Fei Qi

Office hours: Tuesdays 5:00PM - 6:00PM and 8:30 - 9PM (Thursday office hours TBD), Hill 624 or by appointment.

Email: cl.volkov at rutgers dot edu (for friends) / fq15 at scarletmail dot rutgers dot edu (for teaching)

In the Summer of 2017 I will teach 640:311 (Advanced Calculus) for Sections T6.

Here is the syllabus and the tentative schedule.
Please beware of the folllowing
1. This course is very demanding and requires at least 15 hours per week of after-class studying.
2. Once summer lectures start, there is no way to get 100% tuition refund.
So please double-check your summer schedule and make sure you have planned enough time for my class. If you don't, then you should seriously consider if you want to register officially. If you cannot make a decision, you are welcomed to discuss with me.
Course Materials:
  • Lecture 1 (May 30, 2017). Lecture Notes
    For more details, please read Ted Sundstrom, Mathematical Reasoning: Writing and Proof, Version 2.1, Chapter 1 and 2.
    Also you can read Zorich, Mathematical Analysis I, Section 1.1.

  • Lecture 2 (June 1, 2017). Lecture Notes, Workshop 1 (written by Dr. Scheffer), Writing Samples.
    The course materials mainly comes from Chapter 5 and 6 of Sundstrom's book.
    Also you can read Zorich's book, Section 1.2 and 1.3.
    All workshops are due 11:55PM the next Tuesday. So in case you have questions, you can discuss with me either before or after Tuesday's class.

  • Lecture 3 (June 6, 2017). Lecture Notes
    For more details, please read Zorich, Section 2.1.
    An slightly different argument showing root 2 is not rational can be found in [Z], 2.2.2.c. The argument in the notes is modified from [A], Theorem 1.4.5.
    The construction of real numbers using Dedekind cuts can be found in [A], Section 8.6.

  • Lecture 4 (June 8, 2017). Lecture Notes, Workshop 2
    Since I wasn't able to cover the density theorem, the workshop problem 5 is removed from this week's assignment.
    By now you should finish reading [A], Section 1.1 - 1.3 and Thompson-Brucker-Brucker, Elementary Real Analysis, Section 1.1 - 1.7.

  • Lecture 5 (June 13, 2017). Lecture Notes
    It is very important that the Nested Interval Property applies only to closed intervals that are bounded. Think: which part of the proof fails when the intervals are not bounded.
    One can prove under the assumption of Archimedean Property, Nested Interval Property can imply Axiom of Completeness. Please see James Propp's paper Real Analysis in Reverse for more details. In the coming Chapter we will see a lot more such properties.

  • Lecture 6 (June 15, 2017). Lecture Notes, Workshop 3
    In case you are interested in solving the optional workshop problem, please see the Notes on Countable Sets and Cantor's Diagonalization.
    The idea of Cantor's Diagonalization is to construct a decimal that is outside of the range of the function from the naturals to reals. Please see [A], Section 1.6 for details. In the note above you will find the most essential argument.
    By now you should finish reading Section 1.4 - 1.5 and 2.1 of the textbook, and Section 1.8 - 1.10, 2.1 - 2.4 of the TBB book
    About cardinalities, please read [Gamow] One Two Three Infinity, Chapter 1 and 2.

  • Lecture 7 (June 20, 2017). Lecture Notes
    Here you should learn the technique of finding the N from the given conditions of convergence, instead of from the estimates.
    Also, to use the Algebraic Limit Theorem, it is important to make sure that all the limits involved exist. Otherwise you might make some serious mistakes.

  • Lecture 8 (June 22, 2017). Lecture Notes, Workshop 4
    For the Order Limit Theorem, it is important to make sure that all the limits involved exist. Otherwise you might make some serious mistakes.
    Monotone Convergence gives a very convenient way of proving convergence, but usually does not tell you directly what the limit is. In general, getting the actual limit is usually difficult. In this class we only deal with some simple cases.
    Please make sure you can recall how to prove AoC implies MCT. Make a brief summary definitely helps.
    By now you should finish reading [A] 2.2 - 2.4, [TBB] 2.5 - 2.10.

  • Lecture 9 (June 27, 2017). Lecture Notes
    In case you are struggling with the Workshop 4, Mr. Yang kindly wrote a guide to all the problems and agreed to share. Note that this is just a guide. The thinking process has been elaborated presented. Yet it does not make a proof. You still need to organize these thoughts into a proof.

  • Lecture 10 (June 29, 2017). Lecture Notes, Workshop 5
    In case you are not satisfied with certain grade of the quizzes, or you have missed it due to any reason, please finish a write-up of the homework of the previous lecture and present your solution to me in person.
    For example, if you are not happy with your grades for Quiz 7, then you should do all the homework problems assigned in Lecture 7.
    I'll check a random problem to see if you really have good understanding towards it. If you have, then your quiz grade will be made to 8/10. To make up quizzes 1 - 9, your solutions must be presented before July 13th. After July 13th, the grades for Quiz 1 - 9 cannot be changed any more.
    By now you should finish reading [A] 2.5 - 2.6, [TBB] 2.11 - 2.12.

  • Lecture 11 (July 4, 2017) No lectures today. Happy holiday!

  • Lecture 12 (July 6, 2017). Midterm Exam, Workshop 6 (Written by Dr. Scheffer)
    Second chance policies: In case you didn't do well in the midterm, here is what you should do:

    • Study the course notes and other materials to make sure you know how to solve every problems in the exam.
    • Arrange a time for a Russian styled oral exam. I will pick a random problem in the exam.You will have 10 minutes for preparing the solutions. Then you should present the solution on the blackboard.
    • Books, pre-written notes are not allowed. The only thing you can refer to is the notes you generated in that 10 minutes.

    If your presentation is satisfactory, your midterm grade will be exonerated from the final grading computation. In other words, your grade will be computed as 60% Final + 20% Workshop + 10% Oral Quiz + 10% Written Quiz.

  • Lecture 13 (July 11, 2017). Course Notes
    For those who missed tonight's lecture, please make sure you are capable of proving every single entry in the table on Page 9. In class I explained those examples on the blackboard. However the proof was only given orally. Please let me know if you are having trouble proving any items. I will be happy to supply an argument.
    The written quiz tonight is replaced as a Questionnaire regarding the midterm. Please find it in Sakai Assignments.

  • Lecture 14 (July 13, 2017). Course Notes, Workshop 7
    Note: You don't need to worry the compactness part in either [A] or [TBB]. I did use the examples in [A] and the motivating comments in [TBB]. For Workshop 7, you don't need to know anything other than the currently posted course notes.
    By now you should finish reading [A] 3.2, [TBB] 4.1 - 4.4.

  • Lecture 15 (July 18, 2017). Course Notes
    I have set up the system, so Workshop 6 can be (re)submitted until Aug. 4. Workshop 7 can be (re)submitted until July 25th.

  • Lecture 16 (July 20, 2017). Course Notes, Workshop 8
    By now you should finish reading [A] 3.3, [TBB] 4.5 (Note that the Cousin's Property was not covered). You should start reading [A] 4.2 and [TBB] 5.1.
    Sorry for having delivered a stupidly organized lecture tonight. Hopefully the reorganized notes look better. Please let me know if you have troubles.

  • Lecture 17 (July 25, 2017). Course Notes
    Here are the pictures of the blackboard: Sequential Criterion - What we want, Sequential Criterion - What we know

  • Lecture 18 (July 27, 2017). Course Notes, Workshop 9
    By now you should finish reading [A] 4.1 - 4.3, [TBB] 5.1, 5.2, 5.4 and 5.5.
    On the second page of Workshop 9 you will find some comments to the exercises in [A]. Please at least attempt those problems I boldfaced.

  • Lecture 19 (Aug. 1, 2017). Course Notes
    As we are about to finish Chapter 4 on Thursday, it is a very good point to review everything. If you have a good understanding on the materials in Chapter 1 to 4, you should feel no difficulty at all to understand Chapter 5, and most of the parts in Chapter 6 (until you arrive at the issue of uniform convergence of sequences and series of functions). If you are taking 312 next semester, your life will be easy for a while. So please do so without hesitation.
    For those who didn't do well in the quiz tonight, please answer the following questions:
            1. How many exercises did you attempt in 3.2, 3.3, 4.2, 4.3, 4.4?
            2. What kind of difficulty did you experience?
            3. Anything I can do to help?
    Please send your answers through emails. The grade for the quiz will be adjusted to 8/10 or your actual grade, whichever is higher.

  • Lecture 20 (Aug. 3, 2017). Course Notes, Workshop 10
    Please attempt to prove those facts in Part 3 by yourself and do not read my argument unless you have no clue. My argument might be too complicated than it should be. The easiest way to simplify any complicated argument is to work your own argument without reading a word from the original one.
    The reason I chose these two easy problems for this last workshop assignment is to provide more free time for you to review the materials and attempt all the other problems in the book. Don't be lazy. You are not studying analysis for me, but to prepare for your future studies. The exercises in [A] is really the minimal amount you have to go through in order to master the skills.
    By now you should finish reading [A] 4.4 - 4.5 and [TBB] 5.6 - 5.9.

  • Lecture 21 (Aug. 8, 2017). Course Notes.
    As you can see, if you have a solid understand for Chapter 1 through 4, there is no trouble for you to understand at least the theory of derivatives. The main challenge for this Chapter is how to use the results in real life. Please see Zorich's exercises for more practice.

    Lecture 22 (Aug. 10, 2017). Course Notes. Review of Chapter 1 to 4
    The exam will be held on next Tuesday. There will be 13 problems with 200 points. 150 points are considered as a perfect score. Please find more details on Sakai.
    By now you should finish reading [A] 5.1 - 5.3. If you have time, please also read [TBB] 7.1 - 7.7. We don't have enough time covering all these materials however the knowledge will be assumed in 312.

In the Spring of 2017 I taught 640:244 (Differential Equation for Physics and Engineering) for Sections 20 - 22.
I taught the same class in the past. Here are the materials I taught in Summer 2015. And here are the materials I used for teaching recitations of 244 in Spring 2015, Fall 2014, Spring 2014 and Fall 2013.

Please find Dr. Shtelen's syllabus, schedule and homework assignments here.

Please find the information concerning maple labs here.

All announcements are to be posted on sakai. Please make sure that you have a working email address registered to the system.

You may find the following resource useful for this course:
Recitation Materials:

For 244 students, I have two requirements

If you have difficulties in these algebra issues, a series of link is provided for help.

  • If you don't know how to manipulate logarithm, please find
    Please read Section 10.5 on page 45 in the pdf file (page 733 in the book), try all example problems, and do Exercise 44 - 61 on page 51 in the pdf file (Page 740 in the book).

  • If you are not very fluent with the quadratic equations (e.g. always use the root formula), please find
    Read Section 8.1, 8.2, try all example problems, and do Exercise 66 - 83 on page 23 in the pdf file (Page 573 in the book). Make sure you understand all the related methods

    In particular, if you have never seen criss-cross factorization before, please check the youtube videos
    Criss-Cross Method 1, Criss-Cross Method 2, Criss-Cross Method 3 and Criss-Cross Method 4.

  • If you have never seen matrices before, please find
    Read Section 3.6, try all example problems, and do Exercise 15 - 23, 46 - 49 on page 51 - 52 in the pdf file (page 227 - 228 in the book).
    Read Section 3.7, try all example problems, and do Exercise 2 - 7, 20 - 25, 35 - 40 on page 63 - 64 in the pdf file (page 239 - 240 in the book).
    After you work on this topic, try the problems of the attendence quiz at Lecture 15 and you will find it easy to play.

  • If you keep on making mistakes on exponentials, please find
    Read Section 1.8, try all example problems, and do Exercise 59 - 84 on page 88 in the pdf file (page 88 in the book).

  • If you don't know how to divide a polynomial, please find
    Read Section 5.3, try all example problems, and do Exercise 27 - 42 on page 31 in the pdf file (page 339 in the book).
    After you have done the work, please compare to the technique I used on dealing with t/(t+1) or -2-t/(t+1) in class. You will see that this is actually the simplest example of division.

  • If you are not fluent on simplifications of rational functions, please find
    Read Section 6.1 - 6.4, try all example problems, and do Exercise 29 - 48 on page 61 - 62 in the pdf file (page 463 - 464 in the book).

  • If you are not fluent on playing with trigonometric functions, please find
    Read Section 4.3, make sure you memorize the table of the values of sine, cosine and tangent on usual special angles on page 23 of the PDF file (page 279 in the book)
    and do Exercise 17 - 26 on page 28 of the pdf file (page 284 in the book)
    Read Section 4.5, make sure you can recognize, distinguish different graphs of the trignometric functions and manipulate them by scaling and translation, and do Exercise 3 - 14, 23 - 16 on page 48 in the pdf file (page 304 in the book)

  • If you are not fluent on factorizing polynomials, please find
    Read Section 5.4, try all example problems and do Exercise 51 - 70 on page 40 of the pdf file (page 348 of the book) .
    Read Section 5.5, try all example problems and do Exercise 9 - 46 on page 52 of the pdf file (page 360 of the book).
    Read Section 5.6, try all example problems and do Exercise 43 - 70 on page 61 of the pdf file (page 369 of the book).
    Read Section 5.7, try all example problems and do Exercise 1 - 66 on page 67 of the pdf file (page 375 of the book).
    If you really do all the exercises, then after that you may probably find yourself addicted to playing such a game. I don't recommend to resist such an addiction. Just do all other exercises and it will accelerate your speed greatly in solving problems in homogeneous ODEs.

Notice: The books seem to be developing and page numbering might change. But the section number together with the exercise numbering will be invariant. If you did not find the exercise and the reading materials at the pages I told you, just scroll around, or use Ctrl+F to find sections.

In the Fall of 2016 I taught 640:311 (Advanced Calculus I) for Section 3 and 4.
Due to the migration of server at the beginning of the semester, this website was not updated for the whole semester. All the course materials was put on sakai only.
All the course materials are avaiable here. The TeX source of what I wrote is available here.

In the Summer of 2016 I taught 640:311 (Advanced Calculus I) for Section T6.
Here are the Syallbus and the Schedule.
All announcements and workshop assignments are made on sakai only. Please check the announcement tab and the email archive tab to make sure you don't miss anything.
For 311 students, I have two requirements
  • Please make sure you have a solid understanding on the math 300 class (Introduction to Mathematical Reasoning). You can review the knowledge using the following material
    Dr. Sussmann's notes on Math 300, Lecture 2, 3 and 4
    This set of notes summarizes the most essential knowledge in that class. On his course website you'll find more related material for reviewing.

  • Please recall the knowledge of Calculus I, especially the graphs of the most commonly seen elementary functions. You can check the following file to recall the knowledge:
    Table of Common Graphs
    Although the main focus is to formulate rigorous argument, in many cases this process is facilitated by the intuition from the graphs.
    Also I'll assume a solid basis of computational skills for this class. Please try problems in Chapter 1 and 2 of famous Russian book
    3193 Problems in Mathematical Analysis
    to test your skills.

Lecture Materials:
  • Lecture 1 (May 31, 2016): Lecture Notes, Homework (due June 2nd): 1.2.1, 1.2.5, 1.2.6, 1.2.7.
    None of the homework problems today needs any kind of induction! Please make sure you don't use it.
    For workshop submissions, Overleaf is a very nice online LaTeX editor that you can use. You can use the template to write your solutions. A LaTeX introduction and an Online Tutorial to LaTeX are provided here to help.
    In case your book hasn't arrived yet, here is the scan of the first few sections.

  • Lecture 2 (Jun. 2, 2016): Lecture Notes, Homework (due Jun. 7th): 1.2.10, 1.2.13, 1.3.3, 1.3.5.
    I have created a Discussion Page on Overleaf, collecting interesting problems I was asked via email. Please check here from time to time.
    Dr. Chris Woodward has agreed to share his lecture notes on math 300. Please find it on sakai.

  • Lecture 3 (Jun. 7, 2016): Lecture Notes, Homework (due Jun. 9th): 1.3.1, 1.3.11, 1.4.2, 1.4.8
    Someone asked about the Well-Ordering Principles and how to prove it. Dr. Sussmann provided a proof here

  • Lecture 4 (Jun. 9, 2016): Lecture Notes, Homework (due Jun. 14th): 1.4.6, 1.4.7, 1.5.2, 1.5.5
    In order to prepare you better for the future classes regarding countability, I'll go over this set of notes in the next lecture, but mostly by handwaving.

  • Lecture 5 (Jun. 14, 2016): Lecture Notes, Homework (due Jun. 16th): 2.2.2, 2.2.3, 2.2.4

  • Lecture 6 (Jun. 16, 2016): Lecture Notes, Homework (due Jun. 21st): 2.3.1, 2.3.2, 2.3.4, 2.3.5

  • Lecture 7 (Jun. 21, 2016): Lecture Notes, Homework (due Jun. 23rd): 2.3.7, 2.3.12, 2.4.1, 2.4.2

  • Lecture 8 (Jun. 23, 2016): Lecture Notes, Homework (due Jun. 28th): 2.5.1, 2.5.2, 2.5.6, 2.5.7
    Some hints to 2.5.1d is provided in the lecture notes. Hopefully it helps.

  • Lecture 9 (Jun. 28, 2016): Lecture Notes, Homework (due Jun. 30th): 2.5.5, 2.6.2, 2.6.3, 2.6.4
    In this set of notes I added the simplification to the arguments to Problem 2.5.2d. My argument for 2.5.2c refuses to be simplified. And it is useful for the homework problem 2.5.5. Please study them well before you attempt 2.5.5.
    The purpose of including series is to show an example on how Cauchy's criterion is applied. Infinite series won't be the main topic for the exam.

  • Lecture 10 (Jun. 30, 2016): Lecture Notes. No homework today. Attempt all other problems in the book and prepare for the coming midterm.
    Here is the Collection of Workshop Solutions so far. In case you are stuck, please find help here.
    The detailed proofs of theorems concerning limit superior and limit inferior is available in my old workshop notes.

  • Lecture 11 (Jul. 5, 2016): Midterm 1, Solutions
    People not doing well in Midterm 1 are welcomed to attend the Second Chance Club. Please find the details here

  • Lecture 12 (Jul. 7, 2016): Lecture Notes, Homework (Due Jul. 12th): 3.2.1, 3.2.2 (excluding d), 3.2.4 (excluding e)

  • I stopped updating this website due the server migration process (that messed a lot of things up). Everything was put on sakai. In order to prepare the future semester better, I put something back here.
  • Lecture 13 (Jul. 12, 2016): Lecture Notes, Homework (Due Jul. 14th): 3.2.4, 3.2.8, 3.2.14, 3.3.1, 3.3.5.

  • Lecture 14 (Jul. 14, 2016): Lecture Notes, Homework (Due Jul. 19th): 3.3.4, 3.3.6, 3.3.9, 3.3.11.

  • Lecture 15 (Jul. 19, 2016): Lecture Notes, Homework (Due Jul. 21st): 4.2.1(a), 4.2.2, 4.2.4, 4.2.5, 3.3.2.

  • Lecture 16 (Jul. 21, 2016): Lecture Notes, Homework (Due Jul. 26th): 4.2.3, 4.2.6, 4.2.7, 4.3.1. Also if you got 3.3.11 wrong, please resubmit it.

  • Lecture 17 (Jul. 26, 2016): Lecture Notes, Homework (Due Jul. 28th): 4.3.6, 4.3.8, 4.3.9, 4.4.2.
    Comments to other problems: 4.3.2 - 4.3.8 are very good exercises for you to get acquainted to the knowledge. 4.3.9 - 4.3.12 are important facts in the theory. 4.3.13 and 4.3.14 are fun but not so essential.

  • Lecture 18 (Jul. 28, 2016): Lecture Notes, Homework (Due Aug. 2nd): 4.4.2, 4.4.3, 4.4.4, 4.4.6, 4.5.2 (skip (e)), 4.5.7.

  • Lecture 19 (Aug. 2, 2016): Midterm 2, Solutions

  • Lecture 20 (Aug. 4, 2016): Lecture Notes, Homework (Due Aug. 9th): 5.2.2, 5.2.5, 5.2.7, 5.2.9.

  • Lecture 21 (Aug. 9, 2016): Lecture Notes, Homework (Due Aug. 11th): 5.3.2, 5.3.4, 5.3.6, 5.3.7.

  • Lecture 22 (Aug. 11, 2016): Lecture Notes. No more homework.

  • Lecture 23 (Aug. 16, 2016): Final Exam.

Supplementary Reading

In the Spring of 2016 I taught workshops for 640:311 (Advanced Calculus I) for Section H1 and 02.

Please find Dr. Cramer's course material on Sakai.

Please find Dr. Cakoni's course material on Sakai.

Workshop Materials:

  • Week 7 (Mar. 2, 2016):
    For Section H1: No notes available this week.
    For Section 02: Workshop Notes.
  • Week 8 (Mar. 9, 2016):
    For Section H1: Workshop Notes. Please check the notes next week for a more complete version.
    For Section 02: Workshop Notes.
  • Week 9 (Mar. 16, 2016): Spring break. No class.
  • Starting from Week 11 I switched to blackboard teaching. No further iPad notes available.

In the Fall of 2015 I served as the TA-at-large for 640:421 (Advanced Calculus for Engineering), Section 1 and 2.

ANNOUNCEMENT: Due to technical reasons, the first online office hour on September 6th is cancelled. Instead I'll stay in my office, holding in-person office hours.

Since there is no recitation meetings, I'll put some related materials here for reference.

About Laplace Transformation:

About Linear Algebras:

Review Materials:

Fei Qi
Room 624, Hill Center
Department of Mathematics
Rutgers University
110 Frelinghuysen Road
Piscataway, NJ USA 08854