Classical Analysis

Course Meeting Time and Location
Monday, Wednesday and Friday
11:00 - 11:55 am
107 Downs Physics Laboratory (DWN)


Course Instructor Contact Information and Office Hours
Dr. Kirill Lazebnik
lazebnik@caltech.edu
210-6 Math Building (Building 15)
Office hour: 5pm Wednesday


TA Contact Information and Office Hours
Zijian (Jack) Tao
cubicle 110L, Math building
Office hour: Tuesday 4-5pm 

Zachary Chase
zchase@caltech.edu


Course Schedule and Textbook

The general plan is to work through the textbook Principles of Mathematical Analysis (3e) by Rudin. More specifics to follow as the class progresses. I am also listing as a reference the textbooks Analysis I, Analysis II by Terence Tao. The course structure and homework will follow Rudin, however I recommend Tao's books if you have time to read them. My own favorite analysis text as an undergraduate was Elementary Classical Analysis by Marsden and Hoffman. 

 DateTopic 
 9/25/17 Introduction, Chapter 1.
 9/27/17 Chapter 1
 9/29/17 Chapter 1
 10/02/17 Chapters 1, 2
 10/04/17 Chapter 2
 10/06/17 Chapter 2
 10/09/17 Chapter 3
 10/11/17 Chapter 3
 10/13/17 Chapter 3
 10/16/17 Chapter 3
 10/18/17 Chapter 3
 10/20/17 Chapter 4
10/23/17 Chapter 4
 10/25/17 Chapter 4
 10/27/17 Chapter 5
10/30/17 Chapter 5
 11/1/17 Chapter 5
 11/3/17 Chapter 5
11/6/17 Chapter 6
 11/8/17 Chapter 6
 11/10/17 Chapter 6
11/13/17 Chapter 6
 11/15/17 Chapter 6
 11/17/17 Chapters 6, 7
 11/20/17 Chapter 9
11/22/17 Chapter 9
 11/27/17 Chapter 9
 11/29/17 Chapter 9
 12/01/17 Chapter 9

Course Policies
Assignments for the course will consist of weekly homework. The problems will be posted some time Friday afternoon and due 11AM Thursday in the dropbox in Downs. Late work will never be graded/accepted, but in documented and extenuating circumstances the 0 will not count in the determination of a final letter grade. There will be a midterm (date to be determined) and a final exam. The homework assignments comprise 30% of your final grade, while the midterm makes up another 30% and the final exam 40% of your final grade. 

For homework assignments - I ask per your honor code that you do not consult any online resources such as solution manuals or math forums until after the due date has passed for the homework in question. For example you are not allowed to look at online resources that would assist you in solving problems from the homework set assigned on 9/29/17 until after 11am on 10/5/17. After 11am on 10/5/17 you are welcome to peruse any solutions manual or math forum you would like pertaining to these problems, but please do not look at solutions for problems which you think may be assigned in the future (i.e. solutions for problems in upcoming chapters). Feel free to work amongst each other, but never copy someone else's work or write something down which you don't understand. If there are ever any questions/issues about what is/is not allowed please do not hesitate to ask me. Also - I will give bonus points to students who submit their homework written in LaTeX - this is a valuable skill worth learning early. 


Assignments
 Date PostedAssignment Due Date 
 9/29/17 Finish reading chapter 1, start reading chapter 2 from Rudin. Write up problems #2, 5, 6, 7, 9 from chapter 1 of Rudin (third edition). Longer problems will be weighed more heavily than shorter problems. For #7 part (a) make the observation that (b^n-1)=(b-1)(b^(n-1)+b^(n-2)+...+b+1) - you should remember this identity. Proofs should be written carefully and rigorously. You do not need to cite such axioms/properties as x<y for x,y real numbers implies that tx<ty if t>0, but for example you should justify uses of the least upper bound property. Unjustified deductions or unclear lines of argument will be graded harshly. Homework written in LaTeX will be given a bonus 10 points (out of 100). 11AM on 10/5/17
 10/6/17 Finish reading chapter 2, start reading chapter 3 from Rudin. Write up problems #6, 10, 11, 16, 20, 22, 25, 29. Again proofs should be written with rigor/clarity, and work written in LaTeX will be given a bonus 10 points.  11AM on 10/12/17
 10/13/17Continue reading chapter 3 from Rudin. Write up problems #2, 3, 11, 14, 21, 22, 23 from chapter 3. As always, solutions should be written rigorously (e.g. limit calculations should be justified using the "epsilon definition" of convergence) and work written in LaTeX will be given a bonus 10 points.  11AM on 10/19/17
10/20/17There is no written homework this week. There is a midterm that will be posted by the end of the day on Monday, 10/23/17. You will have a week during which to take the midterm, but it is only a 3 hour exam that should be taken in a single sitting, and it is a closed book exam. It will consist of 5 questions. The exam will cover what we have done up to this point (10/20/17) in lecture (i.e. the first three chapters of Rudin together with the first couple of sections from Chapter 4). The exam should be handed in (in the usual dropbox) by Monday 10/30/17 at noon. 
 

10/27/17Start reading chapter 5 from Rudin. Write up problems #4, 6, 20 from chapter 4. As usual - bonus points for work written in LaTeX. A reminder that you have a midterm due on Monday.  11AM on 11/02/17


11/4/17Start reading chapter 6 from Rudin. Write up problems #3, 9, 11 from chapter 5. As usual - bonus points for work written in LaTeX. I would also encourage you to work through problem 18, though you don't need to hand in a written solution. 11AM on 11/09/17

11/10/17Finish reading chapter 6 from Rudin. Write up problems #2, 5, 7, 8 from chapter 6. As usual - bonus points for work written in LaTeX. 11AM on 11/16/17
11/17/17Write up problems #11, 12 from chapter 6 and problem #9 from chapter 7. As usual - bonus points for work written in LaTeX. Note that this week the homework is due 11AM on Wednesday 11/22/17 rather than Thursday. We will start material from chapter 9 on Monday, in case you want to start reading ahead. 11AM on 11/22/17

11/28/17We will finish up the course by working through some material about the differentiation of functions in several real variables (chapter 9 of Rudin). There will be no more written assignments. More information on the final to come. Some problems you might consider looking at from chapter 9 as practice are #6, 8, 17, 19. Problem 17 concerns the notion of a Jacobian which we have not covered, but is simply the determinant of the derivative (remember the derivative is a linear transformation), so that one can check the hypotheses of the inverse function theorem if one can compute the Jacobian of the function in question. Similarly for #19.  

Midterm and Final Exam
Midterm Exam is available here. The password has been emailed to you. If you have any issues accessing the file, please contact Meagan. It is due at 12 pm (noon) on Monday October 30th.
Final Exam is available here. The password has been emailed to you. If you have any issues accessing the file, please contact Meagan. It is due at 1 pm on Thursday December 7th.

For future reference I am now (December 8th) listing on this webpage the midterm exam and the final exam for this quarter. 



Collaboration Table
 HomeworkExams
You may consult:  
Course textbook (including answers in the back)YESNO
Other booksYESNO
Solution manualsNONO
InternetNONO
Your notes (taken in class)YESNO
Class notes of othersYESNO
Your hand copies of class notes of othersYESNO
Photocopies of class notes of othersYESNO
Electronic copies of class notes of othersYESNO
Course handoutsYESNO
Your returned homework / examsYESNO
Solutions to homework / exams (posted on webpage)YESNO
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESNO
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersNONO
Post about problems onlineNONO
For computational aids, you may use:

CalculatorsYES*NO
ComputersYES*NO

* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.