Toomas Karmo (Part B): Practical Tips for Maths-Physics Studies

Quality assessment:

On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3'5 is "mediocre pass", 4.5 is "good", and 5/5 is "excellent"): 4/5. Justification: There was enough time to develop the relevant  points to  reasonable length.

Revision history: 

UTC=20160719T0001Z/version 1.0.0: Kmo uploaded base version (and planned to upload in the ensuing four-hour interval, without formal documentation in this revision history, nonsubstantive revisions, as versions 1.0.1, 1.0.2, 1.0.3, ...) . 

[CAUTION: A bug in the blogger software has in some past weeks shown a propensity to insert inappropriate whitespace at some late points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual blogger "Posted by Toomas (Tom) Karmo at" appears.]

4. Means to the Goal: (c) Applying Available Physical Supports

I have already referred briefly to the filing of papers. Now it is time to expand on this - referring, by way of illustration, to my work on rotation of three-dimensional "xyz" rectangular ("Cartesian") coordinate axes. 

Pleasant though the thought of university libraries (and on suitable occasions even of humble municipal libraries) might appear, for maths and physics one has to have one's own books, and these have to be extensively pencil-annotated. Pleasant though the thought of working from a screen might appear, for maths and physics one has to be brutal toward the forests, by taking hardcopy printouts (and on these, too, making copious pencil annotations). 

My general procedure is in fact to use paper and pencil twice. 

A mechanical pencil with the 0.5 millimetre Grade B black lead optimal for maths - optimal because neither too hard nor too soft - goes first to the margins of the printed book or the computer-printer hardcopy. But the book or online resource is only a guide to what one has to work out on one's own, to the extent feasible. Writing consequently needs to be done at length, in a folder of commentary. A Talmudic scholar might call this a folder of "Midrashim". Looking at my work on Euler rotation-angle triples and rotation of xyz coordinate axes, I find that I have created a Midrashim folder, indexed in the trusty Debian GNU/Linux computer by putting a zero-byte file, with filename UNNN____HARDCOPY, into the directory 

/home/karmo/AANN____workworld/GNNN____studies_private/

  Q_______science/QA____maths/QA00000039____precalculus/
     /EOON____20160610T031324Z____study__wikip__euler_angles__taitbryan__matrices_etc
  

Q and QA and QA...39 in the directory names come from the Library of Congress call-number formalism, used for shelving books, while the EONN____20160610T031324Z indicates an initiative started at Universal Coordinated Time (UTC) 20160610T031324Z. The corresponding filing folder has this same directory-tree information, in an abbreviated form, on its tab, allowing me to retrieve the folder from my shelving even a few years hence, on the strength of what is recorded in the computer. Within the folder, I find many sheets of paper, each with the same directory-tree information at the top (so that loose sheets of paper can be put back into the right filing folders at desk-tidying time). 

This is a folder, then, intended eventually to contain quite a few Midrashim (in the Hebrew plural).

One particular Midrash (to use now the Hebrew singular) starts on a sheet bearing, in its upper-right-hand corner, the annotations "Euler angles p.::01@0.1::01" and "20160709T1944Z" - meaning that I am herewith working not just on Wikipedia articles relevant to Euler-angle triples, but in this instance on the particular article whose actual title is "Euler angles". (There are easily three other relevant articles - for instance, one with the quite different title "Rotation matrix", at https://en.wikipedia.org/wiki/Rotation_matrix.) In   "p.::01@0.1::01",  the first "::01" means that I am presently on the first page of the article, and the "@0.1" that I am presently in the first tenth of the printed text on that page. The second "::01" means that this is the first of the approximately four sheets I shall ultimately have to use for this particular Midrash. There will eventually have to be at least one sheet in this particular Midrash for the Euler extrinsic rotation-angle triples, and least one for Euler intrinsic rotation-angle triples, and then (because the Euler scheme needs to be contrasted with the historically later Tait-Byan scheme, much used in aeronautical discussions of "Yaw, Pitch, and Roll", and correlatively of "Heading, Elevation, and Bank") at least one for Tait-Bryan extrinsic rotation-angle triples and at least one for Tait-Bryan intrinsic rotation-angle triples. 

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A singularly troubling aspect of my work - to this day, I have not succeeded in learning whether others are similarly troubled, and I would accordingly be anxious to receive e-mail advice through Toomas.Karmo@gmail.com - is mental fatigue. Catholic philosopher Simone Weil underscores the frightening moral aspect of the problem in her essay "Reflections on the Right Use of School Studies" (on p. 72 of the English-translation paperback Waiting on God, as issued in 1950 by Fontana): There is something in our soul which has a far more violent repugnance for true attention than the flesh has for bodily fatigue. This something is much more closely connected with evil than is the flesh. That is why every time that we really concentrate our attention, we destroy the evil in ourselves.  

It is as though some Noonday Demon is perpetually pulling us down from the lofty task at hand. The Demon seems to me, in my own particular circumstances, to be obsessed with downward excursions into books of fiction, into books of British or Estonian or Roman history, and above all into Web surfing. Further, although I am here less tempted than many others are, I do imagine the Demon gloating with a special glee over the advent in recent decades of computer gaming, and in recent years of Facebook and Twitter. 

This specially troubling Demon is peculiar to studies in maths and physics. 

Computer programming, at any rate as I knew it in 1990s University of Toronto coursework, can go on for hours and hours, deep into the night, with the Demon mysteriously absent. Latin reading, and the writing of prose in English or Estonian, are likewise untroubled by the Demon, and so can go on for hours. Indeed programming, Latin, and (very especially) prose composition tend in my case to take on something of the brain-stimulating character of coffee. 

And yet in my case - I am, I reiterate, anxious to receive reports from other workers, via Toomas.Karmo@gmail.com - maths seems to become brain-toxic after just sixty or seventy minutes. To my humiliation, I am forced to get up from the maths folders and walk in fresh air. A ten-minute walk does not usually do the trick. What is safe is, rather, a full half hour, or else a good twenty minutes with also some minor tidying-up of living quarters. Then it is possible to work again, albeit with somewhat diminished vigour, for another sixty minutes. There next has to be another working break with walk in fresh air, or something similar, again for half an hour or so, in the unpleasant realization that the next working session is liable to feature a still more gravely diminished level of vigour. 

After (at most - that is to say, on some rare day of high mathematical productivity) three hours of working, and two breaks, something else is bound to intervene, to relieve the strain - for instance, the evening meal, or bed time. 

There is also the problem of the Wandering Mind. This, however, I deal with rather efficiently. I keep around my neck a "nano-desk". The "desk" is a thin wooden board, of length and width around 15 millimetres and 5 millimetres, drilled with a pair of holes for its neck string. The board forms a rigid writing surface for some half-dozen blank white business cards (these can be bought from the same printing houses as will run off political fliers or advertising brochures), secured to it by little binder clips in spring steel. If I catch my mind wandering for, say, four minutes, I make a note on the card, and under the guidance of my admonitory note head for the restorative fresh-air walk not at (as it might be) UTC=201060709T1950Z, but instead at (as it might be) the mildly punitive time of UTC=201060709T1954Z. 

****

I have mentioned the trusty 0.5-millimetre B-hardness lead, in trusty mechanical pencil. Apart from this, it is pretty well essential to be able to write in colours, both in ink and in erasable pencil. For colours, the wooden pencils marketed as "Eberhard Faber Col-Erase", with also a basic cheap sharpener, prove good. 

Sometimes coloured pencils are a mere luxury, in Soviet Estonian parlance a mere kodanlik eputis (a "bourgeois frippery"). In the corresponding parlance of the late, unlamented Occupation Authorities - I already remarked on this within the present blog, back in April - this would possibly be a буржуазная фривольность. Sometimes, however, coloured pencils are mission-critical, as when one is working on something in geometry, and one's drawing is already crowded with a dozen lines and ten labelled points, and one now has to focus attention on two particular (perhaps partly overlapping) congruent triangles.  

I also use coloured ink pens, as I shall remark in a moment. But these are less often needed. 

Physical models are surprisingly necessary. 

From time to time, one needs a sphere. For instance, one needs an actual, physical, sphere in thinking about the dependence of unit vectors (the Einheitsvektors) on angles, in spherical coordinates. The Geneva-promulgated international standard ISO 80000-2:2009 prescribes writing theta for zenith angle and phi for the azimuthal angle. Adopting this approved convention, one then asks such questions as the following two: 

• Is the unit vector associated with increasing zenith angle direction-dependent both on choice of theta and on choice of phi, or on choice of only one of these two quantities - and if the latter, then on which one? 
• Is the unit vector associated with increasing azimuthal angle direction-dependent both on choice of theta and on choice of phi, or on choice of only one of these two quantities - and if the latter, then on which one? 

This is, in fact, mildly sticky. The two questions do not get the same answer. - A surprisingly large part of one's work consists simply in improving the too-terse notation of the textbooks. I myself like to use "e" with an upstairs hat and a basement tilde for an Einheitsvektor, thereupon adding such subscripts as "tang" to indicate a tangent-to-curve Einheitsvektor in line-integral work, "norm" to indicate a normal-to-surface Einheitsvector in flux-integral work, "2" or "y" for the second-direction Einheitsvektor in xyz coordinates, and so on. Proceeding in this spirit, we may well subscript  the e-tilde-hat  with "zen" in the case of the troublesome zenithal Einheitsvektor, and with "azi" in the case of the troublesome azimuthal Einheitsvektor (and, for that matter, with "rad" in the case of the perhaps-mildly-troublesome radial spherical-coordinates Einheitsvektor). But how, then, in an adequately careful notation, do we mark the angular dependence of Einheitsvektor directions? We do best to proceed under the general guidance of ISO 80000-2:2009 (the document is duly aware of the problem), writing parentheses, as when a function is applied to one or more arguments. Do we want, then, a notation in the two-variable function style "foobar(theta, phi)" (where "foobar" is the e-tilde-hat-with-appropriate-subscript), or in the univariate-function style "foobar(theta)", or in the univariate-function style "foobar(phi)"? To say that the two just-posed questions do not get the same answer is to say that of the three stylings "foobar(theta, phi)", "foobar(theta)", and "foobar(phi)", one is appropriate in one of the two just-posed cases, whereas some other one of the three is appropriate in the other of the two just-posed cases. 

At the moment, I have, for relatively error-free visualization of such situations, a styrofoam sphere, 6 or 7 centimetres in radius. 

The sphere has in the last few days proven useful for settling a specially troublesome, albeit small, question, ultimately arising from my studies in rotation of xyz coordinates: Suppose we have on a spherical surface three specks ("Blue", "Green", and "Red"; henceforth "points B, G, R"), comprising the vertices of a spherical triangle each of whose three internal angles is a right angle. (It is as though the three points are on the surface of the Earth, idealized to a perfect sphere, with one point perhaps in Texas, another in the Black Sea region, and a third not  far from Cape Horn.) For definiteness, think of these three points as in the following sense obeying a right-hand rule: if three rods are given, each of length equal to the radius of the sphere, and if each of these rods has one end at the centre of the sphere, and if the rods are mutually perpendicular, and if the rods meet the surface of the sphere at B, G, and R respectively, then the B, G, and R rods, taken in that order, form a right-hand triple.  (In more vivid language: If the four fingers of the right hand are straightened along the B rod, pointing toward the B speck, and the thumb is outstretched, perpendicular to the four fingers, along the R rod, pointing toward the R speck, then the natural motion for the fingers when they are allowed to relax into a curl will be the motion from the R rod (not away from, but rather toward) the G rod.)  Suppose we also have, somewhere on that spherical surface (perhaps on the surface in the interior of the triangle, perhaps somewhere on the three great-circle arcs which are its sides, perhaps somewhere on the surface in the exterior of the triangle), a black cross, marking the "North Pole". Suppose the latitudes of the G and R specks to be given. What inferences, if any, can now be made regarding the latitude of the B speck?  (Under what circumstances is the latitude of the B speck fully determined by the latitudes of the G and R specks?  Under what circumstances is there still some room for vacillation in B-speck latitude? What kind of vacillation is possible - through a continuous range of latitutdes, or through merely some small discrete set of permissible latitudes?  If the latter, then how many discrete distinct latitudes are in the set?)

So, as I say, styrofoam. 

I did manage to answer my question, marking the sphere up in coloured inks - first with, among other things, my R and G speck, and then with appropriate circles centred on those two specks. It proved easy to score the desired circles in the soft styrofoam with an ordinary compass, such as is used for drawing on a plane, and then to ink  them in, selecting gel pens in appropriate colours. 

But in working the question with this tactic, the styrofoam is permanently marked, in a way liable to impede upcoming, unrelated, investigations over the coming months. 

I do now know what equipment is optimal. One googles on the string Lenart sphere wikipedia. And having skimmed, with delight, the Wikipedia writeup, and examined its delightfully eloquent pair of photographs, one then googles for vendors. 

Back before the summer of 2008, when we all got kicked out of the David Dunlap Observatory, our auditorium housed something nearly as good as the current commercial spherical-geometry offerings, namely an ancient (1935?) melon-sized "spherical blackboard" in something like slate. This material would surely tolerate markup in coloured chalks, easily cleared with a damp rag. But István Lénárt's Google-researchable commercial solution is more elegant. Indeed it is marketed - as I write, I await a communication from a marketing rep, so that I can consider even a purchase - not only with István Lénárt's book suggestively entitled Non-Euclidean Adventures on the Lenart Sphere, but also with such useful plastic accessories as sphere-hugging arc  rulers. 

The celebrated 1967 Dustin Hoffman film "The Graduate" features, I gather, the following deadpan dialogue, as a commercially successful "Mr McGuire" tells the callow young hero what to aim for when planning a career: 

MR McGUIRE:      I just want to say one word to you. Just one word.

BENJAMIN:      Yes, sir.

MR McGUIRE:      Are you listening?

BENJAMIN:      Yes, I am. 

Mr McGUIRE:     Plastics. 

One might also hope some day to find a plastic hyperbolic paraboloid ("saddle") surface, with explanatory "Adventures" booklet, appropriate surface-hugging (deformable?) plastic rulers, and the like, marketed for people who desire to explore non-Euclidean geometry beyond even the positive-curvature domain of István Lénárt. Geometry in two and even three or perhaps even four dimensions, possibly in spaces of so-called "negative curvature" as opposed to the mere "positive curvature" of the two-dimensional sphere-surface space, is said to be the geometrical setting for General Relativity. 

But let us return to ordinary Euclidean three-dimensional space. 

For chirality, or "handedness", in electromagnetics - as when one ponders the difference between the voltage induced by some time-varying magnetic field in a right-hand-wound coil and the voltage induced by the same time-varying magnetic field in a left-wound coil - little opposing spirals of wire prove helpful. I think I did mine in copper single-strand hookup wires, choosing usefully contrasting colours of plastic insulation. 

For rotation of three-dimensional Euclidean xyz axes, one needs some physical model, to play with day upon day upon brain-wearying, Demon-defying day , as one might obsessively play with a Rubik Cube. It would be good to find such models sold commercially, under Mr McQuire's astute inspiration, in some suitable grade of tough plastic. At my own desk, however, I improvise, resorting to three wooden sticks and three drinking straws, all held together with a lump of plasticene - or, rather, resorting to six wooden sticks, six drinking straws, and two lumps of plasticene, to yield a pair of rotation-of-coordinates models. I additionally have tiny special-purpose sticks, originally used to bind pickled herring into duly compact spirals. Those little herring sticks have occasionally proven helpful in thinking of such things as a Cone of Constant Zenith Angle. In each of the two sets of three sticks, the three are labelled (at only one end, thought of as marking the direction of the increasingly-positive) "Eins", "Zwei", and "Drei", and are this month thought of as the fixed x, y, and z axes. In each of the two sets of three straws, three are labelled (at only one end) in blue, green, and red ink, respectively, and are this month being thought of as the manipulable x, y, and z axes. 

I may as well finish this essay by writing up, by way of illustration, some of my findings on Euler rotation-angle triples, in the context of a sticks-and-straws-in-plasticene model. The writeup, although more than a little arid, does serve as a case study in the sort of thing which has to go into a Midrash, and so for this and other reasons may prove helpful to some maths students:  

(1) We examine the extrinsic Euler scheme "z, then x, then z", with Euler-angles triple alpha, beta, gamma (and with, for definiteness, the positive sense of all angles being given by the usual right-hand rule: to rotate something by, say, +37 degrees around, e.g., the Eins axis is to rotate it in such a way that as the fingers of the right hand curl along the rotation, fingernails leading, the thumb points toward the "Eins" label). In terms of the physical model, this is a scheme on which we rotate the Coloured triple through alpha about Drei, then through beta about Eins, and then through gamma about Drei. We set ourselves the goal of showing that any orientation of Coloured axes with respect to German axes can be achieved by selecting some alpha in the half-open interval [0, 2 pi[, some beta in the closed interval [0, pi], and some gamma in the half-open interval  [0, 2 pi[. (This is one acceptable precisification of the claim that "the extrinsic Euler scheme captures any desired rotation of axes.") 

Think of the Drei stick as the polar axis, in the sense of terrestrial globe-makers, with the "Drei" end of the stick marking (say, for definiteness)  the North pole. Imagine the three mutually perpendicular wooden German-labelled sticks being so carefully cut and positioned that their six ends are confined to a sphere, and imagine the three coloured drinking straws being cut to exactly half the length of the three German-labelled wooden sticks - so that when their uncoloured ends are embedded in the central (idealized as a mere point) blob of plasticene, their free, coloured, ends lie on that same sphere. 

Impose some arbitrary imaginary reference meridian on the sphere, yielding longitudes. 

Think of the entire alpha-beta-gamma sequence of rotations as an "Operation", comprising three "Manoeuvres". 

Any desired latitude for the Green and Red straw tips can be achieved through some appropriate choice of alpha in [0, 2 pi[  and beta in [0, pi]. 

Further (this is crucial) whatever Red and Green latitudes are thereby selected, as the second Manoeuvre in the three-manoeuvre Operation is completed, is safe against alteration when the third Manoevre is undertaken. In the third Manoeuvre, we keep the Red and Green latitudes unaltered, and we can in this concluding Manoeuvre achieve any desired Red and Green longitudes. 

Further, once the entire Operation is complete, the desired position (in both latitude and longitude) of the Blue tip is achieved. The orientation of the Blue tip on the sphere is not independent of the orientations of the Green and Red tips: on the contrary, the position vector of the Blue tip is always  predictable from the achieved positions of the Green and Red tips, being in fact the vector cross product of those two respective position-vectors. (In that order: "vector-B" is the cross product of "vector-G" with "vector-R", and is not equal to, but rather is antiparallel to, the cross product of "vector-R" with "vector-G".) 

(2) We examine the intrinsic Euler scheme "z, then x-prime, then z-prime-prime", with Euler-angles triple alpha, beta, gamma (and with, for definiteness, the positive sense of all angles being given by the usual right-hand rule: to rotate something by, say, +37 degrees around, e.g., the Blue axis is to rotate it in such a way that as the fingers of the right hand curl along the rotation, fingernails leading, the thumb points toward the "Blue" label). In terms of the physical model, this is a scheme on which we rotate the Coloured triple through alpha about Red, then through beta about Blue, and then through gamma about Red. We set ourselves the goal of showing that any orientation of Coloured axes with respect to German axes can be achieved by selecting some alpha in the half-open interval [0, 2 pi[, some beta in the closed interval [0, pi], and some gamma in the half-open interval  [0, 2 pi[. (This is one acceptable precisification of the claim that "the intrinsic Euler scheme captures any desired rotation of axes.") 

The argument is similar to the argument for (1) above. It perhaps in some sense suffices to say "Merely reverse the roles of the German and the Coloured axes." This is, however, a bit quick. It is in fact the kind of thing that we might conceivably hear in a bad lecture on campus, where the prof lacks the time, or alternatively lacks the inclination, to be explicit. Proceeding more carefully, we run through the same argument as in (1) above, but now with Red as the polar axis. We argue, in the style of (1) above, that any desired Drei and Zwei latitudes can be achieved by performing the first two Manoeuvres in the three-manoeuvre Operation, and that the third Manoeuvre, while delivering any desired Drei and Zwei longitudes, has the crucial virtue of leaving that pair of latitudes unperturbed. 

I bought my first computer, a 64-kilobyte Osborne 1 with 52-character lines on its screen, and with the eight-bit-registers Zilog Z80 chip for a CPU, in 1982 or so. Upon demonstrating this device at some length to two friends in Melbourne - look, I said,  it is even programmable in BASIC! - one of the two exclaimed, "Gee, I'm sure glad this isn't boring." The reader, having plodded through or past my discussions of Eins, Zwei, Drei, Blue, Green, and Red,  may now well echo that robust Antipodean sentiment. 

Among the inaccurate things parents and teachers like to tell children are these three: Members of Parliament serve the public interest. The police are your friends. Science is fun. But the fact that these statements are inaccurate does not negate the spiritual benefits of parliaments, policing, and science. 

[This concludes  the present "Practical Tips for Maths-Physics Studies". I have not yet decided what to write on for the next big blogger upload, scheduled for the Universal Coordinated Time interval 20160726T0001Z/20160726T0401Z. Perhaps something on my various adventures, down through the years, in police handcuffs, notably at the David Dunlap Observatory?]