# Sergei Yakovenko's blog: on Math and Teaching

## Sunday, February 19, 2017

### Lecture notes

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:57
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## 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.

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## Saturday, February 11, 2017

### Exam

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

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.

# Corrections

## 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.

## 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
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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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