Email: firstname dot lastname at umanitoba dot ca
Research Interest: Vertex algebraic structures in Mathematics and Physics, Mathematical Constructions of Quantum Field Theory
Work in progress
Publications:
In Winter 2021 I will teach Math 8510, Topics in Algebra 1: Introduction to Vertex Algebra and its Representation Theory. This is a PIMS network-wide graduate course.
I am basically following my Yale notes. For the points where the notes are not clear enough, I will add comments and supply remarks in the side notes. All the commented notes can be found here. The side notes can be found here.
In Winter 2021 I will also teach Math 1500, Introduction to Calculus, in University of Manitoba.
Lecture 1 (Jan. 19, 2021)
PreLecture Activities: Video 1-1, Video 1-2, Video 1-3, Review Question Slides.
Lecture activities: Recording, Notes .
Important resources: Course Syllabus, Course Schedule, Textbook.
Please check out the Math Help Center for Math Bootcamps and Saturday Workshop Series.
Other Resources: Stewart Calculus Chapter 1. Stepized slides for 1.1, 1.2, 1.3, 1.4.
For solving polynomial inequalities, please go over Page 3 to 5 of my old note and the MathIsFun Note on solving quadratic inequalities.
In case you registered recently and did not see my previous emails, you can browse the Email Archive. The file will be updated frequently.
Lecture 2 (Jan. 21, 2021)
Pre-lecture activities: Video 2-1, Video 2-2, Problem Slides Please attempt up to Page 25.
Lecture activities: Recording, Notes , Conceptual Slides on Limits
Some past Slides that might be useful: Stewart Calculus: 2.2,
Lecture 3 (Jan. 26, 2021)
Pre-lecture activities: Video 3-1, Video 3-2, Problem Slides Please attempt up to Page 39.
Lecture activities: Recording, Notes , Conceptual Slides on Limit Laws
Some past Slides that might be useful: Stewart Calculus: 2.3
Lecture 4 (Jan. 28, 2021)
Pre-lecture activities: Video 4-1, Video 4-2, Video 4-3, Problem Slides Please attempt up to Page 69.
Lecture activities: Recording, Notes
Lecture 5 (Feb. 2, 2021)
Pre-lecture activities: Video 5-1, Video 5-2, Problem Slides Please attempt up to Page 85.
Lecture activities: Recording, Notes, Office Hour Notes, Conceptual Slides on Squeeze Theorem, Conceptual Slides on Continuity.
Lecture 6 (Feb. 4, 2021)
Pre-lecture activities: Video 6, Problem Slides Please attempt up to Page 101.
Lecture activities: Recording, Notes, Office Hour Notes, Conceptual Slides on Limits at Infinity.
Supplementary: Solving inequalties using the graph.
Lecture 7 (Feb. 9, 2021)
Pre-lecture activities: Video 7-1, Video 7-2,
Video 7-3, Problem Slides Please attempt up to Page 21.
Lecture activities: Recording, Notes (Part 1), Notes (Part 2).
We will not have time in class to go over these Conceptual Slides. These slides are made from on Section 2.7, 2.8 of Stewart Calculus, corresponding to 3.1, 3.2 and 3.4 of our textbook. They should help in further understanding the related notions.
Lecture 8 (Feb. 11, 2021)
Pre-lecture activities: Video 8-1, Video 8-2, Problem Slides Please attempt up to Page 34.
Lecture activities: Recording, Notes, Proofs.
The proof slides include the most important proofs for the course. We went over differentiability implies continuity, sum rule and product rule in class.
Lecture 9 (Feb. 23, 2021)
Pre-lecture activities: Video 9-1, Video 9-2, Problem Slides Please attempt up to Page 42.
Lecture activities: Recording, Notes, Additional Problems from the Textbook, Office Hour Notes.
Lecture 10 (Feb. 25, 2021)
Pre-lecture activities: Video 10, Problem Slides Please attempt up to Page 48.
Lecture activities: Recording, Conceptual Slides, Example Solutions, Problem Slides, Additional Problems from the Textbook, Office Hour Notes.
Lecture 11 (Mar. 2, 2021)
Pre-lecture activities: Video 11, Inverse function and logarithmic function, Problem Slides Please attempt up to Page 67.
Lecture 12 (Mar. 4, 2021)
Pre-lecture activities: Video 12, Problem Slides. Please attempt up to last page. Chapter 3 Review. Please think about the problems before coming to class.
In Winter 2020 I taught Math 3472, Real Analysis III, in University of Manitoba.
Lecture 1 (Jan. 7, 2020): Syallabus, Boardworks (missing the last part), Student Notes, Exercises and Homework Problems.
We covered 12.1 - 12.3 of the textbook. We showed that the existence of directional derivative along every direction is not a good definition of differentiability for multivariable functions, as it does not imply continuity.
Lecture 2 (Jan. 9, 2020): Boardworks, Student Notes, Exercises and Homework Problems.
We covered 12.4 except the example on the linear functions. Originally I also planned to cover 12.5 but did not have time.
Lecture 3 (Jan. 14, 2020): Boardworks, Exercises and Homework Problems, Hints to 12.6.
We covered 12.5, first half of 12.7, and everything in 12.8. We introduced the matrix of a linear function. This part is pretty abstract. Two optional exercises are provided for practice. Then we use Theorem 12.5 to obtain the matrix of the total derivative, namely the Jacobian matrix. Finally we discussed an estimate that will be used in proving the chain rule.
Lecture 4 (Jan. 16, 2020): Boardworks, Exercises and Homework Problems.
Today we only managed to cover 12.6. Originally I was planning to finish 12.7 and start partially 12.9. We will have to wait until next week.
Starting from next week, all written quizzes will be given only on Tuesdays.
Lecture 5 (Jan. 21, 2020): Boardworks
Today we only managed to cover 12.9 and proved the chain rule. We briefly mentioned the conclusions in 12.7 and 12.10, but did not have time to prove anything.
As we did not cover enough materials today, no homework assignments for today.
Lecture 6 (Jan. 23, 2020): Boardworks, Exercises and Homework Problems
Today we finished everything in 12.7 and 12.10 regarding the matrix of linear functions and Jacobian matrix of composite functions. We also finished the first part of 12.11, proving the mean value theorem.
Lecture 7 (Jan. 28, 2020): Boardworks, Exercises and Homework Problems
Today we finished the everything in 12.11 and started 12.12. We will finish the proof next time.
Lecture 8 (Jan. 30, 2020): Boardworks, Exercises and Homework Problems
Today we finished 12.12 and most of 12.13. Next lecture we shall finish Chapter 12 and start Chapter 13.
The midterm is scheduled on February 25. You will need to finish four problems. Three of them will be based on Type I and II homework problems (most likely identical, modification will only be done to lower the difficulty). Another one will be from somewhere else. I will also provide an optional extra credit problem that will be challenging.
Lecture 9 (Feb. 4, 2020): Boardworks, Exercises and Homework Problems
Today we finished Chapter 12 and started Chapter 13.1.
Lecture 10 (Feb. 6, 2020): Boardworks, Exercises and Homework Problems
We finished 13.1 and started 13.2. We finished proving the first theorem in the section. Next lecture we will discuss its consequences.
Lecture 11 (Feb. 11, 2020): Boardworks
We finished 13.2 and 13.3. We discussed the ideas for the proof of the inverse function theorem. Please organize these ideas into a formal proof.
Lecture 12 (Feb. 13, 2020): Boardworks, Exercises and Homework Problems
We finished 13.4. The midterm exam after the reading week will be based on the materials up till today.
Here is a very brief Review. It is organized with my own way of wording. Please also attempt to make a summary using your own words.
Lecture 13 (Feb. 25, 2020): Midterm
Second chance policies: In case you didn't do well in the midterm, here is what you should do:
Lecture 14 (Feb. 27, 2020): Boardwork, Exercises and Homework Problems
Today we covered 11.1 to 11.5. We proved the approximation theorem, Bessel inequality and Parseval formula.
Lecture 15 (Mar. 3, 2020): Boardwork, Exercises and Homework Problems
Today we covered 11.6 to 11.8. We proved Riesz-Fischer theorem and Riemann Lebesgue Lemma.
Lecture 16 (Mar. 5, 2020): Boardwork, Exercises and Homework Problems
Today we covered 11.9 and gave an overview to 11.10 - 11.12. We studied the Dirichlet integrals and studied two theorems concerning their limits.
There are some subtleties regarding the integral sin(x)/x over [0, \infty). I'll talk about them next Tuesday.
Lecture 17 (Mar. 10, 2020): Boardwork
We spent a lot of time discussing the oral quiz and thus only covered 11.10, the integral representation of partial sum of Fourier series. Please make sure you can answer these questions orally.
Here is the solution to Problem 11.7(b).
Lecture 18 (Mar. 12, 2020): Boardwork
Today we covered 11.11, 11.12 and most of 11.13. We used Riemann-Lebesgue lemma to study the integral representation, interpreted the convergence problem in terms of limits of Dirichlet integrals, and obtained two sufficient conditions for the convergence of Fourier series. We also studied the Cesaro summability of the Fourier series, stated the Fejer's theorem and briefly discussed the idea. Next lecture we will prove it.
Obvious some students are stressing out. So I will not leave any homework this week. Instead, please use the time study the proofs and summarize the key facts and steps of proofs.
It is highly recommended that you read the Stein-Sarkachi, Chapter 2, where you will see a more conceptual and modern treatment of the subject. Comparing different version of the same facts will facilitate the process of internalization.
Due to the outbreak of COVID-19 in Winnipeg area, the rest of the lectures will be delivered electronically via self-paced PowerPoint Presentations.
Please download the file on your computer and view it via Microsoft Office. The University of Manitoba offers free institutional access.
Currently the presentation does not work without Microsoft Office. I have no way to fix it at this moment.
Lecture 19: Presentations, Exercises and Homework Problems
This lecture covers 11.13 - 11.15. We gave a proof to Fejer's theorem, discussed its consequence, and proved Weierstrass approximation theorem. In addition, I also included the proof of mean convergence of Fourier series generated by square-integrable functions. This result is still much weaker than Carleson's theorem, but it is at least a step forward.
Lecture 20: Presentations, Exercises and Homework Problems
This lecture covers 11.16 - 11.19. We discussed the Fourier series in complex exponentials, and generalized both the trignometric and exponential Fourier series to general periodic functions. We also studied the Fourier integral and its convergence theorem for nonperiodic integrable functions, and interpreted the theorem in term of Fourier transform.
Lecture 21: Presentations, Exercises and Homework Problems (TBD)
This lecture covers 11.20 - 11.22. We defined the convolution f * g for two integrable functions f and g, proved some sufficient conditions for its existence, boundedness, continuity and integrability, and showed that the Fourier transform of the convolution f * g coincides with the product of Fourier transforms of f and g. Finally, we proved Poisson's summation formula and discussed two applications.
In Fall 2019 I taught Math 1500, Introduction to Calculus, in University of Manitoba.
Lecture 1 (Sep. 5, 2019): Slides for Syllabus, Full Syllabus Please do read the latter!
Slides for 1.1 I finished up to Slide 14. The rest of the slides will be gone over next lecture.
All the numberings of the examples coincide with those in the textbook. You will find more elaborately written solutions in the textbook
Here are information regarding Open Math Competition, Math Bootcamp, Saturday Workshop Series, Additional Resources
Please also go over the Slides for 1.2, preferably also the textbook. I am not sure if we will have time for it during the next lecture.
Lecture 2 (Sep. 10, 2019): Slides for 1.1, Slides for 1.3.
We finished the slides for both 1.1 and 1.3, but did not spend any time reviewing 1.2, Please make sure that you are comfortable with the knowledge.
In case you are not feeling confident in solving inequalities, please read Section 1.7 of the Precalculus Textbook and try some exercises.
Here is a refresher for completing a square. Please also check other related topics on the same website for review.
Lecture 3 (Sep. 12, 2019) Slides for 1.4, Slides for 2.2.
We went over everything in the slides.
I only gave a very brief summary to Section 2.1. You are very encouraged to read through the detailed discussions in the textbook on average velocities and instantaneous velocities. It will help in connecting the math and the real world.
The lecture today is quite heavy on the conceptual side, which took human-kind two centuries to develop. I went over them quickly, as a detailed discussion is not within the expectation of this computation-based course. Next lecture when we start the computations, the pace will be much slower.
Lecture 4 (Sep. 17, 2019) Slides for 2.3.
We went over everything in the slides.
I included the solutions to some examples, as my writing in class is somehow different to that in the textbook.
The solution to the attendance quiz is also included. For the squeeze theorem exercise, it seems no one realized that the sign of x is indeed a problem. Please read the solutions carefully.
Lecture 5 (Sep. 19, 2019) Slides for 2.5.
We stopped at Slide 14. The rest of the slides will be gone over during next lecture.
To review the knowledge of trignometric functions, please consult Appendix D of the textbook. Please find a scan here.
To review the knowledge of inverse functions and inverse trigonometric functions, please consult Section 1.5 of the textbook. I am still making slides for that section. It should be ready by the next lecture.
The proof to those limit laws are not expected in this course. However, you should know how to justify each step in the computations. Please read Example 2.3.1.(a), 2.3.2 and 2.5.4 in great detail, and attempt Exercise 2.3.1 - 2.3.9. Here are
the solutions to some of them.
Lecture 6 (Sep. 24, 2019) Slides for 2.5. Slides for 2.6. Side Notes for Example 4 and 5
We stopped at 2.6 Slide 14. Only two examples on Slide 15 were left. I attached the solutions on Slide 15 and 16. Please go over them by yourself.
Please also go over the additional slides on infinite limits and indefinite limits by yourself. Here is an unanimated version. Please be aware that the knowledge in 4.7, namely the L'Hospital rule, is not allowed to use in the tests and quizzes for the current course.
The Slides for 1.5 is ready. Though we are scheduled to cover this section in class, I am not sure how much time can be used for going over the details. You are strongly suggested to go over the set of slides and the textbook before that.
Lecture 7 (Sep. 26, 2019) Slides for 2.7. Slides for 2.8. Side Notes.
We finished 2.7, but stopped at 2.8 Slide 7. Next class we will start from 2.8 Slide 8
Some of the discussion of the tangent line exists in the slides of 2.2. It's a good time to review now.
If you are not fluent on simplifications of rational functions, please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch06_SE.pdf
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)
In case the links on this webpage becomes inaccessible, please find the slides in the following dropbox folder (not updated in real time).
Lecture 8 (Oct. 1, 2019) Slides for 2.8, Slides for 3.1. Precalc used.
We finished all the slides, but skipped the discussion on accelerations, jerks, and Example 3.1.7. Please go over them by yourself. We will not have more lecture time for these topics.
Test 1 will cover up to 2.8. In particular, it will not evaluate your skills on the differentiation rules we introduced today. Please make sure you have read the textbook of all the sections, reconstructed the solutions of the examples, and attempted enough exercises.
Lecture 9 (Oct. 3, 2019) Slides for 3.2. Slides for 3.3. Full Solution to Example 1(b), Section 3.2.
We finished all the slides, but did not have time to go over Example 1, 3, 5, 6 in Section 3.3. These are all very easy computations. Next lecture I will briefly go over them.
These two sections are not difficult to understand. But it takes some efforts to gain the skill to perform fast and accurate computations. Please attempt enough exercises.
Lecture 10 (Oct. 8, 2019) Slides for 3.4.
We finished all the slides and had time for the attendance quiz. I added some more details regarding the last three examples, together with solutions to the attendance quiz.
I found in this morning that the server did not update my webpage last Thursday. So I decided to share my OneDrive folder. In the future if the webpage did not update in time, please find the slides in this folder. And do not forget to report by email.
Please be aware that the future slides in the folder are not ready to use until the very last moment. If you want a preview, I would recommend using the textbook.
Lecture 11 (Oct. 10, 2019) Slides for 3.5, Slides for 3.9
We finished everything except Example 5 in 3.9. I have included the solutions in the slides.
Please make sure you attempt enough exercises in 3.4 before you work on 3.5 and 3.9. The book solutions assumes the reader to be fluent enough using the chain rule and thus did not offer too much explanations in the computations.
Next lecture we will rush through Section 1.5 to compute the derivatives of the logarithmic functions. Please do take a look at Slides for 1.5 before coming to the lecture.
Lecture 12 (Oct. 15, 2019) Slides for 1.5, Slides for 3.6
We finish up to Slide 6 of Section 3.6. Next lecture we will start from there.
In case you didn't do well for Test 1, the good news is, the computational skills you need for Test 2 are independent of those for Test 1. If you manage to follow up the materials in Chapter 3 and attempt enough exercises, there should not be much troubles to score well in Test 2.
It turns out we are almost two lectures ahead of schedule. After finishing 3.6 next lecture, we will go for a review and solve some additional exercises in class.
Lecture 13 (Oct. 17, 2019) Slides for 3.6
Review Slides
We stopped at Slide 5 and finished almost all the boxed questions in the review slides, except 36 and 50. Next lecture we will finish these problems, together with 4, 16, 42 and 44 for more practices on quotient rules.
In case you want to attempt other even-numbered problems, please use Wolfram Alpha to check your solutions.
I will update the slides by next Monday to include some problems among 51 - 112.
Lecture 14 (Oct. 22, 2019)
Review Slides, Review of proofs
The solutions to Problem 50 and all the related rates problems in the slides are included. The solutions to the attendance quiz are not included. If you attempt them and want to see if you got correct answers, please send me your work. I'll mark them.
In the review slides for proofs, I included an idea part to show the thinking process of the proof. And the actual proof is written with more rigor. I wrote it in this way so as to make it easier to understand. You are not required to write a proof in the same way.
Lecture 15 (Oct. 24, 2019)
Slides for 4.1
We finished everything on the slides.
I attached the solution to Problem 3(a) of the attendance quiz for the previous class at the end of the review slides. Please make sure you do not commit the crime of abusing algebra in the tests.
In case any links of the webpage fails, please do not hesitate to report.
Lecture 16 (Oct. 29, 2019)
Slides for 4.2, Slides for 4.3,
We finished everything in 4.2 and stopped at Slide 7 of 4.3.
All the proofs to be required in the tests in Chapter 4 are covered in today's lecture. Please make sure you can reconstruct them.
4.3 is the core knowledge of the class. On Thursday we will finish it. Please attempt enough exercises after class.
Lecture 17 (Oct. 31, 2019)
Slides for 4.3,
We finished everything in 4.3 except Example 7 and Example 8. I decided not to post the full solutions, but to finish them in the next lecture. In case you cannot wait until the next lecture, please feel free to read the book.
Next lecture we will systematically introduce the technique of curve sketching. To successfully sketch a curve, we will have to use everything we have learned so far. Please preview the Sildes for 4.5 and review the related concept accordingly.
Lecture 18 (Nov. 5, 2019)
Sildes for 4.5, Why xe^x approaches to 0 when x gets negatively large
We only finished the first three examples of 4.5 and did not cover the two leftover examples from Lecture 17. Next lecture at the beginning I will rush through them.
The Test 3 will cover everything in Chapter 4 up to 4.5. Section 4.7 to be introduced next lecture will not be included in Test 3, so as to make it easier for you.
As you have seen today, to sketch the curve of a function you need to know everything we have learned from this class. It is a good time to start a review. Please make good use the reading week.
Lecture 19 (Nov. 7, 2019)
Sildes for 4.5, Slides for 4.7
We only finished Example 1 in 4.7 and stopped at Example 2. Next lecture I will not work on Example 2, as the solutions are all in the slides. Please attempt Example 3, 4, 5 by yourself. Next lecture we will rush through these three examples to conclude Chapter 4.
Next lecture we will also start a new module regarding antiderivatives and integrations. The new module is very indirectly related to the previous chapters. Please plan the reading week well for reviewing Chapter 1 - 4.
Lecture 20 (Nov. 19, 2019)
Slides for 4.7, Sildes for 4.9
We finished 4.7 and stopped at Slide 5 of 4.9. Next lecture we will go over the examples in 4.9 and 5.4.
The review suggestion for Test 3 can be found on Slide 13, Section 4.7.
Lecture 21 (Nov. 21, 2019)
Sildes for 4.9, Sildes for 5.4
We finished everything in the slides.
For 5.4 you are now ready to attempt 1-20. The rest of the problems depends on the knowledge of 5.3 that will be taught next week.
Regarding review suggestions I posted on Slide 13, 4.7, please put Task 5, 6, 7 as your priority. In case you feel it difficult to handle Task 7, please turn to Task 1 and Task 2 to improve your skills, then come back to Task 7.
Next lecture we will be studying 5.1 and 5.2. It will be very heavy in concepts. You are advised to preview those sections using the Slides for 5.1, 5.2
Lecture 22 (Nov. 26, 2019)
Slides for 5.1, 5.2
We finished 5.1 but did not finish 5.2. We stopped at Slide 20. I have supplemented some of the contents I went over on the blackboard.
Here is the complementary note on how to obtain sums of squares. It is not required in the final exam. There is no pressure to study them.
Please see the suggested homework problems to have an idea of the type of problems that might appear in the final exam. This comment applies to all sections.
Lecture 23 (Nov. 28, 2019)
Slides for 5.1, 5.2
We finished everything in the slides, including 5.1, 5.2 and the second half of 5.3.
Next lecture we will see the proof of the fundamental theorem of calculus. Then we will attempt some additional exercises in Chapter 5 to strengthen the skills. You can preview the slides following this link.
Lecture 24 (Dec. 3, 2019)
Slides for 5.3
We discussed the proof of the fundamental theorem of calculus and finished up to Slide 17.
A better solution to Problem 42 on Slide 16 is supplemented on Slide 17.
Next lecture we will first finish the problems on Slide 18 - 23. Then we will go over the conceptual review and true/false quizzes in Chapter 5, Chapter 1 and Chapter 2. Please preview the slides following this link.
Lecture 25 (Dec. 5, 2019)
Review Slides for Chapter 5, 1 and 2, Solutions to T/F Quiz Questions in Chapter 5, Chapter 1 and Chapter 2
Three remarks are added on Slide 6 of the review regarding the variables appearing in the definite integrals.
Here are the slides of required proofs in Chapter 2 and 3. For Chapter 4, the required proofs are Theorem 5 in 4.2 and Increasing/Decreasing Test in 4.3. Please find the proofs in the textbook.
Here is a note regarding the derivative of integrals with varying upper and lower limits.
I will be open for appointments until the exam day. Please make sure you make an appointment 24 hours before the time you want to come.
In Spring 2019 I taught Math 755, Vertex algebras and representations in Yale University.
In Fall 2018 I taught Math 115a, Calculus of Functions of One Variable II in Yale University.
Lecture 1 (Aug. 29, 2018): Syllabus (annotated), Lecture Notes
Answers to the Practice, First Day Survey (designed by Dr. Smith), Worksheets of finding the area of the unit disk (designed by Dr. Smith)
Before coming to class on Friday, you should finish reading 5.1 and 5.2. Also, you can watch Ghrist Lecture 25
Please also make sure that you can receive the announcements sent in Canvas.
Lecture 2 (Aug. 31, 2018): Lecture Notes, Details of discussion
In case you are still not confident in playing with the sigma notation, please go over this introduction, then work on Section 2 of this worksheet.
Before coming to class next Wednesday, you should finish reading 5.3 and 5.4 and finish the corresponding explorations.
Also, you can watch Ghrist Lecture 26 and Ghrist Lecture 26 - Bonus
Lecture 3 (Sep. 5, 2018): Lecture Notes, Details of discussion
In case you need more practice with trigonometry functions, please use this booklet (from Sydney University). Please make sure you are absolutely comfortable with the first three chapters.
Problem Set 1 is due tonight at 11:59PM. Please let me know if you have any issue uploading your work to Canvas.
Lecture 4 (Sep. 7, 2018): Lecture Notes. All details are in the textbook. Please read!
In addition to the book problems, you can also play with Problem 1191 - 1200 of the famous Demidovitch Problem book.
You are now ready to work on Problem Set 2. You are advised to start working now.
Lecture 5 (Sep. 10, 2018): Lecture Notes, Details (done in D & I), Details (done with implicit substitution)
The problems I have done in the lecture are in general trickier than what you will see in the exam. If you are comfortable with these questions, you will have no problem getting a perfect score with integration by parts.
For Problem 5.3.4.(f), you may use the method introduced in Example 2.8.1 of the textbook
Lecture 6 (Sep. 12, 2018): Lecture Notes, Details
Here is a handout from University of Adelaide recording the usual trig. identities.
The sum and difference formulas at the end of Section 7.2 is not required for tests and exams. Still, it doesn't hurt to know more.
Lecture 7 (Sep. 14, 2018): Lecture Notes, Details
I have been using implicit substitutions in class, which is based on the knowledge of linear approximation and differentials. Here is a scan of Section 3.10 in Stewart's textbook that will help you to develop a conceptual understanding to differentials. You can also learn the knowledge from Ghrist Lecture 12 and Ghrist Lecture 15. Some practice problems can be found in Demidovich Section 2.6.1. Problem 712 to 745 are relevant, with solutions available.
Lecture 8 (Sep. 17, 2018): Lecture Notes, Details
For practice problems in the textbook, you can use Wolfram Alpha to check your solution.
Caution: Do not attempt to use the step-by-step solutions generated by Wolfram Alpha in your problem set. Computers work on these integrals with an approach that is very different from what we human would do. It is very easy for us to recognize that.
Lecture 9 (Sep. 19, 2018): Lecture Notes, Details
Please see Section 9.4 for an explicit solution to the logistic model. You will find that it is less informative than what can be seen from the qualitative method.
Lecture 10 (Sep. 21, 2018): Lecture Notes, Details
Due to the conference I am currently attending, I had limited time and thus did not include a lot of details. In case you need me to include more details, please send me an email. I'll update the notes later when I have time.
As I can observe from class and from email correspondences, people are starting to sacrifice sleep for other aspects of life. I strongly urge you not to do that, as I don't want you to regret in the same way I do when you reach my age.
Plese read the recent research by Dr. Spira et. al regarding the relation between daytime sleepiness and alzheimer. Please also wathc the TED talk for the background concepts.
Lecture 11 (Sep. 24, 2018): Lecture Notes. Please find details in the textbook.
Please go over the graphs of common functions and figure out a way to understand them. Please also go over function transformations to see how to shift, stretch, compress and reflect the graph of a function.
Please find the graphs of x^n on the following Desmos illustration (which is far better than my iPad drawing.)
For solving polynomial inequalities, please go over Page 3 to 5 of my old note and the MathIsFun Note on solving quadratic inequalities.
Lecture 12 (Sep. 26, 2018): Lecture Notes. I did not do any example in class. Please try the examples in the book.
Please find Newton's law of cooling in Example 3 of Section 3.8 of Stewart. Here is a scan for your convenience.
Please also take some time to try the Lab Project of Taylor Polynomials on Page 258 of Stewart. Here is a scan for your convenience. We will use Taylor polynomial to explain the error bounds on Friday. The knowledge will appear again when we discuss Taylor series. So it does not hurt to get a headsup now.
Lecture 13 (Sep. 28, 2018): Lecture Notes, Supplementary Notes.
The derivation of error bounds of left endpoint rule, right endpoint rule and midpoint rule can be found here (the writing should be futher improved.)
Please find the derivations of errors bounds of trapezoidal rules here (organized by Ed Bender, UCSD.)
Lecture 14 (Oct. 1, 2018): Lecture Notes. Details
Please review L'Hospital's rule by going over Stewart 4.4. Here is a scan of the section.
Lecture 15 (Oct. 3, 2018): Lecture Notes (did not actually cover in class).
Here is the solutions to some problems I was asked during the office hours.
The first midterm will be held on Thursday night 7PM - 8:30PM in Davis Auditorium. Good luck!
Lecture 16 (Oct. 5, 2018): Lecture Notes (did not actually cover in class).
I just managed to cover Part 1 to Part 3.3. Part 3.4 is too important to rush through. I'll leave it for the next lecture.
Lecture 17 (Oct. 8, 2018): Lecture Notes (last time). Lecture Notes
We just started to talk about geometric series. This special type of series is of great importance. Please make sure you have good knowledge to it.
Lecture 18 (Oct. 10, 2018): Lecture Notes (last time). Lecture Notes (supposed to be for today)
We did not even finish the lecture notes for last time, not to mention the notes supposed for today.
Please read the textbook 11.1 to 11.3 at home. All the example problems I have been doing is from the textbook. For the next lecture, I intend to cover 11.3 and 11.4 altogether. I will go over the notes for 11.3 in a fast pace. Please treat it as a review instead of a lecture.
Here is some hints to Problem 7.8.2 and 11.1.36. Hopefully they help.
The scan of the exams will be sent to you at 7PM. Please report any issues on grading by 11:59PM. For fairness to students in other sections, issues reported after that will result in no change of grades.
Lecture 19 (Oct. 16, 2018): Lecture Notes.
Lecture 20 (Oct. 22, 2018): Lecture Notes.
Lecture 21 (Oct. 24, 2018): Lecture Notes. Sorry for the delayed website update. My fall break is basically spent in bed struggling with cold, fever and serious coughing. I wish your fall break had been much better than mine.
Lecture 22 (Oct. 26, 2018): Worksheet (by Prof. Addicott). Solutions
Here is a collective table of facts that can be used to test the convergence of a series. This is probably better than the class handout, as comments are provided for each method. Also please find a flow chart that shows the general procedure in determining the convergence of a series. This helps you to determine where to start and what to do when stuck.
Lecture 23 (Oct. 29, 2018): Lecture Notes.
Lecture 24 (Oct. 31, 2018): Lecture Notes.
Lecture 25 (Nov. 2, 2018): Lecture Notes.
I wrote a set of notes that summarizes the common inequalities used in this class. Hopefully it helps to gain some confidence.
Lecture 26 (Nov. 5, 2018): Handout, Details I should cover in class.
As Nov. 6 is the election day, the problem set for this week will be due on Friday, instead of on Wednesday.
Lecture 27 (Nov. 7, 2018): Lecture Notes
I made some modifications in this final version. Please compare it with your notes in class.
Please find the scan of review for Chapter 11 and go over the concept check section. Also you can use Problem 40 - 58 to check your understanding towards power series. In case you are stuck or have a question, don't hesitate to ask.
Dr. Smith will be holding a review session on Sunday night. You are strongly advised to attend.
Lecture 28 (Nov. 9, 2018): Lecture Notes
Lecture 28 (Nov. 9, 2018): Lecture Notes
Lecture 29 (Nov. 12, 2018): Lecture Notes
Lecture 30 (Nov. 14, 2018): Lecture Notes (same as previous, as I did not finish the notes)
Lecture 31 (Nov. 16, 2018): Lecture Notes
Lecture 32 (Nov. 26, 2018): Lecture Notes
Lecture 33 (Nov. 28, 2018): Lecture Notes
Lecture 34 (Nov. 30, 2018): Lecture Notes
Lecture 35 (Dec. 3, 2018): Lecture Notes
Lecture 36 (Dec. 5, 2018): Lecture Notes
Lecture 37 (Dec. 7, 2018): Lecture Notes
Past Courses in Rutgers University
Semester | Role | Course and Materials | Number |
Sections & |
Comments |
Spring 2018 | Online TA | Introduction to Linear Algebra | 250 | 5, 6 | |
Fall 2017 | On fellowship, no teaching | ||||
Summer 2017 | Instructor | Introduction to Real Analysis I | 311 | T6 | Most up-to-date course notes |
Spring 2017 | TA | Differential Equations for Science and Engineering | 244 | 20, 21, 22 | Most up-to-date recitation notes |
Fall 2016 | TA | Introduction to Real Analysis I | 311 | ||
Summer 2016 | Instructor | Introduction to Real Analysis I | 311 | T6 | |
Spring 2016 | TA | Introduction to Real Analysis I | 311 | H1, 2 | |
Fall 2015 | Online TA | Advanced Calculus for Engineering | 421 | 1, 2 | |
Summer 2015 | Instructor | Differential Equations for Science and Engineering | 244 | C1 | Most up-to-date course notes |
Spring 2015 | TA | Differential Equations for Science and Engineering | 244 | 23, 24, 25 | |
Fall 2014 | TA | Differential Equations for Science and Engineering | 244 | 9, 10, 11 | |
Summer 2014 | Instructor | Calculus I | 135 | C6 | Most up-to-date course notes |
Spring 2014 | TA | Differential Equations for Science and Engineering | 244 | 8, 9, 10 | |
Fall 2013 | TA | Differential Equations for Science and Engineering | 244 | 1, 2, 3 |
Fei Qi
Room 451, Machray Hall
Department of Mathematics
University of Manitoba
186 Dysart Road
Winnipeg, MB R3T2N2, Canada