General Information for Math 4035, Fall 2021

Covid-19 Cautionary Comments

We are entering the Fall 2021 semester under a plan to resume in-person instruction at LSU, knowing that the covid-19 pandemic is raging in our community, our state, our nation, and the world. The syllabus that appears below has been written on the assumption that we will succeed in protecting our campus from serious outbreaks of disease that might derail the current plans. We must all be ready for the need to make reasonable changes in plans depending on what happens and on what changes the university might or might not announce as the semester unfolds.

In order to make the semester as successful as possible, it is very important that each of us observe all the precautions and procedures the university has announced. This includes masking, social distancing, vaccinations and/or covid-19 tests, quarantining if needed, etc. The single most important thing each of us can do to make this semester successful is to be vaccinated against covid-19. For the record, my own entire family and I have been vaccinated. It is easy to do, convenient, and free. It is our most effective step in preventing infection, hospitalization, and death due to covid-19. Do not underestimate covid-19: even a healthy young person may need many months to recover fully from this infection. It is also a fact that if we do not catch covid-19 then we cannot spread it to others. Thus the best way to protect our community from this disease is to protect ourselves. Please do everything you can to safeguard your own good health. If our LSU campus community pulls together, we can make this return to in-person instruction on campus successful.

Time MWF 11:30 AM -- 12:20 PM The class meets starting on Monday, August 23. The final exam will be Wed., Dec. 8, 7:30am - 9:30am
Location 116 PRESCOTT
Leonard Richardson Office 386 Lockett Hall
Office Hours 1:30 PM--2:20 PM, MWF, and 1 PM--2 PM T,Th. Office Hours are strictly ONLINE ONLY, at this link:
There is a Zoom Waiting Room in case more than one person comes at the same time. I am available at many other times. Email first to make sure I'm able meet with you. I answer email many times daily---usually quickly.
Office phone 578-1568
Text Richardson, L., Advanced Calculus: An Introduction to Linear Analysis, John Wiley & Sons, 2008. ISBN 978-0-470-23288-0. There is a list of errata. If you find an error not on this list, please tell me. The text is also available as a free e-book through the LSU Library at this link.
Syllabus We will cover most of Chapters 8, 9, 10, and 11.
Graduate Assistant

You will turn in homework in class on the due-date. If you must be absent on the due-date, you may email the scan files of your homework to the grader directly by class-time: Mr. Colton Sandvik at , who will grade those homework problems that are to be turned in---the ones that are assigned in red boldface in the table below.
The grader will be available to answer questions about the homework grading at at his Zoom Room on Wednesday from 12:30 PM to 1:20 PM.

Please be sure to write your solutions neatly and carefully so that they can be read.

Math Major Requirements and Recommendations

Math 4031, followed by either Math 4032 or Math 4035, satisfies the Advanced Calculus requirement for the Mathematics major with a mathematics concentration. It prepares students for graduate study of mathematics and its applications. The Department strongly recommends that Mathematics majors planning graduate study in Mathematics take all three Advanced Calculus courses: Math 4031, 4032, and 4035.


Mathematics 4031 and 2085 or the equivalent.


Problems, mainly proofs, will be assigned frequently. These will be collected on paper in class, corrected, and returned to you at the next class meeting, if possible. You are encouraged to seek hints to help you get started with these problems! Please turn in every assignment! The key to learning to prove theorems lies in how you study Advanced Calculus. It is very important to understand thoroughly how and why the proofs presented in the book and in class work. Please read the Introduction to your textbook! We will go over every collected homework problem in class, to help you prepare for tests. At the end of the course, you will have the benefit of whichever of the following two final average adjustments gives you the best final average.
  1. Adjustment #1: Add your average homework score on a 10-point scale to the average of your 3 hour tests with the final exam. For example, if your average on the homework is 5 points out of 10, and you have an 85% exam average, Adjustment #1 raises your final average to 90%, which is an A-.
  2. Adjustment #2: Replace your lowest hour-test grade with half of your 200-point final exam score.
    Please turn in every homework assignment!
Proofs assigned for homework are a very important learning experience. Some students try an easier technique - copying the correct proofs from class after the homework has been graded, without turning in their own efforts. This tends to produce low grades on Part I of each test, because the student's own conceptual errors have not been turned in and thus have not been corrected. So please turn in every assignment!


These will be in-class closed book tests with no notes or communication devices allowed. Part I of each hour test will consist of a choice of 8 out of 12 short answer questions, and Part II will offer a choice of 2 out of 3 proofs. (The Final Exam will be equivalent to two hour tests.) The proofs will be very similar to the collected homework problems. The short questions will be small variations of homework problems---including those not collected---together with examples from the lectures and notes. Thus if you have done the homework conscientiously, you should be prepared well for all tests. If you must miss a test, it is your responsibility to speak to me as soon as possible to determine whether or not your excuse is acceptable.


There will be three hour tests, worth 100 points each, and a two hour final examination, worth 200 points. Your test average will be the sum of all your grades divided 5. Your Final Average will be the better of the two grades calculated by adjusting the average using either Adjustment #1 or Adjustment #2 described above under Homework. The minimum grade for each letter grade is as follows: A+, 97 A, 93 A-, 90 B+, 87 B, 83 B-, 80 C+, 77 C, 73 C-, 70 D+, 67 D, 63 D-, 60 F, below 60. You should save all your graded work for future study and in case you think your final grade is in error. You should save all your graded work for future study and in case you think your final grade is in error.

General Advice: Questions and Attendance

  • Many students need help to learn how to write proofs. If you feel confused, it is important to see me for help as soon as possible. If you don't know how to start a homework problem, ask for a hint---either in class or in my office hours---or by email. I can guarantee you it is possible to learn to write sound proofs: Learning begins with your efforts and your persistence.
  • Attend class and ask questions. The roll will be called at the beginning of each class. If you are not present you are greatly missed. Please come to class, unless you have a valid reason such as illness or a quarantining requirement! Non-attendance or lax attendance is usually the first sign of impending academic difficulty. Sometimes a student feels frustrated because of not understanding the classwork. If that is the case, it is necessary to ask more questions. Advanced calculus is a subject you can learn---but you must participate in this work.
  • Come to class on time. However, anyone may need to arrive a bit late on some occasions for reasons beyond ones own control. If you are in that situation, just come right in and take your seat. You should not miss any more of the class time than is necessary for reasons beyond your control. Because of the covid-19 pandemic, it is recommended that you choose the same seat each day in the classroom. This is to minimize the number of students who might possibly expose you to infection. Of course, everyone who is exposed or ill should get tested and follow LSU's reporting and quarantining instructions.
  • Assignments are collected at the beginning of class. . If you need to be absent and turn it in directly to the grader by e-mail, do not turn it in later than that, because it is not fair to the graduate teaching assistant, who will be busy enough with the work of grading the assignments that are turned in at the proper time.
  • LSU offers extensive academic support services to help students adjust to the demands of university studies: List of Frequently Used Services.

Homework Assignments and Downloads

We will update the list of assignments and tests below as the semester progresses. You will know an assignment has been updated if a due-date appropriate to this semester appears in the left-hand column. However, sometimes we will assign a problem for a certain date and then postpone it because we don't cover as much as planned in class. So check regularly for updates as to what is due and when. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question, not giving your name of course!


Academic Honesty The University has clear policies requiring academic honesty. If you email me about a pending assignment, I may send a hint to the whole class in answer to your question, not giving your name of course! If on the other hand you get a good idea from another book, or from talking with a friend, academic honesty requires that you acknowledge your sources openly. Above all, never copy directly from another person's written work as though it were your own. Remember that your own good name is irreplaceable. This is a sound principle which will serve you well throughout your life. Moreover, on a practical level, it is very foolish claim as your own an argument from a former student in this class or from a textbook. The arguments which are copied can be recognized very easily as not coming from the student, and often the precise source can be identified readily. This means that the honorable course of action is also the practical one.
Due Date Assignments: Hand in problems in red for grading. The problems in red are required. Assignments must be written neatly so that the grader can read them. But there is also a class of optional problems, called Bonus Problems. These are worth up to 20 extra homework points per problem. Bonus problems need to be turned in directly to me on a separate sheet from the regular homework, clearly marked Bonus Problems at the top. Bonus problems are due, if you choose to do one of them, one full week after the date listed, unlike normal graded homework, which is due the date listed. Bonus problems will be graded more strictly for logical rigor than the required homework. Bonus problems are more challenging than the required problems.
August 23 Today is the first day of class. You should have your texts for your classes and be ready for work.
August 24 Today only, my office hours will be a half hour earlier: 12:30 PM -- 1:30 PM.
August 25 Read Pp. 245 -- 246, through Example 8.1. Be sure to practice reading with a pencil!

August 26

Today only, my office hours will be earlier: 12:00 PM -- 1:00 PM.

August 27 Read Pp. 246 -- 249. Be sure to practice reading with a pencil! Do problems 8.1 -- 8.4; 8.6 -- 8.8; 8.10. These are not to hand in, but you should write your solutions on paper in order to learn from the work. In this section, 8.1, we use Definition 2.5.3 (page 63) of convergence in a normed vector space and of a Cauchy sequence in a normed vector space.


Hurricane Ida + Labor Day. Makeup Days:

Saturday, Sept. 25, Friday, October 22, and Saturday October 30.
September 8 Because I cannot be with you in person on Wednesday, September 8, I have recorded a one-hour lesson for you at this LINK. Since our class won't meet in person Wednesday, you may use this recorded lesson at any time at your own convenience. If you have any questions or problems with this method of making up for my absence Wednesday, please be sure to email me so I can do anything I am able to do to accommodate your needs. This applies to any questions or difficulties you have for any reason. Your first written homework is now due Friday, as you can see below. I am looking forward very much to being back with you in person on Friday, and for the rest of the semester as well. Thank you.
September 10 Do problems 8.9, 8.11, 8.14(a)--(c) and turn them in to me at class time!
Mr. Sandvik will grade your homework. Please be sure to write your solutions neatly, so that he can read them!

The first Bonus Problem B1 is 8.14(d). If you choose to do it, turn it in to me--NOT to the grader-- one week from today, separately from the regular homework. I grade Bonus problems strictly for rigorous reasoning.
September 13 8.16-17; 8.21-23. Terminology: A set is called closed if and only if its complement is open.
September 15 Sorry-No office hours Wednesday or Thursday this week.
September 17 8.18-19; 8.24-25; 8.30--31; 8.33.
September 20 8.20, 8.26, 8.28. The Bonus Problem B2 is 8.32. If you choose to do it, turn it in to me one week from today, separately from the regular homework. Bonus problems are graded strictly for rigorous reasoning.
I may be late for my office hours today. If I am not there when you visit via Zoom, please email me and I will arrange a meeting for you when I return. Thank you.

September 21

I will miss my office hours today for a dental reason. If I am not there when you visit using Zoom, please email me and I'll arrange a time for you when I get back.

September 22 8.34 --8.38; 8.42.
September 24 8.39, 8.40, 8.41. (Hint for problem 39: You need not suppose false. Just follow the rest of the stated hint in the book to show that the complement of E must be open if E is compact.)
If you get an idea from a friend, acknowledge your sources openly. Above all, never copy directly from another person's work as though it were your own. I am sorry to report that there are already two cases of students copying from a third student. If you are one of the three involved, please don't let it happen again.
SATURDAY, September 25 This is our first makeup class for the days missed because of Hurricane Ida. We will present solutions to the problems turned in Friday. And if there is time we will begin to learn about connected sets.

September 30

Sorry-no regular office hour today. If you need help please email me and I will arrange a time for you.
October 1 8.44, 8.46, 8.48 -- 8.53.
October 4 8.45, 8.47, 8.54.
October 5 SORRY-NO REGULAR OFFICE HOURS TODAY. Please email if you need help today and I'll make a time for you later today.
October 6 Bring Review Questions today!!
The next Bonus Problem B3 is 9.11, due one week from October 8 if you do it. Turn it in separately from the other homework. In one direction, 9.11 is quite easy, so that bonus credit requires both directions.

Here is another Bonus problem, suggested by a student. I will call it B2.5. You have a week from October 8 to turn it in if you like. Suppose A and B are disjoint connected sets, but suppose A-closure intersect B-closure is nonempty. Prove that A-closure union B-closure is connected.

October 8 First Hour Test today! For practice, please download Hour Test #1, 2011, Solution Sketches and Class Statistics.
October 9 Please download Hour Test #1, 2021, Solution Sketches and Class Statistics.

October 12

No regular office hours today. Email if you would like me to open a Zoom office hour for you later this afternoon--probably after 3 PM. Thank you.

October 13 9.2, 9.3, 9.6, 9.7.

October 15 9.4, 9.5. Thursday I tried to hold office hours but Zoom did not work. I was on the phone with the University's IT service for 2 full hours trying to find out what was wrong. But I think I can get back on Zoom now so hopefully office hours will be normal Friday.
October 18
October 20 9.12, 9.14--9.15, 9.19--9.26.
October 22 Today is a virtual makeup class for the Friday missed during the week following Hurricane Ida. That the class is virtual (meaning online only) is the University's decision which we are obliged to follow. This means that although the homework problems "due" today should be done by today, you won't be able to turn them in to me until Monday when we meet again in person. But class will meet today using my Zoom office hour link. I will let you in from the waiting room, so please try to be on time if possible, so I don't have to interrupt the lesson in the middle to let late students in. Thank you for your cooperation in this and please be sure to turn in the problems in red below on Monday!
9.13, 9.16, 9.28.
Here is the link to the cloud recording of today's virtual class.
October 25 9.29 -- 9.32, 9.35, 9.36, 9.38. Note: In 9.36, there is a missing letter "t" in the argument of a function. Also: Remember to turn in today the problems that were "due" on the virtual class day, October 22!
October 27 9.33, 39, 41.

October 29

9.44 -- 9.48 The next Bonus Problem B4 is 9.42. Due one week from today if you decide to do it.
SATURDAY, October 30 9.43, 9.49, 9.50 Today is our last make-up class for the week lost to hurricane Ida.
November 3 10.1--10.3, 10.5--10.7, 10.12.
November 5 10.4, 10.8, 10.16, 10.20.

November 9

Office Hours must be moved today to another time. If you need help today, please Email me and I will arrange a time today.
November 10 10.25 -- 10.29, 10.31, 10.37, 10.42. Typos: In 10.31 E3 should be E2. In 10.42, the domain of the mapping is incorrectly stated. It should read D : En --> Em.
November 12 10.30, 10.32, 10.38, 10.39.
November 19 10.56. Second Hour Test Today. This test will cover everything we have done since the first hour test.
November 21 Please download Hour Test #2, fall 2021, Solution Sketches and Class Statistics.
November 22 10.44, 10.46 -- 10.49, 10.51. My office hours today will be late: 2:30-3:30 instead of the usual time.
November 24 10.45, 10.50, 10.52.
November 29 10.59 . Hint for 10.59: For a linear transformation T: En --> Em, compare the dimension of the image of T with the dimension of the domain of T. Also: 10.57--10.59.
December 1 10.60--10.66.
December 3 Bring questions to review for the Final Exam! Don't forget to review from the beginning of the course! Also, please remember to fill out the end-of-course evaluation form that is available to you online! Your anonymous feedback is very important: It helps your teacher and the University to serve your needs to the best of our ability. Remember: Your opinion matters: You are the reason we are here! Thank you.

December --

Exam Week Office Hours TBA Office Hours Monday and Tuesday will be announced soon. I am available at many other times. Email first to make sure I'm able meet with you. I answer email many times daily---usually quickly.

December 8

Final Exam: The final exam will be Wed., Dec. 8, 7:30am - 9:30am.
The Final Exam will have a choice of 12 out of 18 short questions for 96 points and 4 out of 6 proofs for 104 points. At least one of the proofs and 3 of the short questions will come from the hour tests. There will be two proofs and 6 short questions from each one-third of the semester.

Advice: Be sure to review the first two-thirds of the course. Often students who are doing well overlook the need for this review. Safety Net Policy: Each student is guaranteed at least the grade determined by the methods explained above. However, as an incentive to do well on the final exam, no final grade will be worse than one letter below the final exam grade. As an extreme example, one might have a failing final average and an A on the final exam, earning a B in the course. (The safety net helps some students each semester, but I have never seen an instance of the extreme example just given.)

Please download Final Exam, Fall 2021, Solution Sketches and Class Statistics.