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.
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 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 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.
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.
Week 2 (Jan. 25): Recitation Notes, Quiz 1.
In case you have time, please also watch MIT Lecture 1 to further understand the geometric interpretation of ODE.
Regarding the first order linear ODE, you can also check MIT Lecture 3 and read Dr. Z's notes for 2.1 for further understanding.
Here are my own notes for Section 2.2 and 2.4
Week 3 (Feb. 1): Recitation Notes, Quiz 2, Solutions.
In case you have time, please also read Dr. Z's notes for 2.2 and Dr. Z's notes for 2.4 for further understanding.
More modeling examples with ODE is available in MIT Lecture 7 and MIT Lecture 8.
Here are my own notes for Section 2.3, Section 2.5 and Section 2.6
Maple Lab 1 has been assigned and will be due on Feb. 24, 2017. Please find the instructions here and the seed file here. Please submit your work on sakai.
Week 4 (Feb. 8): Recitation Notes (Part 1), Recitation Notes (Part 2) (allow me to reuse the notes in the past). Quiz 3
In case you have time, please also watch MIT Lecture 5 and read Dr. Z's Notes on 2.5 (Note that Dr. Z used a different method).
Here are my own notes for Section 2.7 and Section 3.1.
Week 5 (Feb. 15): Recitation Notes, Quiz 4
As we won't have time to go over the problems assigned this week, here is the old recitation notes for everything in 3.1 - 3.4 and solution to some tricky problems.
Please make sure that you can answer all the Review Questions.
In case you have time, please watch the MIT Lecture 2 for a much better understanding to numerical methods, and MIT Lecture 9, MIT Lecture 10 and MIT Lecture 11 for a better understanding to 3.1 - 3.4.
Here are my own notes for Section 3.3 and 3.4 and Section 5.4 (starting from Page 7). You can also read Dr. Z's notes on 2.7, 3.1, 3.2, 3.3 and 3.4
In case you are not familiar to complex numbers, please watch MIT Lecture 6 to get acquainted.
Some old study guides, midterm exams and solutions can be found in the websites of my past teaching: Summer 2015, Spring 2015, Fall 2014. Hopefully they help.
Week 6 (Feb. 22): No recitation notes this week. Aside from those exam problems, I just went over the notes I announced in the previous week.
The Quiz this week is take-home. Please carefully review Section 2.6 and 3.4.
Week 7 (Mar. 1): Recitation Notes, Yet another take-home Quiz
The principle I talked about in the recitation notes applies to Chapter 4 as well. You should keep in mind that
1. First try templates, as well as exponential powers, are determined ONLY by the right hand side of the ODE.
2. To determine how many times your template fails, you have to look at the characteristic roots, which are determined ONLY by the left hand side of the ODE.
Please understand this set of recitation notes thoroughly.
For 3.5 and 3.6, Dr. Z's notes may also be helpful: Notes on 3.5, Notes on 3.6
My own notes on 3.5 (Part 1), 3.5 (Part 2), 3.5 (Part 3), 3.6, 3.4 and 3.7 (Course Plan), 3.4 and 3.7 (Notes Part 1), 3.7, 5.4 (Notes Part 2), 3.6, 3.8
Week 9 (Mar. 15): Spring break. No recitation today. Enjoy!
Week 10 (Mar. 22): Recitation Notes, Quiz 8, Quiz 8 Make-up
Maple Lab 3 is due next week. Late submissions are allowed up to next Friday (Mar. 31, 2017).
In case you have time, please read Dr. Z's notes on Section 4.1, Section 4.2, Section 4.3.
My own notes on 4.1, 4.2, 4.3. Please find my notes on 3.8 above.
Week 11 (Mar. 29): Recitation Notes for Linear Algebra, Quiz 9, Recitation Notes for 7.5, 7.6 and 7.8
(Although these notes were written a while ago, it should be able to help)
For the linear systems, Dr. Z's notes on 7.1, 7.4, 7.5, 7.6 and 7.8 should also be helpful.
Please go over the (updated) Review Questions and make sure you are comfortable on everyone of it. I think it would help you better than any practice exam.
Week 12 (Apr. 5): Recitation Notes Part I: Phase Portraits (real eigenvalues), Part II: Phase Portraits (repeated and complex eigenvalues), Quiz 10
Maple Lab 4 is due on next Tuesday (Apr. 11). Late submissions are accepted until next Friday (Apr. 14).
Related MIT Lectures: Lecture 24, Lecture 25, Lecture 26, Lecture 27
Week 13 (Apr. 12): Quiz 11
Aside from exam problems, all I talked about in class are in the recitation notes or previous week. Please go over it and especially make sure you know how to deal with complex eigenvalues.
Week 14 (Apr. 19): Quiz 12
Here are my summer course notes on Chapter 9: 9.1, 9.2, 9.3, 9.4, 9.4 leftovers (Shared by Ms. Shawnie Caslin). Also please watch MIT Lecture 31 for how to deal with nonlinear systems.
I wasn't able to type up the notes for finding global trajectories. In case you have taken neat notes, please don't hesitate to share.
Maple Lab 5 is supposed to due yesterday. Late submissions are accepted until next Tuesday (Apr. 25).
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
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
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
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.
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 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 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 12 (Jul. 7, 2016): Lecture Notes, Homework (Due Jul. 12th): 3.2.1, 3.2.2 (excluding d), 3.2.4 (excluding e)
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 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.
Please find Dr. Cramer's course material on Sakai.
Please find Dr. Cakoni's course material on Sakai.
In the Fall of 2015 I served as the TA-at-large for 640:421 (Advanced Calculus for Engineering), Section 1 and 2.
Since there is no recitation meetings, I'll put some related materials here for reference.
About Laplace Transformation:
About Linear Algebras:
Room 624, Hill Center
Department of Mathematics
110 Frelinghuysen Road
Piscataway, NJ USA 08854