Sergei Yakovenko's blog: on Math and Teaching

Friday, July 6, 2018

Calculus on manifolds 2017/8 – Exam

Filed under: Calculus on manifolds course,problems & exercises — Sergei Yakovenko @ 5:12

Finally the summer is here, and with it the exams.

The revised lecture notes are available here. The problems for the exam are at the end of the notes.

The rules of the game are extremely simple.

    1. The grading is by the pass/fail scheme.
    2. There is no predefined minimum of problems/items to get “pass”. Everything is decided in a purely subjective matter. Please avoid writing nonsense just for the sake of writing something: if accumulated beyond certain critical mass, it may result in the unlikely “fail”.
    3. All textbooks are at your disposition: actually, you are encouraged to consult as many of them as you feel like it. Solutions of the form “this is theorem 123 from [XXX]” are accepted, provided that they indeed are coherent (e.g., not based on a different set of definitions).
    4. You are welcome to talk to each other during preparation, but please spare me from checking identical submissions. The art of writing math texts is in itself important and you can learn it only by trial and error/corrections.
    5. The deadline for submission is August 3, 2018 as set by the Feinberg Graduate school. I can cover you for some extra time in case you have unexpected circumstances, but please don’t try to stretch the deadline beyond reasonable limits.
    6. If you find an error, invalidating one of the exam problems (errare humanum est), please post a comment to this post, and I will try to correct the formulation (or cancel the problem outright if it is beyond repair, – things happened…)
    7. All questions concerning the exam/notes can be posted here in the comments or (if you prefer) in an email to  (that’s me, in case you didn’t notice). I am essentially around here until the deadline, so if you prefer a personal communication, you are welcome to lay an ambush when I am in my office (Zyskind 150, usually every day between 15:30 and 18:00). But in any case, I’d prefer to be notified about request for a meeting in advance.
    8. The choice of the language for submission is yours. My preference is the pidgin English typeset in LaTeX, but this is by no means mandatory. On the other hand, if you plan to write in Mandarin or Basque, I’ll have to disappoint you. Please double check if your the language of your choice is covered.
    9. Problems marked with asterisks are considerably more difficult then the rest. If you love challenges, this is for you to try, but don’t get frustrated if you fail. If not, just register them as subtle math results which are so close to a general math course.

Good luck, and … and good luck!



Monday, January 29, 2018


Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 5:05


The exam is posted online on Jan 30, 2017, and must be submitted on March 16, 2018 (the first day of the new semester).

Its goals are, besides testing your newly acquired skills in the Analysis, to teach you a few extra things and check your ability for logical reasoning, not your proficiency in performing long computations.
If you find yourself mired in heavy computations, double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers. They really may change the meaning of what you write!

Problems are subdivided into items. The order of these items is by far non-random, you have to solve them from the first through the last, (solution of one item may be a building block for the next one). A complete solution of a problem is a proof of some important theorem in Analysis, so you will be discovering these results on your own. The Remarks will help you to place the freshly proved theorem on the general picture.

To get the maximal grade 100, it is not necessary to solve all problems. Problems are of varying length, variable complexity, various level of abstraction. No apriori points are assigned for solution of each problem, no summation at the end. You can get extra points for short and elegant argument or have some points removed for writing an obviously stupid things (honest errors will simply bring you zero points). You have all the time, try to solve as many problems as you can, we will appreciate and assess the results as objectively and honestly as possible.

You are assumed to work individually, which is, of course, impossible to verify, but please in any case avoid submitting isomorphic solutions: this is a bad taste for take-home exams.

You are absolutely free to write in English (easier for me) or Hebrew, submit handwritten pages or compuscripts, in a hard copy or by email (even scans will work). I we will encounter difficulties reading your submission, we’ll let you know.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (errare humanum est), the problem will be either edited (in case of minor omissions) or cancelled (on my account). In case of any doubt don’t hesitate to leave your questions as talkbacks to this post, so that other people will be able to follow. Asking questions is never penalized!

Good luck to everybody!

Sunday, February 19, 2017

Lecture notes

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:57
Tags: , ,

Lecture notes for the course

The set of notes, including extra bibliography and the exam problems, is available here.

These are very raw, extremely informally written and mostly very sketchy notes, consume with moderation at your own risk. Perhaps, one day they will be turned into something more reliable and close to the standards.

Saturday, February 11, 2017


Filed under: Calculus on manifolds course,lecture,problems & exercises — Sergei Yakovenko @ 5:22

Problems for the take-home exam

Here is the file with the problems for the exam.

The rules are simple:

  1. The submission is due on the last day of the exam period (including the vacations) as per the FGS rules, that is, March 26, 2017.
  2. English is preferred over Hebrew, typeset solutions to the handwritten ones, although no punishment for the deviant behavior will ensue. Hardcopy solutions should be put into my mail box in the Zyskind building,  otherwise feel free.
  3. Nobody is perfect: if you believe you find an error and the problem as it is stated is wrong, don’t hesitate to write a talkback to this post. All bona fide errors will be corrected or the problem cancelled outright.
  4. I tried to make the exam as instructive as the course was. Most of the problems are things that I planned to include, but didn’t have time to. To simplify your life, they were split into what I believe are simple steps. Don’t hesitate to consult textbooks, but let me see that you indeed read and digested them. The presumptively harder items are marked by the asterisk.
  5. To get the perfect grade 100, you don’t have to submit solutions to all problems. The grade will be based on my purely subjective assessment of your exam and in any case will not be additive neither multiplicative.  Please be aware that writing patently stupid things may be more detrimental to the outcome than just skipping an item that you cannot cope with.
  6. I hope to post on this site the aggregated and slightly polished lecture notes in hope they might help you.
  7. I hope to be able to answer any questions you might have concerning the problems, better posted here than emailed to me. Moreover, I encourage open discussions here as long as they don’t result in posting complete solutions. Sometimes one stumbles over the most stupid things and needs to talk to other to overcome that. That’s fairly normal. To enter math formulas, you type the dollar sign $ immediately followed by the word “latex”, and after the blank space type in your formula. Don’t forget to close with another $.
  8. If you cannot meet the deadline for serious reasons, write me. Everything is negotiable.

Good luck and merry ט”ו בשבט!

Monday, February 1, 2016

Finally, exam!

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 3:41


The exam is posted online on Feb 1, 2016, and must be submitted on the last day of the exams’ period, February 26. Its goals are, besides testing your acquired skills in the Analysis, to teach you a few extra things and see your ability for logical reasoning, not your proficiency in performing long computations. If you find yourself involved in heavy computations, better double check whether you understand the formulation of the problem correctly. Remember, small details sometimes matter!

Please provide argumentation, better in the form of logical formulas, not forgetting explicit or implicit quantifiers \forall and \exists. They really may change the meaning of what you write!

Problems are often subdivided into items. The order of these items is not accidental, try to solve them from the first till the last, and not in a random order (solution of one item may be a building block for the next one).

To get the maximal grade, it is not necessary to solve all problems, but it is imperative not to write stupid things. Please don’t try to shoot in the air.

The English version is the authoritative source, but if somebody translates it into Hebrew (for the sake of the rest of you) and send me the translation, I will post it for your convenience, but responsibility will be largely with the translator.

If you believe you found an error or crucial omission in the formulation of a problem, please write me. If this will be indeed the case (errare humanum est), the problem will be either edited (in case of minor omissions) or cancelled (on my account).

That’s all, folks!© Good luck to everybody!

Yes, and feel free to leave your questions/talkbacks here, whether addressed to Michal/Boaz/me or to yourself, if you feel you want to ask a relevant question.


Correction 1

The formulation of Problem 1 was indeed incorrect. The set A' was intended to be the set of accumulation points for a set A\subseteq [0,1]. The formal definition is as follows.

Definition. A point p\in [0,1] belongs to to the set of limit points A' if and only if \forall\varepsilon>0 the intersection (p-\varepsilon,p+\varepsilon)\cap A is infinite. The point p itself may be or may be not in A.

Isolated points of A are never in A', but A' may contain points p\notin A.

Apologies for the hasty formulation.

Correction 2: Problem 3(b) cancelled!

The statement requested to prove in Problem 3(b) is wrong, and I am impressed how fast did you discover that. Actually, the problem was taken from the textbook by Zorich, vol. 1, where it appears on p. 169, sec. 4.2.3, as Problem 4.

The assertion about existence of the common fixed point of two commuting continuous functions f,g\colon [0,1]\to[0,1] becomes true if we require these functions to be continuously differentiable on [0,1] (in particular, for polynomials), but the proof of this fact is too difficult to be suggested as a problem for the exam.

Thus Problem 3(b) is cancelled.

Sunday, February 5, 2012

Take-home exam

The last effort

Here you can find the problems for the take-home exam. The rules of the game are outlined in the preamble, I copy them here for your convenience.

The following problems are suggested for the home exam, to be submitted no later than by March 8, 2012. Almost each problem consists of several subproblems, arranged in a specific order. This order is not accidental and should be considered as an implicit hint: solutions of subsequent problems are based on the preceding ones. Please take care to avoid the words “obvious”, “clearly” etc., use as few “plain” words as possible and instead write the intermediate assertions in a closed and precise form using the quantifiers and standard set theoretic notations.

The problems have different complexity: some are easier, some require additional ideas, but none of them is “computational”: if your solutions involves too many identical transformations and/or other computations, have a second look, whether you indeed answer the question that was asked, or something different.

To get the full score 100, it is not necessary to solve all problems and answer all questions: the grade will be awarded based on your demonstrated understanding of mathematics and not on your familiarity with some theorems.

Don’t forget to consult the lecture notes: sometimes you may find useful hints or examples there.

For your convenience Dima will soon post the Hebrew translation of these problems.

Don’t hesitate to ask questions in the comment field: we’ll try to answer them to the extent permissible for an independent home assignment 😉

Good luck!

UPD (Feb 06, 2012, 8:30 am) A small correction of Problem 9 made (sign corrected + more accurate wording).

Monday, February 8, 2010

בחינת בית

בחינת הבית של הקורס מצורפת. אינכם חייבים לפתור את כל השאלות בכדי להצליח בבחינה, אך נסו לפתור שאלות רבות ככל האפשר (גם רעיונות שלא הצלחתם לעבד לכדי הוכחה מדוייקת מומלץ לרשום). שאלות לגבי הבחינה אפשר לשאול כאן או במייל שלי או של סרגיי. בהצלחה לכולם!

Tuesday, January 22, 2008

Exam for the Winter Semester 2007/8

Filed under: Analytic ODE course,problems & exercises — Sergei Yakovenko @ 4:59

Contrary to the pessimistic expectations voiced in the previous post, the strike is over and everybody can come out of the closet.

The course “Geometric and Analytic Theory of Differential Equations” is declared a guided reading course based on this weblog. The winter semester for this course is over: the classes will resume some time on the last week of February, 2008.

 Those interested in grades or in controlling how well they digested  the material, are welcome to pass the exam. The rules of the game are simple: the exam is take-home, the deadline for submission is February 28.

Problems for Semester I are available online.  Any questions (if they appear)  can be left in the comments to this post.

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