# Sergei Yakovenko's blog: on Math and Teaching

## Integration of differential forms and the general Stokes theorem

We defined integrals of differential $k$-forms over certain simple geometric objects (oriented cells, smooth images of an oriented cube $[0,1]^k$), and extended the notion of the integral to integer combinations of cells, finite sums $\sigma=\sum c_i\sigma_i,\ c_i\in\mathbb Z$, so that $\langle \omega,\sigma\rangle=\displaystyle\int_\sigma \omega=\sum_i c_i\int_{\sigma_i}\omega=\sum c_i \langle \omega,\sigma_i\rangle$. Such combinations are called $k$-chains and denoted $C^k(M)$.

Then the notion of a boundary was introduced, first for the cube, then for cells and ultimately for all chains by linearity. The property $\partial\partial\sigma=0$ was derived from topological considerations.

The “alternative” external derivative $D$ on the forms was introduced as the operation conjugate to $\partial$ so that $\langle D\omega,\sigma\rangle=\langle\omega, \partial \sigma\rangle$ for any chain $\sigma$ with respect to the pairing $\Omega^k(M)\times C^k(M)\to \mathbb R$ defined by the integration. A relatively simple straightforward computation shows that for a $(k-1)$-form $\omega=f(x)\,\mathrm dx_2\land\cdots\land \mathrm dx_n$ we have
$D\omega=\displaystyle\frac{\partial f}{\partial x_1}\,\mathrm d x_1\land \cdots\land \mathrm dx_n$, that is, $D\omega=\mathrm d\omega$. It follows than that $D=\mathrm d$ on all forms, and hence we have the Stokes theorem $\langle \mathrm d \omega, \sigma\rangle=\langle \omega,\partial\sigma\rangle$.

Physical illustration for the Stokes theorem was given in $\mathbb R^3$ for the differential 1-form which is the work of the force vector field and for the 2-form of the flow of this vector field.

The class concluded by discussion of the global difference between closed and exact forms on manifolds as dual to that between cycles (chains without boundary) and exact boundaries and the Poincare lemma was proved for chains in star-shaped subdomains of $\mathbb R^n$.

There will be no lecture notes for this lecture, since the ideal exposition (which I tried to follow as close as possible) is in the book by V. I. Arnold, Mathematical methods of classical mechanics (2nd edition), Chapter 7, sections 35 and 36.

## Tuesday, June 15, 2010

### Vladimir Igorevich Arnold: 13 years ago

Filed under: Uncategorized — Sergei Yakovenko @ 10:35
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Today Vladimir Igorevich Arnold is put to rest… ברוך דיין אמת

Below are several (scaned) prictures taken in Toronto, on the island in the Ontario lake, where VIA celebrated his 60th birthday in a big company of his students, friends and colleagues.

June 12, 1997.

## Saturday, June 12, 2010

Filed under: Analytic ODE course — Sergei Yakovenko @ 3:12
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Today, June 12, 2010, Vladimir Igorevich Arnold, or VIA, as we used to abbreviate his name between ourselves in writing, should have turned 73. Today, as many times on this day in the past years, I should have been writing a short informal  “Happy Birthday” email that never was acknowledged – VIA was not known for wasting time on polite conversations, – yet I knew he would have read it. If I were in Paris, I would call and drop by, – as all of his students would do.

Instead, today we are waiting for our Teacher to be laid to rest: the funeral in Moscow is scheduled for June 15. VIA died of foudroyant pancreatitis on June 3, 2010.

The mere thought of Arnold being ill contradicts his personality as we remember him. All his life VIA projected strength, confidence, perfection, beauty, elegance. He was all motion, all burst. I remember him teaching  the second-year class on Ordinary Differential Equations in the huge hall 16-24 in the Moscow University main building. At the beginning of each class he rushed in, with his trademark briefcase with the last soundbite of the bell,  starting the first phrase of the lecture while still 3-4 meters from the blackboard. In a fraction of second his briefcase was thrown on the table, a piece of chalk appeared in his hand, and when the first phrase was completed, we already saw a carefully drawn picture on the blackboard and a few formulas written in his calligraphic handwriting near it. His lectures were practically impossible to write down, as impossible it is to record by a cell-phone  a superb performance of your favorite music.  Besides, it was very difficult to record the insight: as Arnold speaks, draws, writes, you suddenly see how different things are getting connected and the whole picture transpires through the initial fog. Fortunately, at that time his famous textbooks were already published; in these books he succeeded in doing the impossible and recording these revelations…

Later I started attending the famous Arnold’s Seminar (with the capital “S”). It will certainly be described by many people who were both closer to VIA and have sharper pens, yet this phenomenon was so unique that no detail should fall into oblivion.  The Seminar was scheduled so that people could attend it after the standard office hours, as many (probably, the majority) of the participants were not officially affiliated with Moscow University.  Arnold rushed in the room and took his permanent seat in the middle of the front row next to the blackboard.  The seminar did not begin until VIA got from the briefcase a bunch of recent preprints and reprints and handed them out to the elder participants of the Seminar: “Vitya (to Vassiliev)! The author claims that he proved so-and-so, but I could not find any appearance of the contact structure in his computations. This simply cannot happen, we both know that it should be somewhere there!” (And in a couple of weeks Vitya would indeed return the manuscript to VIA with margins peppered by remarks explaining where the “missing” structure was concealed and showing how its explicit use may simplify the proof…). This “home assignment” could take quite a bit of time, yet at some moment Arnold opened his “school-like” copybook, entered the name and the title of the talk, and the Seminar began.

The choice of speakers and the titles, apparently, reflected the current interests of VIA; for me (at that time a 4th year undergraduate student) neither was telling, yet this was largely irrelevant since each Seminar was a one-man performance. A typical scenario was as follows. For the first 15-20 minutes the speaker talked “practically uninterrupted” , – no more than once in 1-2 minutes, when VIA asked questions seemingly technical or even bordering on chicanery. Gradually the “exposition” turned into an agitated conversation between the speaker and Arnold; this ping-pong could last for quite long for the rest of the audience to get completely lost. Then a culmination occurred: VIA jumped from his place to the blackboard and shouted “No, this is impossible to understand your way. The right picture should be as follows…” And then he explained in 5 minutes both the origin of the initial problem treated by the speaker, its links and connections to other problems (and at this moment it became crystal clear why Arnold invited this speaker to talk on this subject), and what the main result is proving (or disproving, or corroborating)… In a few next moments Arnold would explain how he would try to prove this result, and often the speaker, changing colors from red to white, would nod in acquiescence…  At such moments Arnold was literally shining from pleasure and suddenly would chuckle with his inimitable laughter, as  a child who “just did it!”.

This might well look like a derision of the speaker, yet it was not. The “revenge” could come instantly, when Arnold would start fantasizing about possible ramifications, generalizations and further developments that may come out of the result just learned. The speaker, regaining his balance by that time, could cut short these fantasies: “This corollary is indeed true, but the proof is by no means that simple as you think, VIA, for such and such reasons. And the generalization you suggest is simply wrong: just two weeks ago I constructed a counterexample” (of which the speaker did not plan to talk at all). At such moments VIA’s excitement rose to a maximum: he jumped again and started explaining why he was wrong and what underwater rocks and unexpected phenomena manifest themselves in “so innocent a problem”.  It was these moments which justified attending the Seminar for two hard hours (sometimes longer). Even the youngest participants (like me) left the room exhausted yet with some clear mathematical message to take home.

This childish chuckle, instantly transforming the face of Arnold, in my eyes reflected some part of his mathematical personality. He was very much like a prodigy child in the Aladdin’s treasure vault: enjoying mathematical reality in all its brilliance. Mathematical anecdotes mention great mathematicians whom examples only distracted from developing general theories. Arnold was the opposite: examples were alpha and omega of his approach. Of course, it was impossible to look inside this beautiful mind, yet I have a feeling that he knew mathematical objects (small dimensional varieties, Lie groups, fundamental dynamical systems, …) the way a zoologist knows and loves his bees, beasts, birds etc. This was based on his tremendous erudition and, in turn, allowed him to see connections between seemingly very distant things. Probably, about any natural number less than one hundred, he remembered all mathematical results and constructions in which this number occurs.

The impact of VIA on the generation of Moscow mathematicians who are now approximately between 65 and 40, is enormous. His direct students exhibit a quasi-religious feelings towards him: no adjective (alone or in a combination) suffices to convey the impression he left. Lightning-fast thinking, sharp reaction, incredible intuition, … – all attributes of a superhuman; he himself contributed to this image, stressing his physical skills like swimming, hiking, skiing, which also were well beyond “ordinary” capacity. Yet the child inside him was pretty much human: like many children, he loved to tease people, and many who didn’t know him closely were understandably offended. For his students he often did (without saying) things that prove a deep personal involvement he felt towards them (as recounted by Sasha Givental). Such stories will certainly grow manifold as VIA’s “scientific children” overcome their pain and get to their computers, first in Russian yet hopefully also in English

But even for those who “simply” happened to witness Arnold the Mathematician in action and enjoy the beauty and elegance of his view of the subject, the impact was “catastrophic” in the bifurcational sense of the word. At the time when I decided about the field of mathematical specialization, because of the unique atmosphere of the Moscow University those days, the choice was tantalizing. Algebra and algebraic geometry with Yurii Ivanovich Manin, Geometry or mathematical physics with Sergei Petrovich Novikov, Probability and Dynamical systems with Yakov Grigorievich Sinai, Complex analysis with Anatoly Georgievich Vitushkin, Representations theory with Alexander Alexandrovich Kirillov-Sr., all in their prime, all bursting with energy, all doing beautiful mathematics… And of course, there was the proverbial figure of Israel Moiseevich Gelfand! Instead I chose the subject which “before Arnold” many considered as boring, dull and non-inspirational; “A theorem on one property of one solution of one differential equation”, quoting VIA’s mocking description of “bad” Differential Equations. Since then I had not a single regret for falling in love with so wonderful part of Mathematics: its centrality and most diverse connections with almost all other areas is what I learned to enjoy, featuring a clear imprint of VIA’s taste. My professional career was practically predetermined by the fact that it began  in the epoch of Vladimir Igorevich Arnold.

According to Arnold, the last words of Isaac Barrow, the adviser of Isaac Newton, were “Oh Lord! Soon I will know solutions to all differential equations”. Vladimir Igorevich, I wish you to know that the seeds you planted all your life will yield hundred-fold harvest. Any other outcome would be unfair, ugly and hence simply wrong, as the truth is always beautiful…

P.P.S. The expanded version of this obituary is available here. It was published in the memorial book, Chapter 22, under the title Vladimir Igorevich Arnold: A view from the rear bench.

ARNOLD: Swimming Against the Tide,

Edited by Boris A. Khesin: University of Toronto, Toronto, Ontario, Canada, and Serge L. Tabachnikov: ICERM, Brown University, Providence, RI and Pennsylvania State University, State College, PA.

2014; 224 pp;  Softcover
MSC: Primary 01;
Print ISBN: 978-1-4704-1699-7
Product Code: MBK/86

List Price: $29.00 Individual Member Price:$23.20

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