OK, so this blog is nominally about physics and associated topics, but…
Having watched Doctor Who this evening, and tolerating no spoilers whatsoever, Steven Moffat is an arsing genius and I love him to bits. That is all.
A subtle variant of the “inconsistencies” in the last post can arise where a concept is introduced and used in a simplified form, which can catch you unawares in more general settings. For example (and while I’m trying to keep these posts agnostic of any particular field, it’s hard to describe this without example; but don’t worry if the details aren’t clear, it’s the idea that matters):
A vector space has associated with it a scalar field. The most obvious one is the field of real numbers; they are also commonly associated with complex numbers. Given a vector space over either of these fields, you can turn this into an inner product space by defining the inner product of two vectors and as , where is a member of the scalar field. The inner product must satisfy a number of properties; one of these dictates a form of commutation: , where denotes complex conjugation—a process that leaves real numbers unchanged. You will therefore, very occasionally, see an author avoid mentioning complex numbers if they are making only brief use of an inner product space with real scalars, by quoting this property as . If not aware of this, you may miss the fact that conjugation is required in the complex environment.
Again, the only real protection to this is to make sure you read any introductory chapters where terms ought to be properly defined. If in doubt, and the text makes assumptions of prior knowledge, Wikipedia is generally a good place to look for definitions, as it will tend to quote the most general form.
One of the purposes of this blog was to discuss the additional challenges presented by self-study, over structured courses such as those offered by Universities and the like. I touched upon the lack of an obvious “road-map” in the Background page, and I’ll expand on that later. First I’d like to talk about a problem that is not unique to self-study, but that is exacerbated by going it alone.
Simply put, once you get past the “general physics” books, aimed at late high-school to early undergraduate (such as the rather fine “Fundamentals of Physics” by Halliday, Resnick and Walker), you will be flicking between books specialising in particular subjects, and no two of them will use the same notation or terminology.
Starting with conceptual differences, there are three main issues:
- Different names being used for the same concept,
- The same name being used for different concepts,
- Different definitions for equivalent concepts.
In some cases you’ll see two or more of these at the same time (you could say that the general problem is a linear combination of these basis difficulties, but you might strain a friendship or two if you do).
Some examples I’ve met so far:
- Where most books I’ve encountered talk of Hermitian operators (or matrices), the book I’m currently reading on Fourier Analysis calls these symmetric operators. This sort of thing is not too tricky to negotiate, unless compounded with a different, but equivalent, definition for the concept in question (more of which later).
- Whereas most books in the subjects I’m currently studying will use adjoint to mean “conjugate transpose” of a given matrix, I did spend some time brushing up on my differential equations using a book that defined the adjoint of a matrix as . The fact that three chapters later he switched to using it as the conjugate transpose without thinking to warn us was, I think, just to piss me off. (Entertainingly, this was not a very new book; in the chapter on improving the efficiency of approximation algorithms we were warned that computer time can cost up to $1200 per hour.) Fortunately, in this case, a previous book had warned of the clashing terminology and confirmed that these were entirely unrelated usages, or I might have spent an evening scratching my head.
- In a nicely circular conclusion to this tale of three, in the book above that used “symmetric” where other books use “Hermitian”, the defintion of a symmetric operator was an operator for which inner products and were equivalent, while another book defined an operator to be Hermitian if it was self-adjoint, i.e. . As you may have guessed (or already know), these definitions are equivalent—as long as you use the right adjoint!
Once you get beyond conceptual differences, you of course have notation to deal with. Will the author use primes for derivatives ()? Or Leibniz notation ()? Perhaps subscripts (), or (for time derivatives of a vector) dot notation (, or )? Often these will be used interchangably within any single book, according to what seems to the author to be most natural for the given problem, but you can be sure that at some point you’re going to fail to recognise a friendly equation in unfriendly clothes.
So how does one deal with this sort of silliness? Well there are a few things you can do to help:
- Never skip the opening chapters of a new book. Pretty much every text will start with a review of topics, and while it can be dull going over the same ground time after time, this is also where the author lays his/her notational cards on the table. The use of different but equivalent conditions for a certain concept will mean that you may follow the text but get hopelessly lost in a proof if you aren’t prepared for it. And if nothing else, you may get treated to an elegant or more enlightening derivation of something.
- The more sources you have with overlapping interests, the better you’ll get at recognizing the well trodden paths; in the example above of the use of “symmetric” in place of “Hermitian”, we’d just introduced a particular concept (eigenvalues), and I was 90% sure we’d be moving immediately to Hermitian operators, simply because it’s the natural next step.
- Make use of sample chapters (for e-books) or “look inside” buttons on web-stores—or you could even (gasp!) try a book shop (try between “iPods For Dummies” and “Book-reading For Dummies”, if it’s anything like my nearest Waterstones), if you’re particuarly keen to follow a particular style.
- Finally, and a theme that will be picked up later, sometimes if you’re struggling with a particular section of a book, take a step back, and just keep reading without focusing on any proofs. Sometimes you will find that a few pages ahead you will realize that the author is actually doing something you recognize but in a different form, and with this understanding it all slots in place.
Of course, any further tips are always welcome!
A while back I grabbed the Penultimate app for the iPad, just ‘cos it seemed fun at the time. I didn’t really envisage using it for studying—the fingertip control is pretty dicey to write anything small enough to fit more than two or three short lines of calculation. Besides, why bother when you can just grab a pad of paper and a pen(cil)? (if you’ve not seen Peter Serafinowicz’s iPad video, it’s worth a giggle).
That said, since the first road-test of the pad was taking it on holiday, I did actually find that I would occasionally drop from Kindle to Penultimate, to scratch out a determinant verification or the like. And with practice I became a little better at squeezing more in a page of scribbling. Not that I think anyone but me could have decyphered more than a character or two!
With that in mind, I decided to do some experimenting with a home-made stylus. Since the touchscreen is capacitance-based, it was going to have to conduct well enough to be more or less equipotential with my fingers. It was also going to need to be tapered to a flattish surface for the screen to register it. In the end, I sliced off the flat end of a pencil to leave the angle at the tip somewhere between 30 and 45 degrees, wrapped the whole thing in kitchen foil, and neatened up the ends.
Finding the right angle to write at takes a little practice. Also, though some people seem to have found the “wrist protection” feature in Penultimate to work well, I find it works some days but not others. Easily solved by just finding a thick enough insulator to rest your wrist on when writing. Results below (note: I have shocking handwriting at the best of times!)
I don’t think I’ll be totally giving up on pen+paper though.
I wrote this article a few months back; at least two people read it. Since it was one of the motivations that kicked this off, I figured I’d pea-roast it here. Note: It’s possible I was feeling a little tetchy at the time—a certain Director General of the BBC had recently published a plan to take my toys away.
Science. I’m all for it. Without science, we may not today have discovered mp3 players, DVDs, or even the humble laser pointer with which to annoy cats.
But there is a darker side to science. Einstein; Oppenheimer; Feynman: had they known the consequences of their thirst for knowledge, would they have continued in their atom-splitting ways?
I am, of course, referring to the terrible blight that our society now lives with: background visual accompaniments to Horizon “science” documentaries. Being somewhat of the geek persuasion, and therefore having nothing better to do of a Friday evening (once I was sure that Lost was recording to irritate me another night), I sat back and queued up this week’s Horizon: “Is Everything We Know About The Universe Wrong?” (to which the answer is, of course, “no, I know for a fact that Mark Thompson is a <censored>”).
First of all, can we please stop accompanying every mention of the word “maths” or “calculated” with close-up footage of people scribbling pseudo-equations on black/whiteboards? Or at least get enough different shots so that a child—or even an arts graduate—can’t see the same scribble being used to represent cosmic expansion, particle energies and the narrator’s tax return?
Into the content itself, and we start out with some background about The Big Bang. And with every mention of The Big Bang, we see a fresh shot of the bulk of the programme’s budget literally going up in smoke via the joy of slow motion pyrotechnics. Of course, the budget only stretches to 20 or so actually unique explosions, so they cycle them such that none gets used more than, say, 4 or 5 times throughout the programme.
But wait! Someone’s spotted a problem with The Big Bang (pop) theory! It simply cannot explain what happened for the first few thousand millennia or so. What’s needed here are some reverse slow motion explosions. And since we have to throw out some of our calculations, how about close up shots of people literally erasing pseudo-equations with their felt pens!
So what are our scientists to do? What if there simply isn’t an explanation for the shortcomings of The Big Bang (pop) that lends itself to a palatable visual aid? Turns out we’re OK. Some physics dude came up with a model of the time immediately following The Big Bang (pop) called “Inflation”. Cue physics dude standing in a warehouse next to a giant flacid red balloon, itself attached to a gas pipe (I don’t blame it). Gas tap is turned, balloon… wait for it… inflates. Oh yes.
And of course they have several such balloons, many camera angles and a whole plethora of camera speeds to demonstrate how pretty pretty a slow-motion balloon is, just in case you get distracted by the science. And it turns out that some calculations (scribble) led to a graph (brief shot of bendy line) that some observations of background radiation between The Big Bang (pop) and the end of inflation (balloon) matched up with on at least 4 points on the bendy line (may have been more than 4, but the person drawing the dots on the bendy line only had so much time to do it in before the science got too scary). Though no-one yet knows why Inflation (balloon) stopped (still shot of balloon) before the Universe expanded too far (balloon being over-inflated and bursting in another demonstration of how slow motion balloons make it all safe).
But now we hear that something’s wrong with gravity (ooh—magnetic ball bearings on a wooden board; that’s new). Galaxies just aren’t the shape they’re supposed to be. I know this, because I saw a close-up of someone chalking a swirly line onto a blackboard. With science again stumped, the programme moved on to pondering whether it can all be fixed using Dark Matter, which is, like, invisible (ARGH! CAN’T SHOW INVISIBLE! RUN THE BALLOON!).
I must confess, I have no idea what happened after that; I turned the damn thing off and started writing this.
But all is not hopeless. The shambles that was Horizon brought back to mind the excellent Wonders of the Solar System, running on Beeb 2 Sundays (repeated Thursdays, available on iPlayer), presented by the altogether lovely Professor Brian Cox: D:Ream keyboardist, regular 6 Music Breakfast Show guest, and occasional member of the ATLAS project at the LHC (http://hasthelargehadroncolliderdestroyedtheworldyet.com/)*. This is what science documentaries should measure up to.
OK, in places the first episode strayed towards travelogue, as we see locals bathing in the Gangees at Varanasi, where Cox was waiting to witness a perfect total solar eclipse (lucky git!). But even in these scenes, we see him occasionally blu-tacking space-probe images of solar eclipses from other planets onto a red stone wall, while tourists wander past doing…well…touristy things. And all the while, he has the smile on his face that you will only ever see from someone who absolutely, bloody loves what (s)he gets to do for a living.
And OK, the programme occasionally falls into the ubiquitous trap of near any reality-based programme whereby we need to see what’s going to happen later, followed by something happening, followed by what just happened. But honestly, it was less guilty of that than most.
But overall, where it did cut to visual aid, it didn’t go half-measures. “This is sunset on Mars, as seen by the robotic rover, Spirit”. And not a balloon in sight. We even get to the bit where he has to do a bunch of sums to figure out the total power output by the sun. “That’s four, times pi, times…”. The sound fades, we get some desert shots, and not a single scribble! Back to Cox: “It’s four hundred, million, million, million, million watts. That is a million times the power consumption of the United States every year, radiated in one second. And we worked that out by using some water, a thermometer, a tin and an umbrella. And that’s why I love physics.”
And I dare you not to.
* It turns out he was also science-advisor for the Danny Boyle film Sunshine. Which I didn’t know when I tweeted about watching that after Wonders in a message that may have come across as, well, twattish. D’oh!
- Has buffer protection become second-nature?
- Why Derren Brown’s “Assassin” made me uncomfortable
- Bruce Willis Is A Ghost
- Inception for geeks
- iPhone 4 versus duct tape
- My brain hurts.
- Deviating standards: epilogue
- Deviating standards
- Replace the touchscreen with paper, and this could take off…
- The Darker Side of Science