(November 2008)
*** WELCOME TO MICHAEL CWIKEL’S “PRIVATE” HEDVA 2M (104011) WEB PAGE ***
We wish you a very pleasant, interesting and successful experience with this course.
The official Hedva 2m site is on “Moodle”. Its address is:
http://moodle.technion.ac.il/course/view.php?id=1056
Here is a way to get to the OLD Hedva 2m site.
(THE MOST RECENT ANNOUNCEMENTS WILL BE INSERTED HERE. PLEASE GO FURTHER DOWN THIS PAGE TO SEE EARLIER ANNOUNCEMENTS.)
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4.7.11 I think that the old Hedva 2M exams of February 2009 and March 2009 must already be somewhere on a Moodle site. But anyway here they are with their solutions.
The exam (Moed Alef) of February 2009 (“Compressed” version)
A solution of that exam.
The exam (Moed Bet) of March 2009 (“Compressed” version)
A “temporary” solution of that exam. This solution may need some revisions in some places but I do not have time to work on it now.
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29.10.09 Here are some links to some animations showing graphs and level curves of some functions of two variables, and level surfaces of at least one function of three variables.
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17.12.08 A more detailed explanation of the last things that I discussed in my lecture today is here. Later it will be moved to Moodle. (Five pages, in English.)
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13.12.2008. Three things today:
(1) Here is a 4 page document about about Schwarz’ theorem (equality of mixed derivatives f”_{xy}=f”_{yx} ) and approximating derivatives by differences (e.g. when working on a computer). This semester we do NOT expect you to remember the proof of Schwarz’ theorem which is on pages 3 and 4 of this document. But if you read and think about that proof this can help you understand the topic better, and so perhaps do other problems more easily.
(2) Here is a document with six HIGHLY RECOMMENDED exercises, which every student who wants to understand this course should try to solve by her/himself. At this stage you have not learned the material for all of these exercises. You should be able to do the first one now, or as soon as you learn about partial derivatives of second (and higher) orders. You should also now be already able to do the last three exercises on page 2 of the document. (They are all taken from “Hoveret [C3]”.) Please do not forget to do the other two exercises on page 1 of this document (taken from “Hoveret [C2]”, connected with vector analysis) later, before the end of the semester.
(3) In 2000 I wrote a document in Hebrew (2 pages) which gives a slightly different approach to part of the material about the chain rule in the document which I posted here on 1.12.05. You can see that document here.
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1.12.08 Here is a revised version of an old document of mine which gives some warnings about mistakes made with limits, and also gives a kind of “flow diagram” (tarshim zrimot) for possible strategies for deciding whether limits exist or not.
It’s in English, (like one or two other documents that you may have to read during your career.) Some of the things in it were mentioned at least briefly in my lecture yesterday.
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19.11.08 The pdf file of today’s lecture (lecture number 4) is now on moodle and here.
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18.11.08 The pdf file of lecture number 3 which I gave yesterday (17.11.08) can now be downloaded from moodle or from here. The additional material to which you are referred on slide number 7 (about the nearest points on two skew lines) is available here. There is at least one small misprint on slide number 13. “O=[0,0,0]” should be changed to “O=(0,0,0)”.
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12.11.08 Here is today’s lecture. I am sorry that the picture quality is not so good. The special additional material mentioned on slide number 13 (about vector products) is available as a pdf file from here.
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10.11.08 A presentation of the lecture that I gave today, with a few more brief extra comments. is now on “Moodle” here. You can also see (also available from “Moodle”) a “movie” about addition of vectors.
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THE MESSAGES BELOW HERE ARE VERY OLD. BUT SOME OF THEM MIGHT STILL BE USEFUL SOMETIMES.
22.9.07 Shana Tova, Gmar Khatima Tova.
In a few day’s time (BUT NOT YET!) you will be able to look at the scan of your exam from 18/9/07. Perhaps you may sometimes see one or more of the symbols “Y1”, “Y2”,…., or “Z11” written in red or light blue in squares by the person who checked your exam. To see what these numbers/letters mean, please read this. (This is a three page document.)
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17.7.07 In a few day’s time (BUT NOT YET!) you will be able to look at the scan of your bakhan. Perhaps you may sometimes see one or more of the symbols “A1”, “A2”,…., or “A14” written in red in squares by the person who checked your bakhan. To see what these numbers/letters mean, please read this.
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8.7.07. Here are two examples, one quite easy and the other quite hard, which show that in some exotic situations you cannot change the order of integration in repeated integrals. These situations are very unlikely to arise in Hedva 2m.
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15.5.07. After the strike etc. etc. when/if we get towards the end of the semester we will learn the Gauss divergence theorem. Then, you might like to look at the link http://ieeexplore.ieee.org/iel1/13/1980/00054865.pdf?arnumber=54865 which connects this theorem with Archimedes’ principle about the force acting on a body immersed in a liquid. (Thanks to Prof. G. Wolansky for telling me about this.)
ALL ANNOUNCEMENTS BELOW THIS LINE RELATE TO THE WINTER SEMESTER November 2006-January 2007, OR EARLIER
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25.3.07 Here is one version of the Moed Bet exam of 20.3.07. (pdf file). We hope you did well.
The pdf file of the solution (of all versions of the exam) is here. In a few day’s time (BUT NOT YET!) you will be able to look at the scan of your moed bet exam. You may sometimes see some numbers or letters in squares written in red by the person who checked it. To see what these numbers/letters mean please read this.
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18.02.07 Here is one version of today’s examination (pdf file). We hope you did well.
The pdf file of the solution (of all versions of the exam) is here. (This is yet another newer version (version 4) with one small correction to version 3, posted on 17/9/07.) When you look at the scan of your moed alef exam, you may sometimes see some numbers or letters in squares written in red by the person who checked it. To see what these numbers/letters mean please read this.
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14.2.07 Here are some clarifications (updated on 15.2.07) about directional derivatives and information about a mistake in the solution to the exam of July 2000 and in a MathNet exercise.
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22.1.07 The results of the mid term test will be announced today. You will also be able to use the usual procedure to view a scan of your own test. You may sometimes see some numbers or letters in squares written in red by the person who checked your test. To see what these numbers/letters mean please read this.
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15.1.07 Here are some notes (5 pages, English) explaining line integrals (integralim kaviyim) and length of curves.
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11.1.07 Here is one of the versions of yesterday’s test. And here is a solution for all versions (updated (version 3) on 15/7/07 and then (version 4 with just one very small correction) on 17/9/07). We hope you did well.
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2.1.07. Here is an exercise about calculating a particular double integral. It turns out that the result of this calculation provides some interesting information about a rather general formula.
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5.12.06. As promised some days ago, I have now written my notes about continuity and connected sets. They appear as new sections added to the end of the second of the two documents originally posted here on 29.11.06
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29.11.06
1. Here are some comments (in English) about limits of functions of two (or more) variables, including a “flow diagram” suggesting strategies for calculating such limits and some warnings about the dangers of using polar coordinates and repeated limits. Some of these things were also mentioned in Monday’s lecture.
2. Here are some notes (two new versions, in Hebrew and in English updated on 5.12.06 ) which recall the definition of the limit of a function with respect to some specific subset, and now also discuss continuity with respect to a subset.
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21.11.06 Here are some comments about my lecture yesterday.
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19.11.06 Here is another slightly different look at the some of the things (“balls”, “distances” etc. in R^n for general n ) that I discussed in last Wednesday’s lecture. (It is in English, but how will you manage in your future careers if you can’t read technical English?)
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16.11.06 Please read this file. (It was updated on 20/11/06). Why do I ask you to read it? In yesterday’s lecture many of you asked me all sorts of interesting questions. I was happy to answer them, but the result was that I did not get as far as I needed to with the topics of the lecture. This means that I will probably not have time to talk in the next lecture about the topic of connected sets (kvutsot k’shirot). So you should learn it from the one page document here.
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15.11.06 Here is an exercise about finding the distance between a pair of skew lines, or rather, finding the closest points on those two lines. I think that the approach here is better than the approach I used in my lecture last Monday.
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9.11.06 Three things today:
1. Here is a quickly written summary of my first lecture last Monday. It contains three exercises which I did not mention then which could be useful for you. It also has a slightly revised treatment of the topic “projection of a vector on a vector” with more usual terminology. (In Monday’s lecture I defined the projection as a scalar, but it is better to define it as a vector.)
2. Here are five more exercises, about vector products. Some, but not all of them are more difficult than average.
3. Here is how to get and see a presentation which I prepared, dealing with vectors, in particular sums of vectors.
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The material below this line is from the 2005/6 winter semester and before.
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26.3.06 Here is one of the versions of today’s Moed Bet exam. And here is a solution for all versions. We hope you did well.
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24.3.06 The time and rooms for Moed Bet on Sunday are clearly announced on the Technion Undergraduate site. But a few students seem to have difficulty finding this site or this information. In any case, the exam will start at 16:30, and, like Moed Alef, it will consist of two parts, each of 90 minutes, with a 30 minute break between them. B’Hatzlakha to you all!
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14.3.06. There is now yet another update of the solution (Version 1.22, dated 14/3/06). It is very similar to previous versions. It includes some new remarks about the motivation behind Question 7. (In earlier revisions, Version 1.2 corrected a missing minus sign in the solution of Question 2 BET (thanks for telling me) and added some additional comments about Question 7. Version 1.21 added remarks to the solution of Question 7 and reworded the introduction to the solution of Question 2.)
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21.02.06 I have updated yesterday’s solution. The new version (Version 1.1) is very similar to the previous version. Some things in Question 6 are explained a bit more explicitly.
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20.02.06 Here is one version of today’s examination. Here is a solution (Version 2, updated again on 28.3.09) of all versions, and here is a picture to help explain the solution to question 6. We hope you did well.
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18.01.06 I have updated the notes which I posted here yesterday. If you already printed them it is probably not worth printing the new version. Details about the changes are here.
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17.1.06 Here are some notes (5 pages, English, UPDATED ON 18.1.06) explaining line integrals (integralim kaviyim) and length of curves.
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3.1.06 There were several versions of today’s mid term test. Here is one of them, and here is the solution for all versions.
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29.12.05 Some students get confused between the related but DIFFERENT problems of finding a GLOBAL maximum/minimum of a function on some set and finding the LOCAL maxima and minima of a function. There are some explanations about that here.
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24.12.05 You should all have received an email message, which I sent yesterday, with some explanations about the mid term test. Since the topics for the test are the first six sections of the syllabus, I am now providing some explanations (in Hebrew) about the meaning of the wording of the syllabus for some of those topics. For details please click here.
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13.12.05 As part of the preparations for our mid term test, I am planning to be available to answer questions, i.e. for extra tirgulim/shaot kabala for the students of my lecture group on the following three days and times.
Wednesday 14 December, 13:30-14:30.
Wednesday 21 December, 13:30-14:30.
Wednesday 28 December, 12:30-14:30.
All these meetings will be held in room 310, Ulmann.
If any of you who are in the other lecture groups also wish to ask questions, then I also invite you to come to these meetings.
But if it turns out that there are many people with many questions to ask, I will have to give priority to students from my group.
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12.12.2005. Here is an old 3.25 page document about about Schwarz’ theorem (equality of mixed derivatives f”_{xy}=f”_{yx} ) and approximating derivatives by differences (e.g. when working on a computer). This semester we Ado NOT expect you to remember the proof of Schwarz’ theorem which is on pages 3 and 4 of this document. But if you read and think about that proof this can help you understand the topic and so perhaps do other problems more easily.
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5.12.05 Three things today:
(1) Here is a document with six HIGHLY RECOMMENDED exercises, which every student who wants to understand this course should try to solve by her/himself. At this stage you have not learned the material for all of these exercises. You should be able to do the first one now, or as soon as you learn about partial derivatives of second (and higher) orders. You should also now be already able to do the last three exercises on page 1 of the document. (They are all taken from “Hoveret [C3]”.) (The formulation of the last one of these exercises continues onto page 2.) Please do not forget to do the other two exercises in this document (taken from “Hoveret [C2]”, connected with vector analysis) later, before the end of the semester.
(2) In 2000 I wrote a document in Hebrew (2 pages) which gives a slightly different approach to part of the material about the chain rule in the document which I posted here on 1.12.05. You can see that document here.
(In the course taught in 2000 I defined the limit of a function with respect to a set E at points in the closure of E. This semester I did not use that limit explicitly, though I defined and used something very similar to it.)
(3) In the document posted here on 1.12.05, I wrote, “You are allowed, if you wish, to use the results of Theorems 4 and/or 10 and/or 11 as tools in such proofs.” I should point out that, in the lectures, we have also already used the result of Theorem 4 (of that document) as a tool in the proof of another result whose proof you are expected to understand and remember, namely:
Theorem: Let E be an arcwise connected open subset of R^n, and let f:E–>R be a continuous function on E. Let p and q be any two points in E, and let c be any number lying between the two number a=f(p) and b=f(q). Then there exists a point w in E with the property that f(w)=c.
I also wrote that we do not expect you to remember the proofs of Theorems 4,10 and 11 in the document of 1.12.05. To avoid any misunderstandings, let me emphasize that, in general, unless we explicitly say otherwise, you ARE expected to remember and understand all proofs which we give in this course.
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1.12.05 I have just posted an auxiliary document here (6 pages, in English). It deals with limits, continuity, differentiability, the chain rule, and Heine’s theorem.
It recalls some of the definitions which I gave in recent lectures, and provides some proofs of some theorems. In some cases I already gave a proof in the lectures, but here I try to give an alternative, hopefully better and sometimes more detailed proof.
Some of these proofs are quite similar to results which you already studied in Hedva 1m. So, for the mid term test and final exam, we will not expect you to remember all the details of the proofs of three of those theorems. (I specify which ones in the document itself.) BUT WE DO EXPECT YOU TO REMEMBER PROOFS OF THE OTHER THEOREMS IN THIS DOCUMENT (either the proofs here, or other correct proofs).
You are of course expected to remember and understand and be able to use all the definitions given in this document.
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23.11.05 Our mid term test will be on January 3, 2006. B’hatzlakha lkulkhem!
(This date has not yet been announced on the Technion undergraduate website, but I have been assured that it is definitely the correct date.)
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21.11.05 To see a “flow diagram” (IN ENGLISH) for the process of calculating limits of functions of several variables click here . (This is now a new corrected version, posted on 24/11/05.)
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1.11.05 Here are the books listed for study in this course.
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30.10.05 Please click here for VERY IMPORTANT information including (1) the format of the tests/exams, (2) the calculation of your tsiyun sofi, (3) homework exercises, (4) miluim during the semester or test or exam, (5) students with learning disabilities, (6) lectures on video, (7) accessing older lecture notes for this course.
To go to the OFFICIAL website for this course click here. That site also contains ESSENTIAL information, syllabus, exercises, previous examinations, etc. PLEASE NOTE: Some of the files (e.g. lists of teachers and their office hours) on the official website still have to be updated from previous semester, but the SYLLABUS from previous semesters is the same one as we will use this semester.
You can click here for my old Hedva 2m webpages from 2001 and earlier.
(Some of the information and notes in those old pages can still be quite useful.)