Archive for mathematics

Writing Vectors

Posted in mathematics, The Universe and Stuff with tags , , , on October 11, 2021 by telescoper

Once again it’s time to introduce first-year Mathematical Physics students to the joy of vectors, or specifically Euclidean vectors. Some of my students have seen them before, but probably aren’t aware of how much we use them theoretical physics. Obviously we introduce the idea of a vector in the simplest way possible, as a directed line segment. It’s only later on, in the second year, that we explain how there’s much more to vectors than that and explain their relationship to matrices and tensors.

Although I enjoy teaching this subject I always have to grit my teeth when I write them in the form that seems obligatory these days.

You see, when I was a lad, I was taught to write a geometric vector in the following fashion:

\vec{r} =\left(\begin{array}{c} x \\ y \\ z \end{array} \right).

This is a simple column vector, where x,y,z are the components in a three-dimensional cartesian coordinate system. Other kinds of vector, such as those representing states in quantum mechanics, or anywhere else where linear algebra is used, can easily be represented in a similar fashion.

This notation is great because it’s very easy to calculate the scalar (dot) and vector (cross) products of two such objects by writing them in column form next to each other and performing a simple bit of manipulation. For example, the scalar product of the two vectors

\vec{u}=\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) and \vec{v}=\left(\begin{array}{c} 1 \\ 1 \\ -2 \end{array} \right)

can easily be found by multiplying the corresponding elements of each together and totting them up:

\vec{u} \cdot \vec{v} = (1 \times 1) + (1 \times 1) + (1\times -2) =0,

showing immediately that these two vectors are orthogonal. In normalised form, these two particular vectors appear in other contexts in physics, where they have a more abstract interpretation than simple geometry, such as in the representation of the gluon in particle physics.

Moreover, writing vectors like this makes it a lot easier to transform them via the action of a matrix, by multipying rows in the usual fashion, e.g.
\left(\begin{array}{ccc} \cos \theta & \sin\theta & 0 \\ -\sin\theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array} \right) \left(\begin{array}{c} x \\ y \\ z \end{array} \right) = \left(\begin{array}{c} x\cos \theta + y\sin\theta \\ -x \sin \theta + y\cos \theta \\ z \end{array} \right)
which corresponds to a rotation of the vector in the x-y plane. Transposing a column vector into a row vector is easy too.

Well, that’s how I was taught to do it.

However, somebody, sometime, decided that, in Britain at least, this concise and computationally helpful notation had to be jettisoned and students instead must be forced to write a vector laboriously in terms of base vectors:

\vec{r} = x\hat{\imath} + y \hat{\jmath} + z \hat{k}

Some of you may even be used to doing it that way yourself. Why is this awful? For a start, it’s incredibly clumsy. It is less intuitive, doesn’t lend itself to easy operations on the vectors like I described above, doesn’t translate easily into the more general case of a matrix, and is generally just …well… awful. The only amusing thing about this is that you get to tell students not to put a dot on the “i” or the “j” – it always gets a laugh when you point out that these little dots are called “tittles“.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector \hat{\imath} is too easily confused with i, the square root of minus one. Introduce a plane wave with a wavevector \vec{k} and it gets even worse, especially when you want to write \exp(i\vec{k}\cdot \vec{x}), and if you want the answer to be the current density \vec{j} then you’re in big trouble!

Call me old-fashioned, but I’ll take the row and column notation any day!

(Actually it’s better still just to use the index notation, a_i which generalises easily to a_{ij} and, for that matter, a^{i}.)

Or perhaps being here in Ireland we should, in honour of Hamilton, do everything in quaternions.

Littlewood on `the real point’ of lectures

Posted in Education, mathematics, The Universe and Stuff with tags , , , on September 3, 2020 by telescoper

We’re often challenged these days to defend the educational value of the lecture as opposed to other forms of delivery, especially with the restrictions on large lectures imposed by Covid-19. But this is not a new debate. The mathematician J.E. Littlewood felt necessary to defend the lecture as a medium of instruction (in the context of advanced mathematics) way back in 1926 in the Introduction to his book The Elements of the Theory of Real Functions.

(as quoted by G. Temple in his Inaugural Lecture as Sedleian Professor of Natural Philosophy at the University of Oxford in 1954 “The Classic and Romantic in Natural Philosophy”.)

Temple concluded his lecture with:

Classic perfection should be reserved for the monograph: the successful lecture is almost inevitably a romantic adventure. It is at once the grandeur and misery of a scientific classic that it says the last word: it is the charm of a scientific romance that it utters the first word, and thus opens the windows on a new world.

Modern textbooks do try to be more user-friendly than perhaps they were in Littlewood’s day, and they aren’t always “complete and accurate” either, but I think Littlewood is right in pointing out that they do often hide `the real point’ so students sometimes can’t see the wood for the trees. The value of lectures is not in trying to deliver masses of detail but to point out the important bits.

It seems apt to mention that the things I remember best from my undergraduate lectures at Cambridge are not what’s in my lecture notes – most of which I still have, incidentally – but some of the asides made by the lectures. In particular I remember Peter Scheuer who taught Electrodynamics & Relativity talking about his first experience of radio astronomy. He didn’t like electronics at all and wasn’t sure radio astronomy was for him, but someone – possibly Martin Ryle – reassured him by saying “All you need to know in order to do this is Ohm’s Law. But you need to know it bloody well.”

On Grinds

Posted in Literature, mathematics with tags , , , , on July 24, 2020 by telescoper

When I moved to Ireland a couple of years ago, one of the words I discovered had a usage with which I was unfamiliar was grind. My first encounter with this word was after a lecture on vector calculus when a student asked if I knew of anyone who could offer him grinds. I didn’t know what he meant but was sure it wasn’t the meaning that sprang first into my mind so I just said no, I had just arrived in Ireland so didn’t know of anyone. I resisted the temptation to suggest he try finding an appropriate person via Grindr.

I only found out later that grinds are a form of private tuition and they are quite a big industry in Ireland, particularly at secondary school level. School students whose parents can afford it often take grinds in particular subjects to improve their performance on the Leaving Certificate. It seems to be less common for third level students to pay for grinds, but it does happen. More frequently university students actually offer grinds to local schoolkids as a kind of part-time employment to help them through college.

The word grind can also refer to a private tutor, i.e. you can have a maths grind. It can also be used as a verb, in which sense it means `to instil or teach by persistent repetition’.

This sense of the word grind may be in use in the United Kingdom but I have never come across it before, and it seems to me to be specific to Ireland.

All of which brings me back to vector calculus, via Charles Dickens.

In Hard Times by Charles Dickens there is a character by the name of Mr Thomas Gradgrind, a grimly utilitarain school superintendent who insisted on teaching only facts.

Thomas Gradgrind (engraving by Sol Eytinge, 1867).

If there is a Mr Gradgrind, why is there neither a Mr Divgrind nor a Mr Curlgrind?

Math versus Maths

Posted in mathematics, Pedantry with tags , on June 8, 2020 by telescoper

I was amused by this discussion on Dictionary.Com of the different abbreviations of mathematics..

I’d like to think that ending is deliberate!

R. I. P. John Conway (1937-2020)

Posted in Biographical, mathematics with tags , , , on April 12, 2020 by telescoper

I’ve just heard the sad news that that mathematician John Horton Conway has passed away at the age of 82.

John Conway made very distinguished contributions to many areas of mathematics, especially topology and knot theory, but to many of us he’ll be remembered as the inventor of the Game Of Life. I’ll remember him for that because one of the very first computer programs I ever wrote (in BASIC) was an implementation of that game.

It’s a great illustration of how simple rules can lead to complex structures and it paved the way to a huge increase in interest in cellular automata.

I think he got a bit fed up with people just associating him with a computer game and neglecting his deeper work, but he deserves great credit for directly or indirectly inspiring future scientists.

Rest in peace John Conway (1937-2020).

Additional Mathematics O-level 1979

Posted in Biographical, Education, mathematics with tags , , , on July 1, 2019 by telescoper

Yesterday a comment appeared on an old post of mine about the O-level Examination I took in Mathematics when I was at School. With a shock that reminded me that it was FORTY years ago this summer that I was taking my O-levels at the Royal Grammar School in Newcastle. That’s a memory lane down which I wasn’t anxious to take a trip.

For any youngsters reading this, the GCE (General Certificate of Education) Ordinary Level Examinations O-levels were taken at age sixteen in the United Kingdom back in the day; they were replaced during the 1980s by the modern GCSE Examination. For readers in Ireland the O-levels were roughly equivalent to the Junior Certificate, just as A-levels are roughly equivalent to the Leaving Certificate.

Anyway, that also reminded me that I never got round to posting the other O-level I took in Mathematics that summer, in Additional Mathematics. I thought I’d remedy that failing now, so here are the two papers I took (on Tuesday 26 June 1979 and Thursday 5 July respectively.

I had forgotten that there was so much mechanics in this actually (Section C of each paper). Is that different from equivalent papers nowadays? In fact I’d be interested in comments about the content and level of difficulty of this compared to modern examinations in mathematics via the box below.

P.S. I did ten O-levels that summer of ’79: Mathematics; Additional Mathematics; Combined Science (2); English Language; English Literature; French; Latin; History; and Geography. I still have all the papers and have only posted a subset. If anyone has requests for any others please let me know and I’ll scan them.

The Problem of the Moving Triangle

Posted in Cute Problems, mathematics with tags , , on August 16, 2018 by telescoper

I found this nice geometric puzzle a few days ago on Twitter. It’s not too hard, but I thought I’d put it in the `Cute Problems‘ folder.

In the above diagram, the small equilateral triangle moves about inside the larger one in such a way that it keeps the orientation shown. What can you say about the sum a+b+c?

Answers through the comments box please, and please show your working!

When Log Tables aren’t Log Tables

Posted in Education, mathematics, Maynooth with tags , , , , , on May 17, 2018 by telescoper

Every now and then – actually more frequently than that – I reveal myself in Ireland as an ignorant foreigner. The other day some students were going through a past examination paper (from 2014) and I was surprised to see that the front cover (above) mentioned  `log tables’.

Now I’m old enough to remember using tables of logarithms (and other mathematical tables  of such things as square roots and trigonometric functions, in the form of lists of numbers) extensively at school. These were provided in this book of four-figure tables (which can now buy for 1p on Amazon, plus p&p).

As a historical note I’ll point out that I was in the first year at my school that progressed to calculators rather than slide rules (in the third year) so I was never taught how to use the former. My set of four-figure tables which was so heavily used that it was falling to bits anyway, never got much use after that and I threw it out when I went to university despite the fact that I’m a notorious hoarder.

Anyway, assuming that the mention of `log tables’ was a relic of many years past, I said to the group of students going through the old examination paper that it seemed somewhat anachronistic. I was promptly corrected, and told that `log tables’ are in regular use in schools and colleges throughout Ireland, but that the term is a shorthand for a booklet containing a general collection of mathematical formulae, scientific data and other bits of stuff that might come in useful to students; for an example appropriate to the Irish Leaving Certificate, see here. One thing that they don’t contain is a table of logarithms…

Students in Physics & Astronomy at Cardiff University are also given a formula booklet for use during examinations. I don’t remember having access to such a thing as an undergraduate, but I don’t object to it. It seems to me that an examination shouldn’t be a memory test, and giving students the basic formulae as a starting point if anything allows the examiner to concentrate on testing what matters much more, i.e. the ability to formulate and solve a problem. The greatest challenge of science education at University level is, in my opinion, convincing students that their brain is much more than a memory device…

R.I.P. Maryam Mirzakhani (1977-2017)

Posted in mathematics with tags , , on July 18, 2017 by telescoper

Very sad news arrived at the weekend of the death of the brilliant Iranian-born mathematician Maryam Mirzakhani of breast cancer at the age of just 40. Let me first of all express my heartfelt condolences to her family, friends and colleagues on this devastating loss.

A uniquely creative and inspirational figure, Maryam Mirzakhani was the first woman ever to win the coveted Fields Medal; her citation for that award picks out her work on the dynamics and geometry of Riemann surfaces and their moduli spaces.  Here’s a short video of her talking about her life and work. It’s fascinating not only because of the work itself, but the insight it gives into the way she did it – using very large sheets of paper covered in drawings and notes!

R.I.P. Maryam Mirzakhani (1977-2017).

 

Isotropic Random Fields in Astrophysics

Posted in The Universe and Stuff with tags , , on June 29, 2017 by telescoper

So the little workshop  on `Isotropic Random Fields in Astrophysics’ I announced some time ago, sponsored via a “seedcorn” grant by the Data Innovation Research Institute, has finally arrived, and having spent most of the day at it I’m now catching up with some other stuff in the office before adjourning for the conference dinner.

 

This meeting is part of a series of activities aimed at bringing together world-leading experts in the analysis of big astrophysical data sets, specifically those arising from the (previous) Planck (shown above) and (future) Euclid space missions, with mathematical experts in the spectral theory of scalar vector or tensor valued isotropic random fields. Our aim is to promote collaboration between mathematicians interested in probability theory and statistical analysis and theoretical and observational astrophysicists both within Cardiff university and further afield.

 

It’s been a very interesting day of interleaving talks by cosmologists and mathematicians followed by an open-ended discussion session where we talked about unsolved problems and lines for future research. It’s clear that there are some language difficulties between the two communities but I hope this meeting helps to break down a few barriers and stimulate some new joint research projects.