Syllabus

Real Analysis (MA-108-1), Fall 2012 Course Syllabus The instructor may make changes to this syllabus throughout the semester.

Course Class Web Instructor Office Location: Phone: Hours:

E-mail

Real Analysis (MA-108-1), MWF 12:45–1:50 p.m., 4 units http://www.westmont.edu/~howell/courses/ma-108 Russell W. Howell, Ph.D. Winter Hall for Science and Mathematics, Room 307 Refer to the campus map: http://www.westmont.edu/_visitors/pdf/map.pdf 805-565-6178; from on-campus just dial 6178 Monday: 3:30–5:00 p.m., Tuesday: 1:30–3:00 p.m., Thursday: 10:20–11:50 a.m., Friday: 2:00–3:20 p.m.; and any time you happen to find me in and available. [email protected]

Required Resources Analysis with an Introduction to Proof, Fourth Edition, by Steven R. Lay A Mathematician’s Apology, by G.H. Hardy Overview and Objectives This course is designed to examine the foundation of topics that are studied in calculus, and to explore new implications of those theories. Thus, theoretical ideas play a central role, and certainly have a larger emphasis than do computational skills. The course will focus on painting a broad picture of what analysis is about, with some explanation of the history of various ideas as well as the different philosophical schools that exist within the mathematical community today. Most of the text will be covered, with some chapters receiving a greater emphasis than others. A companion reading resource is a short book written by G.H. Hardy, one of the greatest mathematicians of the 20th century. It gives an apology (i.e., a defense) of the pursuit of pure mathematics. The course outline appears on page 3. Writing Assignments Real Analysis fulfills the “writing intensive in the major” requirement for Westmont’s GE. There are several reasons why this course is ideally suited to the development of a clear and concise mathematical writing style. Throughout you will be completing substantial writing assignments consisting of wellarticulated mathematical arguments. You will be graded not only on whether your argument is valid, but also on the coherence, organization, and style of your presentation. The writing assignments are a substantial part of your grade. Your score will be reduced for any work turned in late, so it is vital that you complete your assignments on time. Each assignment will consist of two problems (as outlined on page 4) chosen from problems comprising the daily assignments (page 3). Note that one of the two problems appears in bold. It must be typeset using LATEX, or software such as Scientific Workplace that produces LATEX. It is vital to do all the problems given in the daily assignments (not just the two to be turned in); they will form the basis of regularly scheduled quizzes. Quizzes and Essay A short quiz will be given in conjunction with each writing assignment. It will consist of one or two questions selected from the writing assignments (page 4), with one or two additional questions relating to definitions and concepts. There will also be two quizzes based on your reading of Hardy’s book, and you will produce a three to five page essay (due December 5) giving your own apology for mathematics.

Real Analysis, Fall 2012

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Attendance and Participation Your attendance and classroom participation will affect your grade in either a positive or negative way. You are expected to attend every class, and more than three unexcused absences throughout the semester will be considered as a poor performance. Any classes missed just before or after a holiday will likewise reflect poorly on your commitment to the class. In part your participation will be assessed with a brief review each day of material previously covered. Thus, it is important to come to class prepared to answer any oral questions that may be put to you. In general, determining the quality of class participation depends on answers to several questions: Did you come prepared for class by reviewing previous lectures? Did you do a good job on the daily assignments? Did you actively participate in class? Were you mentally alert and attentive during class? Exams There will be two written in-class exams plus a comprehensive final exam. Your final grade will be determined as follows. Item Writing assignments Assignment quizzes A Mathematician’s Apology: quizzes/essay In-class exams Final exam

Percent 16% 16% 12% 38% 18%

Relevant Dates Refer to the schedule on page 4 Refer to the schedule on page 4 September 24, November 2, December 5 September 26, November 5 December 14: 8:00–10:00 a.m.

Students with Special Needs Students who have been diagnosed with a disability (learning, physical or psychological) are strongly encouraged to contact Sheri Noble at the Disability Services office (x6186, [email protected]) as early as possible to discuss appropriate accommodations for this course. Formal accommodations will only be granted for students whose disabilities have been verified by that office. These accommodations may be necessary to ensure your full participation and the successful completion of this course. Academic Dishonesty You are encouraged to work together on all assignments, and when you turn in written work you should list the names all those with whom you’ve collaborated. The mere copying of another’s work, however, is dishonest. If such an event occurs all papers in question will receive a grade of zero. Dishonesty in any form on the tests, quizzes, or final exam will result in a failing grade for the course, whether for giving information or for taking it. Westmont’s policy relating to academic dishonesty is available at http://westmont.edu/_offices/registrar/academic_policies/academic-dishonesty.html. General Comments Real Analysis has been traditionally viewed as a difficult subject. But most topics with the potential for increasing a person’s intellectual capacity seem difficult, for such growth can only occur when accompanied by a certain amount of discipline. It is important that you view this characteristic not as something distasteful, but as a golden opportunity to expand substantially your intellectual horizons. Indeed, Real Analysis can be a very rewarding subject if you’re willing to make the necessary academic commitments. It is hoped that as the semester progresses you will begin to appreciate the beauty, elegance, and creativity involved in this important area of mathematics. In summary, there are three main goals of this course: • to gain a comprehensive grounding in mathematical topics foundational to calculus; • to become proficient in clear, concise, and sophisticated writing in mathematics; • to stretch your imagination and creativity and thus help you appreciate mathematics as an art.


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Course Outline Date August 27 29 31 September 3 5 7 10 12 14 17 19 21 24 26 28 October 1 3 5 8 10 12 15 17 19 22 24 26 29 31 November 2 5 7 9 12 14 16 19 21 23 26 28 30 December 3 5 7 14

Topic Course introduction; Infinity I Infinity II Deep number properties I Deep number properties II; Topology I Topology II Heine-Borel theorem I Heine-Borel theorem II Bolzano-Weierstrass theorem Sequences Sequential limit theorems Cauchy sequences Subsequences Hardy Quiz I; Review Exam I Limits I Limits II Continuous functions Properties of continuous functions No class (fall holiday) Continuity and uniform continuity Theorems of uniform continuity The derivative Vito Volterra and his amazing theorem Volterra’s proof Rolle’s theorem; mean value theorem Derivatives can’t jump Riemann integrability Properties of the Riemann integral I Properties of the Riemann integral II Hardy Quiz II; Review Exam II The fundamental theorem of calculus I The fundamental theorem of calculus II Infinite Series Convergence tests Power series I Power series II No class (Thanksgiving holiday) No class (Thanksgiving holiday) Pointwise and uniform convergence I Pointwise and uniform convergence II Applications of uniform convergence A truly bizarre function (CEDN) Mathematical Apologies Review Final Exam (Friday, 8:00–10:00 a.m.)

Reading Assignment pp 77–86 pp 77–86 pp 117–125 pp 117–125 pp 129–134 pp 138–141

Writing Assignment p88: 3, 4, 5, 17

# 1

p127: 3, 8, 9, 12ab

2

p136: 9, 11, 19, 23 p144: 3, 4, 5, 8

3 4

p164: p173: p180: p188:

6cde, 9, 12, 15 6, 15c, 17, 18 3bd, 4, 11, 14 6, 9, 10, 15

5 6 7 8

p198: 6ab, 13, 16, 18

9

p207: 8, 9, 13, 18 p214: 7, 9, 13a, 15

10 11

pp 216–220 pp 216–220 pp 231–238 Handout

p222: 4b, 5, 7, 11

12

p240: 6, 9, 11, 18

13

pp pp pp pp pp

241–247 241–247 268–274 277–283 277–283

p249: 5acf, 6, 8a, 11

14

p275: 4ab, 11, 13, 16

15

p284: 5, 6, 9, 12

16

pp pp pp pp

286–290 286–290 294–299 302–308

p292: 5, 7, 8, 14

17

p301: 5abdek, 7, 8, 10 p310: 5abcde, 8, 11, 12

18 19

pp 312–316

p317: 3afl, 4, 5ad, 7

20

pp pp pp pp

p327: 8, 13ac, 14, 17

21

p336: 3, 4, 5, 7

22

pp pp pp pp pp

141–143 156–163 165–171 174–179 181–187

pp pp pp pp

190–196 190–196 199–205 209–213

319–325 319–325 329–333 333–335


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Quiz and Writing Assignment Schedule #

Quiz

Writing

Date

1

Assignments 1, 2

p89: 17; p127: 9

Wednesday, September 5

2

Assignments 3, 4

p136: 9; p144: 8

Wednesday, September 12

3

Assignments 5, 6

p165: 12; p174: 18

Friday, September 21

4

Assignments 9, 10

p199: 16; p208: 18

Friday, October 12

5

Assignments 11, 12

p215: 13a; p222: 5

Friday, October 19

6

Assignments 13, 14

p241: 18; p250: 11

Friday, October 26

7

Assignments 17, 18

p292: 14; p301: 8

Monday, November 19

8

Assignments 19, 20

p311: 12; p317: 7

Friday, November 30