Math 414 and Math 514 (Phillips): Introduction to Analysis 2

This is the home page for N. C. Phillips' Introduction to Analysis 2 (Math 414 and Math 514) the University of Oregon, Winter quarter 2019.

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Basic information

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Grading

Homework

Contribution to final grade: approximately 20%. Due every Monday, except as otherwise stated.

The homework will be hard, some of it quite hard, but it is an essential part of the course.

The homework consists of proofs. This means that everything must be proved. In particular, if a problem asks for an example or counterexample, you must prove that your example has the required properties. If a problem asks whether something is true, you must not only decide whether it is, but also provide a proof or counterexample as appropriate; just as above, counterexamples must be proved to have the required properties.

Proofs should be clear, correct, and complete. They should use complete sentences and correct punctuation. They should have all quantifiers in the right places. They should be in logical order. For example, for a proof of convergence of a sequence from the definition, start with "Let epsilon be greater than zero" and end with the statement that something is less than epsilon. If there are subsidiary arguments in a proof, say what is being claimed, say that you are proving it, prove it, and say when the proof of your claim is done. A reader who can't follow any of the mathematics should nevertheless be able to easily tell where the proof of your subsidiary claim starts and where it ends.

I expect to post on the course website at least sketches of some solutions after the homework is turned in.

Some procedural points:

One Midterm

Contribution to final grade: approximately 30%. Date: Monday 11 February (week 6).

The midterm from the last time I taught the course is here (pdf); LaTeX. Its solutions are here (pdf); LaTeX. The format of the midterm this time may be a bit different. A large portion of the midterm will ask for proofs which you will have to devise for yourself during the exam period. This midterm was given during a two hour period outside of class.

I am willing to allow people to start an hour early or continue an hour late, or to arrange to give the entire exam during a longer period outside of the normal class time. (This was done the last time I taught the course.) Such a procedure can only be followed if suitable rooms and times can be found, and if everybody enrolled in the class agrees to it.

Final Examination

Contribution to final grade: approximately 50%. Date: Monday 18 March, 10:15 am--12:15 pm, in our usual classroom. (The final exam schedule is here.)

No early final exams, according to University rules. If you have another final exam scheduled at the same time as our final exam, you need to give me the details (course number and instructor) by Monday of the 7th week of classes.

Similar comments on the content apply as for the midterm. The final exam from the last time I taught the course is here (pdf); LaTeX. Its solutions are here (pdf); LaTeX.

I keep originals of the final exam, but will give out on request a scan or copy of your exam.

Learning Disabilities

Students with documented learning disabilities who wish to use the Accessible Education Center to take tests under specifically arranged conditions should let me know as soon as possible, certainly by Wednesday of the third week of classes. Such students must also be sure to meet the Accessible Education Center's separate deadlines for requests; these are likely to be a week or more in advance of the exam date (much more for final exams). I can't do anything to help a student who misses its deadline. (I have tried in the past.)

Grading Complaints

Complaints about the grading of any exam must be submitted in writing by the beginning of the first class period after the class in which that exam is returned.

Extra Credit

Extra credit will be awarded to the first person to detect any error or misprint in the current edition of the book, in homework solutions, in exams and exam solutions, or on the course website. More extra credit will be given for catching mathematical errors than misspelled words or wrong dates. To get the extra credit, you must tell me what the correct version should be.

There are likely to be a number of mistakes to catch. When going through materials from the last time I taught this course, I found a number of misprints, including a few egregious ones, which were not caught despite a similar offer for extra credit. (A particularly bad one in a homework solution for the fall quarter was "not compact" for "compact".)

The corresponding part of the pdf version of the syllabus lists a few which have already been identified, so are not eligible for extra credit.

Learning Outcomes

The course deals with: The successful student will be able to precisely state the definitions associated with these topics, and will be able to rigorously prove standard facts. The successful student will also be able to rigorously prove or disprove (as appropriate) statements about these topics which have not been encountered before, and which are at a level of difficulty appropriate to a mid-level introductory analysis course. The successful student will be able to write these proofs so that they are correct, complete, clear, readable, and in logical order.

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Academic Conduct

The code of student conduct and community standards is here. In this course, it is appropriate to help each other on homework as long as the work you are submitting is your own and you understand it, and you give the names of any people you cooperated with. It is not appropriate to help each other while taking exams, to look at other students' exams while taking exams, or to bring any unauthorized material to exams. The only authorized material is pens and pencils.

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Schedule

We will cover most of Chapters 5--8 of the book in the winter quarter, possibly getting into Chapter 9. The part of Chapter 5 already done in the previous course (going through L'Hospital's Rule) won't be repeated. We will follow the book fairly closely.

The book usually used in the preceding course is S. Abbott, Understanding Analysis. Much of the material of this quarter is also there, but often restricted to special cases. As a good example from last quarter, continuity is mostly only treated there in the context of functions from subsets of the reals to the reals, and we will consider continuity of functions between metric spaces. and we considered continuity of functions between metric spaces. This quarter, we will often take the codomain for differentiable and integrable functions to be R^n or C^n, and we will consider the Riemann-Stieltjes integral instead of just the Riemann integral.

The schedule gives approximate times that topics will be discussed. It is necessary to read the book; not everything will be done in class. Homework assignments are subject to change.

Week 1 (7--11 Jan.)

Finish Chapter 5 (differentiation). Start Chapter 6 (Riemann-Stieltjes integration).

Homework (due Monday 14 Jan.): Chapter 5: 1, 2, 3, 4, 5, 9, 11, 13. Point values: 10 points per problem, except 4 points per part in Problem 13 and 12 points for Problem 2. Correction in Problem 13: it should be | x |^a in the definition of the function f. (See the pdf version of the syllabus.)

Week 2 (14--18 Jan.)

Continue Chapter 6 (Riemann-Stieltjes integration).

Homework (due Wednesday 23 Jan.): Chapter 5: 12, 22a--c, 26; Chapter 6: 2, 4; Problem A. Point values: 10 points for each problem part. (Chapter 5 Problem 22a--c and Problem A have 3 parts each.) Problem A: see the pdf version of the syllabus.

Week 3 (21--24 Jan.)

Continue Chapter 6 (Riemann-Stieltjes integration). (Short week, due to holiday Monday.)

Homework (due Monday 28 Jan.): Chapter 6: 7, 8, 10, 11. In Problem 10d, do only the improper integrals from Problem 7. Point values: 10 points for each problem part. (Problem 7 has 2 parts and Problem 10 has 4 parts. Total 80 points instead of 100 points.)

Week 4 (28 Jan.--1 Feb.)

Finish Chapter 6 (Riemann-Stieltjes integration); start Chapter 7 (sequences and series of functions).

Homework (due Monday 4 Feb.): Chapter 6: 15; Chapter 7: 1, 2, 3, 4. The notation in Problem 15 is not very good; see the pdf version of the syllabus. Point values: 20 points per problem.

Week 5 (4--8 Feb.)

Continue Chapter 7 (sequences and series of functions).

Homework (due Friday 8 Feb.; short assignment, because of the midterm on Monday): Chapter 7: 5, 6, 7. Point values: 13 points each for Problems 5 and 6; 14 points for Problems 7. (Total 40 points instead of 100 points.)

Review session for the midterm: Friday 6 February 5--7 pm, 205 Deady.

Week 6 (11--15 Feb.)

Midterm Monday 11 Feb., 6--8 pm, 205 Deady.

Finish Chapter 7 (sequences and series of functions).

Homework (due Monday 18 Feb.): Chapter 7: 9, 16, 19, 20, 21. The notation in Problem 20 is not very good; see the pdf version of the syllabus. Point values: 20 points per problem.

Week 7 (18--22 Feb.)

Start Chapter 8 (standard elementary functions; Fourier series).

Homework (due Monday 25 Feb.): Chapter 7: 22, 24; Chapter 8: 2, 3, 4, 5. (In Chapter 7 Problem 22, you may use Problems 6.11 and 6.12.) Point values: 5 points per part in Problems 4 and 5; otherwise, 10 points for each of the two other problems in Chapter 8, and 20 points for each of the problems in Chapter 7.

Week 8 (25 Feb.--1 March)

Continue Chapter 8 (standard elementary functions; Fourier series).

Homework (due Monday 4 March): Chapter 8: 6, 7, 8, 9, 10, 11. (There is a hint for Problem 10, but it is on the page after the statement of the problem.) Point values: 10 points for each problem part, except 15 points each for Problems 8 and 11 and 20 points for Problem 10.

Week 9 (4--8 March)

Continue Chapter 8 (standard elementary functions; Fourier series).

Homework (due Monday 11 March): Chapter 5: 27; Chapter 8: 20, 23, 24, 25. (Also look at Problems 22 and 26.) Point values: 20 points per problem.

Week 10 (11--15 March)

Finish Chapter 8 (standard elementary functions; Fourier series).

Homework (due Friday 15 March): Chapter 8: 12, 13, 14. Point values: 10 points for each problem part. (Total 70 points instead of 100 points.)

Sa 16 March, 2:00--4:00 pm: Review session for the final exam. Room to be announced.

Final exams week (18--22 March)

M 18 March, 10:15 am--12:15 pm: final exam, in regular classroom.

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Publicly Available Documents Associated with this Course

Here is a list of publicly available documents associated with this course. The material is arranged in approximate chronological order: most recent items at the bottom. Links to written homework solutions, exams, and exam solutions will not work until after the corresponding written homework has been turned in or the corresponding exam has been given, and the links to sample exams and their solutions will not work until these items have been prepared.

Most files will be posted in pdf and AMSLaTeX versions.

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Important dates, according to the Academic Calendar at the registrar's office (not guaranteed!)

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This page maintained by N. Christopher Phillips, email. Please email plain text (7 bit ASCII) only (no web page coded files, Microsoft Word documents, binary characters, etc.; see above for more).

Last significant change: 7 January 2019.