 11/18):

10 p210, 12 p214, 14 p224, 15 p226, 19&20 p232

*Reminder:* Second Midterm will be given next Tuesday, November 19, in class.

Material covered: All of Chapter 4 and Sections 1 through 4 of Chapter5.

11/18-22 | Review 5.5 | Study 5.6 in detail; begin thinking about Projects 1&2 | Homework (due 11/28):

22&24 p240, 25 p243, 27 p253, 29&30 p254

*Project 1:* 25 p169

*Project 2:* 28 p253

11/25-29 | Review 5.6 | Study 6.1 in detail; think about Projects 1&2 | Homework (due 12/2):

2 p258, 3 p260, 4 p262

*Happy Thanksgiving!*

12/02-06 | Review 6.1 | Study 6.2 and 6.3 in detail; read 6.4 (making up for a snow day!) | Homework (due 12/9):

5 p269, 6 p272, 8 p275, 10 p280, 12 p284, 13 p286

12/09-13 | Review 6.4 | Study 6.6 and 6.8 in detail, basic ideas of 6.5 | Homework:

14 p292, 15 p295, 21 p308, 35 p348, 36 p364

*Project 3:* Study 6.5 further and work out Exercise 20 on p306 in detail.

*Project 4:* 22 p322

*Project 5:* Study 6.6 and do Problems 29 on p337 and 32 on p339.

*Project 6:* Work out Exercises 37-38 on p365.

*All remaining course work is due by noon, Wednesday, December 18. *

*Course Grades* (Final score/175+35): 9988 A- (130+10), 6574 A (152+25), 1706 C (63+5),

2195 A (151+8), 9033 C+ (82+8), 6167 A (130+12), 3832 A (157+25), 5711 NR,

2697 A (155+28), 6739 A- (133+8), 3766 I,
8194 C (68+4), 2622 B+ (91+12), 3574 I.

Good work! Have a nice vacation.