# Gershon Wolansky A critical parabolic estimate and application to nonlocal equations arising in chemotaxis

## (OLD SITE FROM 2002)

Welcome to my “private” web page for the course on Wavelets.

108914 “Nosim bmatematika shimushit”

Here are a few words about what wavelets are, and some of the things that they are used for.

You can reach me at:
Michael Cwikel (mcwikel@math.technion.ac.il)
Sundays 14:30-15:30 (NOTE the change. I may perhaps have to change again.)

(Amado, room 730). Contact me by telephone (8294179) or email to make an appointment for other times.
If you formally registered for this course you should be on an electronic mailing list to which I already sent a message.
If you are not registered but still want to join (or leave) the mailing list for this course, please send me an email message
with the subject “wavelets-list”.

Some further preliminary information about the course is here.
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29.4.02 Please contact me to receive yet another new version of the lecture notes/exercises. It deals with almost all topics which have been covered in lectures so far. Of course for some details you have to consult the book of Hernandez and Weiss. For those of you who need to get a grade for this course, I will ask you to submit the exercises which appear in this version and which you did not already solve in the version of 7.4.02. (This version is also available from me.) To get full credit you should submit these exercises within 2 weeks from next Sunday, i.e. on or before May 19.
Those of you who did not yet submit the first set of exercises, please, for full credit, do this on or before next Sunday (May 5). It is a good idea to keep a copy of your solutions to the questions. At some stage, before finally giving grades, I will ask each of you to explain some part of the solutions that you submitted. I will not expect you to memorize details. You will be able to take your time and freely refer to your solution and to the text book and my notes etc. when you explain what you have done.
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7.4.02 Please contact me to receive an updated THIRD, but still preliminary version of some exercises and explanations about some of the material in this course. It is the exercises in this version which you are asked to submit on 21.4.02.

# Michael Cwickel FUNKTSIYOT MAMASHIOT

REAL FUNCTIONS = “FUNKTSIYOT MAMASHIOT”.
Welcome!
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To go to the OFFICIAL website for this course click here. That site also contains various information, homework exercises, material from previous semesters, etc.
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6.6.2004 The postscript file (4 pages) which is here has been written for the students in Professor Baruch Solel’s class whom I taught on 30.05.04 and 3.6.04. It gives basic facts about dyadic intervals in R^d (finite dimensional euclidean space), and, with their help, completes the proof which I began presenting in the classes on those two days, that the space of C infinity functions on R^d with compact support is dense in L^1(R^d).
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The most recent version (10.11.02) of my list of exercises is here. It will be updated from time to time with new exercises added at the BEGINNING of the document.A
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The most recent version of my auxiliary notes (26.10.02) is here. It will be updated from time to time with new material usually added at the END of the document.
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18.11.02 Important information about the mid term test and final examination, and how your grade will be determined, etc. etc. is here.
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7.11.02 I have prepared an electronically accessible version of a collection of old examinations to provide you with extra exercises. There are 42 pages in this collection. You should hopefully be able to access all or most of them here. If you have problems accessing any of these pages, you can also try going to here. If you know more html than I do and can suggest a better way of presenting this material, please contact me.
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28.10.02 Two things:
(i) Iddo Ben Ari’s office hours are Mon. & Thu, 13:30-14:30, in Amado, room 925. He can be reached by email at iddo@techunix.technion.ac.il
My own office hour is on Wednesdays from 10:30 to 11:30. My office is room 730 Amado building. (Telephone 829-4179). My email is mcwikel@math.technion.ac.il
(ii) The only change to the list of exercises since 26.10.02 is a very small change to the wording of one of them (on page 2 about the intervals I_k and functions \phi, f, g and h.) Hopefully it will now be clearer.
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27.10.02 One more small exercise has been added to the list of exercises. My auxiliary notes have also been slightly updated. From now on, the latest version of both of these documents will now also be accessible from the TOP of this page.
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24.10.02 Here is an updated version of the list of exercises with several new ones. Some of these may be used in the tirgul which will be given by Iddo on Sunday. Others may later become part of the “official” set of homework exercises. In any case they can all teach you something.
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13.10.02 Here is a message sent to the students’ mailing list today.
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# Michael Cwickel INFI 2

## *** WELCOME TO INFI 2 ***

We wish you a very pleasant, interesting and successful semester.

.25.09.03 The solution (available below) for the examination of 15.09.03 has been slightly modified. (Some extra comments and information have been added.)
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16.09.03 Examination of 15.0.9.03
A solution for this examination.
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19.8.03 The solution of the recent examination has now been updated a third time. Version 1.3 contains a few more new comments. There are no radical changes.
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5.8.03 Solution of the examination of August 3. (Version 1.3. There may perhaps be yet another updated version of this solution later, with some additional comments.)
A picture of the set E in question 6 can be obtained here. But it is very approximate. For example, it is wrong in at least one detail: Since I have taken a to be bigger than b, I should have c closer to 0 than to pi/2.
A limited number of hard copies of the examination itself are now available from the noticeboard outside my office. You can also get a copy here .
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31.7.03 Solution of moed alef. (29 June 2002). YOU WILL LEARN MORE IF YOU FIRST TRY TO SOLVE THE QUESTIONS WITHOUT LOOKING AT THIS SOLUTION. (The examination form itself is available below and also here.)
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9.7.03 Here is a real bargain! You can get your free personal copy (2 pages, one white and one blue) of the examination of 29/7/03 from the noticeboard next to my office. (Room 730). Hurry! This unique offer is for a limited time only. There are now only 2 or 3 copies left.
But maybe you are too tired to go up to the seventh floor of the Amado Building in this hot weather. If so, please use whatever strength may be left in your fingers to click here.
I have prepared a solution for this examination. But I have decided not to publish it yet. Why? Because you will learn much more if you first make a serious effort to solve the questions by yourselves. Later, after you have stimulated your own creativity and originality, I will show you my solution and you can check if you agree with it.
Some day in the future the solution will hopefully appear here.
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19.6.03 I have updated my lecture notes again. The new version is 3.23, dated 19/6/03. It is here (and also below). If you already have version 3.21 of 12/5/03, then you only need to add pages 43-53.
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28.5.03 Solution of today’s bakhan.
One version of the bakhan itself.
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12.5.03 I have updated my lecture notes again. The new version is 3.21, dated 12/5/03. It is here (and also below). If you already have version 3.20 of 10/4/03, then you only need to add pages 39-43.
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25.3.03 The lecture notes about the Riemann Integral have been modified again. The new version (Version 3.16) is here (and also below). If you already have a printed copy of an earlier version, e.g. Version 3.1 (or 3.15 respectively) then you do not have to print all the new version. You simply need to print the last pages, from 25 (or 27 respectively) to the end. On those pages a few small corrections in the earlier pages are mentioned. You can easily make these corrections by hand.
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12.3.03 The lecture notes about the Riemann Integral have been slightly modified. The new version (Version 3.1) is here (and also below). If you already have a printed copy of the earlier version (Version 3.0) then you do not have to print all the new version. You can simply insert the material in the two pages which are available here.
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THE RIEMANN INTEGRAL Lecture notes (ps file, 39 pages, English, Revised: Version 3.20, April 10, 2003.).

To go to the official website for this course click here.
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My old page (1997) for Infi 2 is here.
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# Michael Cwickel SOME IMPORTANT DETAILS ABOUT ALL MID TERM TESTS & EXAMINATION WHICH I GIVE.

SOME IMPORTANT DETAILS ABOUT ALL MID TERM TESTS & EXAMINATION WHICH I GIVE.

This document may perhaps seem unfriendly. But its purpose is, davka, to
be very friendly about things that really matter: to ensure that every
student will get the recognition that she or he really deserves for his
or her knowledge and achievements, and that your future employers will
be able to be confident that a student with a good grade from the
Technion really is a good student.

The time given for my tests/examinations may be different for different
subjects. Usually it either 90 minutes or three hours. If it is more
than 90 minutes then it will be broken into separate parts, each lasting
no longer than 90 minutes.

During each 90 minute session NO STUDENT IS ALLOWED TO LEAVE THE
EXAMINATION ROOM.

(If you have some special problem, e.g. a medical problem, which means
that you may perhaps be unable to stay in the room for 90 minutes, then
you should tell me about it, and provide appropriate documentation, at
least SEVEN DAYS before the date of the bachan or exam, so we can
attempt to find some special solution for you.)

Unless explicitly stated otherwise in advance, all my tests/examinations
are WITHOUT "khomer ezer". This means also WITHOUT calculators and
cellphones, which should be packed away inaccessibly in your bag and
should definitely NOT be on your person. And please SWITCH OFF your
cellphone. There have been cases where some student forgot to switch off
and his phone rang or beeped again and again because its battery was low.
This disturbed everyone, and all the other students were VERY ANGRY with
that student!

(Occasionally in SOME examinations or midterm tests for some subjects I
provide a standard list of formulae on a separate page (which you will
be asked to return). In those cases where this is done, this will be
announced well in advance, and the list of formulae will be published

BHATZLAKHA!!!

Michael Cwikel

p.s. Of course I did not personally invent these things. As you
probably realize, quite a number of other Technion teachers follow
similar procedures.

# Michael Kwickel MEASURE THEORY

## *** Welcome to MEASURE THEORY ***

I wish you a very pleasant, interesting and successful semester.

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30.06.04 A preliminary version of the sixth set of homework exercises is here. A SMALL CORRECTION WAS MADE TO THIS SET ON 28/7/04. (As you might guess, this will be the final set for this semester.) You can also get some notes about about the Marcinkiewicz interpolation theorem here.
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20.05.04 A preliminary version of the fifth set of homework exercises is here. SOME SMALL CORRECTIONS WERE MADE TO THIS SET ON 28/7/04. SOME REMARKS WERE ALSO ADDED.
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21.04.04 A preliminary version of the fourth set of homework exercises is here. NOTE THAT IN QUESTION 3 YOU CAN SUPPOSE THAT THE MEASURE IS REGULAR.
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04.04.04 KHAG SAMEAKH! A preliminary version of the third set of homework exercises is here. IT IS SUFFICIENT IF YOU FIND A SOLUTION TO QUESTION 1 IN THE SPECIAL CASE WHERE X IS SIGMA COMPACT.
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18.3.04 A preliminary version of the second set of homework exercises is here.
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15.3.04 A preliminary version of the first set of homework exercises is here. There is also a picture which is needed for one of the exercises. You can get it here
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If most of us can agree on a suitable date for it, we will also have a mid term test. It will be “magen” (30%) and will consist of the same two categories of questions (90% and 10%) as in the final exam. (It will of course be given under the same conditions already mentioned above and in this .)
Unfortunately I cannot see any way to provide a detailed check of everyone’s solutions of the homework exercises before the mid term test or final examination. But of course I am happy to answer your questions about these exercises and, if I see that you have been thinking about them, give you some hints here and there. I will do this individually or, if you can agree on a suitable time, in the format of extra “tirgulim”.
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To go to the official website for this course click here. (But I will probably not use that site very much.)
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# Michael Kwickel INFI 3

### *** WELCOME TO INFI 3 ***

October 2004
We wish you a very pleasant, interesting and successful semester.

.To go to the official website for this course click here.
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20.4.05 Here is a picture from my lecture of 27/10/04 which, by mistake, was not put on the CD that is available in the Mathematics library. As its name suggests, it should come between the pictures I3030127.JPG and I3030129.JPG in the directory of pictures for October 2004.
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19.3.05 Here is the questionnaire for the examination of 6/3/05. I have not prepared a formal solution of it, but will be happy to answer your questions about the questions.
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1.3.05 Here is a solution for the examination of 17/2/05.

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22.2.05 Here is a compact version of the examination of 17/2/05. “Compact” simply means that the same questions have been squeezed into a smaller number of pages by removing or compressing things which are not necessary when you are not actually doing the examination. (It is a pity to waste paper and trees.)
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1/2/2005 Oops, there was yet another small correction to the fourth set of homework exercises. This has now been fixed (and also explained in an email message.)
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24.1.2005 The fourth set of “paper” homework exercises, has now been slightly revised. The only changes are in Question 3, They are small changes in the notation. More explanations have been given in an email message.
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19.1.2005 The fourth set of “paper” homework exercises is now here. Please submit your solution on or before 6/2/05 if you want feedback. You should receive your corrected solutions by some time on 10/2/05. We chose a relatively late date to give you some time and flexibility but PLEASE begin preparing for the examination well before 6/2/05.
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17.1.2005 I have updated the CD with pictures of the blackboards (and whiteboards) from the lectures up to and including yesterday’s lecture. I have also included a copy of my old handwritten notes from a different course, which may provide some additional more geometric and more intuitive points of view for some of the topics of this course, e.g. Lagrange multipliers. See here for more details.
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3.01.05 Here is a new version of brief notes about classification of critical points using second derivatives.
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26.12.04 The third set of “paper” homework exercises is now here. If you want us to look at your solutions and give some feedback about them, please submit them by 9.1.05. (If more of you had done the first and second set of exercises and gotten feedback on them, perhaps more of you would have been better prepared for the bakhan.)
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9.12.04 You can see a solution of Sunday’s bakhan here. I thank Sedi for very much help in its preparation.
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1.12.04 The bakhan will be on Sunday, 5/12/2004 at 16:30 in Amado 233. Bhatzlakha!!

I have updated the CD in the mathematics library to include pictures from all lectures up to today’s.
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25.11.04 Three things today:
* The topics that you have to know for the bakhan include the open mapping theorem and everything before it, including also of course exercises on all these topics. (For more details about this see the email that I sent today.) BHATZLAKHA!!
* You can look at jpeg files of the blackboards during all my lectures so far on a CD which you can borrow from the mathematics library like a reserve book. (Maybe you can also simply view these files in the library itself using one of the computers there.) Please remember that of course there are also things which I said in lectures but did not write on the board.
* And here are some hints for Question 5 of Homework Exercises #1.
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19.11.04 The second set of “paper” homework exercises is now here. If you want us to look at your solutions and give some feedback about them, please submit them by 30.11.04. To see in advance some of the material which will be discussed next week (differentiability of inverse mappings etc.) click here (3 pages).
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17.11.04 Here are some notes about implicit functions. (This is a newer version, revised again on 12.12.04, of notes which appeared on my earlier sites.)
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8.11.04 The first set of “paper” homework exercises is now here. If you want us to look at your solutions and give some feedback about them, please submit them by 23.11.04
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28.10.04 Two things today: (1) Some basic exercises about topology in R^n and “cubes” etc. in R^n are here. The 5 exercises on the first page, numbered (i) to (v) are not hard, but I strongly recommend doing them. The second page, for anyone who is interested, is a reminder about the Cauchy-Schwartz inequality and how it leads to the triangle inequality for d(x,y) .)

(2) The date which we have been given for our mid term test (Bachan) is Sunday, December 5, 2004. It will be some time in the afternoon. We will be told the exact time later.
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26.10.04 Please click here for VERY IMPORTANT information about (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.

26.10.04 Here is a proof (revised version) of the theorem describing compact subsets of R^n.
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26.10.04 Information about your teachers and how to contact them is here.
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Some of my old webpages for Infi 3 from some years ago can be viewed here (Oct. 2001), and here (March 2001).
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# Michael Kwickel FOURIER SERIES & INTEGRAL TRANSFORMS

## *** WELCOME TO FOURIER SERIES & INTEGRAL TRANSFORMS *** . *** ( F. S. I. T. ) *** .

We wish you a very pleasant, interesting and successful experience with this course.

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#### (THE MOST RECENT ANNOUNCEMENTS WILL BE INSERTED HERE. PLEASE GO FURTHER DOWN THIS PAGE TO SEE EARLIER ANNOUNCEMENTS.) .

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13.10.06 Some explanations about the correction of the exam of 4.10.06.
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8.10.06 Khag Sameakh!
Here is a solution of the examination of 4.10.06 And here is one version of the examination itself.
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20.9.06 Here are some extra details about Question 5 of last week’s examination. Please look at this document, especially if you are thinking of submitting an “irur”.
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13.9.06 Here is a solution of today’s examination. (This has meanwhile been updated (to Version 3) on 6.10.06) And here is one version of the examination itself.
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8.9.06 Here, as requested by Liran, is a solution of Question 5 AND NOW ALSO QUESTION 10, from Homework No. 3. There are some associated pictures here.
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27.8.06. The “printer friendly” version of the file originally posted here on 2.7.06 (see below) has now been updated again. The changes are small. The main change is that the material in that document which is NOT required for the exams this/last semester now appears in blue and in a different font.
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10.8.06 We have decided to increase the weight of the bachan to 35% and the maximum possible weight of the homework magen to 15%. For more details see the email sent to all students today. It is also here.
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3.8.06
Here is a more detailed version of one of the calculations that I did in the video tutorial today.
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25.7.06 Here is a more “reader friendly” version of the document posted on 23.7.06 for those of you who do not need information about the graphics files.
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23.7.06. Here are details about the topics of the course which you are expected to know, and also about how to get the graphics files of pictures of the blackboard for most of the lectures this semester.
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12.7.06. The standard page (two sides) of formulae which you will be given for use during the exam is here. Do NOT bring your copy to the exam. Please be sure to read the email sent to all students today for further information.
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Here is a more precisely formulated version of the Laplace transform homework exercises which we published a few days ago.
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4.7.06 Here are two pictures relevant to the notes for last Monday’s lectures: the semicircle S_R and the curve Gamma_R. There is also a third picture, the graph of a certain function, related to an exercise in those lectures. I suggest that you try first to draw the graph yourself. Then you can see if you and I got the same answer, using the address (URL) given in the lecture notes.
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2.7.06 Here is a “printer friendly” version (Now version 3, updated on 27.8.06) of the file that I will use for tomorrow’s lecture. I suggest that you print a copy and bring it to the lecture.
And here are two homework exercises, an “unofficial” preliminary version of part of a set of homework exercises about the Laplace transform which will be available very soon.
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25.6.06 Here is a pdf file which displays the material which I will present in tomorrow’s lecture about Laplace transforms. (FILE UPDATED ON 26/6/06.)
This file does not include various additional remarks to be made during the lecture vocally and/or on the blackboard, for example, about integrals on [0,infinity) and convolutions. I will decide in consultation with you tomorrow to what extent I will present the lecture using a computer to display these pages, and to what extent I will lecture in the “traditional” way.
If you print these pages and bring them to the lecture you will be able to simply add comments to them instead of having to write all the details again.
Here is another (monochrome) version of this same file which is more suitable for printing. (THIS FILE WAS ALSO UPDATED ON 26/6/06.)
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19.6.06 Thanks Gilad (TODA GILAD!!) for noticing some misprints in Exercise 5 part b, on page 4 from the exercises posted here on 17.6.06. The link from 17.6.06 will now give you the corrected version. If you just want to see the single line on page 4 which has been changed, click here.
For those of you who want to think a bit more deeply about things related to today’s lecture, there is now an extra exercise (no. 11) in that set.
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17.6.06. A final version of those 8 exercises in Hebrew is now available here. I have also added exercises 9 and 10 (and now also 11). Exercises 5 and 10 from this document are part of the official homework assignment 3, about which Yoram has just sent you all a message.
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15.6.06 Here is a collection of 8 rather special exercises (in English). (The Hebrew version of them has meanwhile been posted on 17.6.06.)

CHANGE OF ORDER OF INTEGRATION FOR GENERALISED INTEGRALS ON AN UNBOUNDED INTERVAL.
Here is a formulation of Fubini’s theorem which is useful for changing order of integration in connection with Fourier transforms.
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12.6.06 Here is an exercise (in Hebrew) which will help you check to what extent you understood today’s lecture.
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5.6.06 Here are some explanations about (1) the Lebesgue dominated convergence theorem and (2) the formula of Leibniz for differentiating under the integral sign. We consider both of these results in versions which are suitable for use with the Fourier transform
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16.5.06 One version of the midterm test is here and a solution for all versions is here.
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9.5.06 Three things today:
(a) Here is a summary (Version 4.6, revised on 9.5.06). It deals with TWO topics:
(1) Differentiation and integration of Fourier series, term by term, including a connection with uniform convergence, and
(2) Fourier series on other intervals instead of [-pi,pi].

We are now discussing or will soon be discussing Topic (1). Topic (2) is simpler and may be left for you to study privately.
(b) Topic (1) includes a mention of uniform convergence. (hitkansut b’mida-shava). This is something which you are supposed to remember from Hedva 1M (or Hedva 2M). Although, in general, uniform convergence is very important, it only plays a limited role in this particular course. Here is a quick summary, a “Uniform Convergence Survival Kit” containing the main things you need to know about uniform convergence for this course.
(c) Here are some notes about two sided series. (A new version with small changes was posted here on 17/4/06.)
2.5.06 Since the deadline for submitting Question 5 of the first set of homework exercises has now passed, we are now publishing a solution of this question. (As of 3.5.06 this solution has been slightly modified with a comment added at the end.)
If you have questions before the test you are invited to come and ask them between 17:30 and 19:30 on Sunday 7/5/2006 in Amado 233. Yoram or I will be there during that time.
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15.4.06 Chag Sameakh, vKhofesh Nayim.
The first set of homework exercises is available here. Some small (mostly obvious) clarifications about the formulation of these exercises can be found in a email message sent today to all students.
There is also another set of five exercises about orthonormal systems here. Two of these are the same as two of the homework exercises. Most of these five exercises contain information which will be useful later on in the course.
Even though we have not yet proved that the standard trigonometric orthonormal systems are closed in C[-pi,pi] or E[-pi,pi] you can assume this already in exercises where you may need it, before we give the proof.
Some of my old notes are available here. (English, 4 pages). They give the easy proof of the equivalence of Parseval’s identity to closedness of an orthonormal system, and describe the Gram-Schmidt procedure. (Both of these topics were also discussed fully in the lectures and in the book of Pinkus-Zafrany.) The last two pages are a discussion of some more exotic questions, not compulsory for this course, about the connection between closed orthonormal systems and complete orthonormal systems.
For other perhaps simper versions of exotic counterexamples showing that an orthonormal sequence may sometimes be complete but not closed, see some notes prepared in Hebrew by Dr. Alla Shmukler and also my notes here.
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25.03.06 Some of the topics to be covered soon in this course are the theorem about orthogonal projections, Bessel’s inequality and the Riemann-Lebesgue lemma. Here are some notes about these three topics which I wrote in 2002. The approach in those notes is slightly different from the approach used in the course textbook. The approach which we will use in lectures this semester may also be slightly different.
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23.3.06 Here is a summary of the lecture that I gave on Monday. It also corrects three small misprints in some formulae I wrote on the board, when I was defining piecewise continuous functions. Thanks Yahel for noticing two of these. If you want to see just the corrected versions of those formulae you can click here.
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22.3.06 To go to the OFFICIAL website for this course click here. That site also contains ESSENTIAL information, exercises, text book, previous examinations, etc. The documents there include two pages of general information and instructions and this semester’s syllabus. There is also a link from there, and from here to the page of the Metargel Akhrayi where you can see office hours for your teachers, dates of examinations etc.

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#### To go directly to download or read the text book for this course, written by Allan Pinkus and Samy Zafrany, click here, or here, .

BELOW THIS LINE ARE OLD ANNOUNCEMENTS FROM EARLIER SEMESTERS.
(Some of these can still be quite useful.)6.6.2005. Here is a pdf file which displays the “slides” which I used in today’s lecture about Laplace transforms for the classes of Avi Levy and Gershon Wolansky. (Some misprints, hopefully all misprints have been corrected.) This file does not include various additional remarks made during the lecture vocally and on the blackboard, for example, about integrals on [0,infinity) and convolutions.
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If you want to see why we should not be robots and why I congratulate Yaacov Marko, you can visit my very old Fourier site here and look there at the entries for 17/9/01.
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### ANNOUNCEMENTS FOR THE WINTER SEMESTER 2004/5

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19.3.05 Here is (one version of) the examination of 16.3.05. And here is a solution of it (updated on 24.3.05).
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1.3.05 Here is a probably not final version of some notes which explain an exact way of defining and using Fourier transforms of certain functions which are not absolutely integrable. In fact this definition works for “functions” which are not even really functions, like the Dirac delta “function”. Of course you are NOT expected to know any of this material for examinations in this course, but this material may help you have a deeper understanding of certain topics in later courses such as “Otot uMarakhot”. (A new version (1.9) of these notes was posted here on 2/2/2006, but it is quite similar to the previous version. Several small misprints have been corrected.)

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20.02.04 Here is a ‘compact’ version of the examination of 15/2/2005. And here is a solution of it.
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27.01.04 Here is a list of the topics that you will be expected to know for the examination.
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23.1.05 Here is Homework Exercises No. 4.
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17.1.05 The CD with pictures of the blackboards from my lectures has now been updated. For more details see here.
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11.1.05 Here is the third set of homework exercises.
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6.1.05 Here is the second set of homework exercises.
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27.12.04 You can see a solution of the mid term test on the official course website (page of the Metargel Akhrayi) or here. It can save a lot of time, both for you and for us if you look at this solution BEFORE you come to look at your makhberet from that test. Many thanks to Yoram Yihyie for writing most of the solution.

CHANGE OF ORDER OF INTEGRATION FOR GENERALISED INTEGRALS ON AN UNBOUNDED INTERVAL.
Here is a formulation of Fubini’s theorem which is useful for changing order of integration in connection with Fourier transforms.
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9.12.04 Since some students are still asking questions about what topics are included or not included in Sunday’s test, I have included further clarifications in the old announcement below. You can also see it directly here.
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1/12/04 The midterm test is on Sunday December 12 at 17:30. The rooms are shown in the usual place on the “undergraduate” website. You should be ready to to answer questions on all the subjects treated in our lectures and tirgulim up to and including Dirichlet’s theorem, including of course exercises which use Dirichlet’s theorem. If this does not define what you need to know sufficiently precisely for you, you can read this.
If you have questions before the test you can of course come to the sha’ot kabala of the lecturers and metargelim.
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27.11.04 Three things today:
(a) Here is a summary (Version 4.6, revised on 9.5.06). It deals with TWO topics:
(1) Differentiation and integration of Fourier series, term by term, including a connection with uniform convergence, and
(2) Fourier series on other intervals instead of [-pi,pi].

We are now discussing or will soon be discussing Topic (1). Topic (2) is simpler and may be left for you to study privately.
(b) Topic (1) includes a mention of uniform convergence. (hitkansut b’mida-shava). This is something which you are supposed to remember from Hedva 1M (or Hedva 2M). Although, in general, uniform convergence is very important, it only plays a limited role in this particular course. Here is a quick summary, a “Uniform Convergence Survival Kit” containing the main things you need to know about uniform convergence for this course.
(c) Viewing my lectures.

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22.11.04 A REVISED VERSION of the first set of homework exercises is here. (As announced earlier, there will not be any grade for homework and there will not be any correcting of homework. However on the mid term test and also on the final exam there will be at least one question taken from the homework exercises.)
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28.10.04 Please click here for VERY IMPORTANT information about (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.

26.10.04 To go to the OFFICIAL website for this course click here. That site also contains ESSENTIAL information, exercises, text book, previous examinations, etc. There is also a link from there, and from here to the page of the Metargel Akhrayi where you can see office hours for your teachers, dates of examinations etc.

.To go directly to download or read the text book for this course, written by Allan Pinkus and Samy Zafrany, click here, or here,
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### Click here for old announcements from 2003 and earlier.

(Some of this old information can be quite useful.)

# Michael Cwickel INTRODUCTION TO HARMONIC ANALYSIS

## *** Welcome to INTRODUCTION TO HARMONIC ANALYSIS ***

OLD WEBSITE. 2003-4

I wish you a very pleasant, interesting and successful semester.

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9.5.04 Yet another further updated version of the fifth set of homework exercises is here. (This is now version 4, revised on 9/5/04).
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28.1.04 The sixth and final set of homework exercises is here. Please submit your solutions on March 5. (I will not be here to check them before then.) I will also be sending each of you on the mailing list a preliminary version of some notes about Schwartz Distributions.
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22.12.03 The fifth set of homework exercises is here. (This is now version 3, revised on 20/3/04). It also includes material which will not be covered in lectures, unless you can agree on a time for extra lectures (to replace those which I unfortunately had to cancel.) Please submit your solutions by January 22. (In parallel there will be other exercises on material covered in future lectures.)
3.12.03 The fourth set of homework exercises is here. Please submit it by January 4.
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3.12.03 The third set of homework exercises is here. Please submit it by December 17.
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20.11.03 The second set of homework exercises is here. Please submit it by December 5.
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30.10.03 I regret that I will not be able to give the lecture on Sunday November 2. I will fix a time with you to give another lecture to replace it.
The first set of homework exercises is here. Please submit it by November 15.
I am willing to discuss ways of solving these exercises with you if you have difficulties with them.
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23.10.03 Our most immediate problem is to find a new time for the lectures which is more convenient for all students interested in this course. If we cannot do this in the lecture on Sunday Oct 26 (which quite possibly will not occur anyway because of the strike) please email me at mcwikel@math.technion.ac.il and I will send you back a “voting form” for you to give me the days and times which are good/possible/impossible for you. Then, hopefully, my computer will find the optimal solution for all of us.
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To go to the official website for this course click here.
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# Michael Cwickel HEDVA 2M (104011)

(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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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.)