Archive for calculus

The Shadow of Newton

Posted in History, The Universe and Stuff with tags , , , , , , , on November 7, 2013 by telescoper

Yesterday I overheard some Electrodynamics students talking about the fact that all the famous names attached to pioneering laws or theorems in that subject seem to be either French (Biot-Savart, Laplace, Poisson..) or German (Gauss, Helmholtz…). Why are there no British names in this list?

Well, there was Faraday, of course. But Michael Faraday was primarily an experimentalist rather than a theorist, which sets him apart from the others already mentioned. So why is it that British theoretical was behind continental Europe in the early part of the 19th Century when all this important work on electricity and magnetism was being done.

There was also Maxwell, but he came along a bit later; he published his theory of electromagnetism in 1861/2. So why were the British so slow to enter this field?

Well, my theory of this is that it’s all the fault of Isaac Newton. I came to this conclusion when reading about the work of British mathematician and physicist George Green, who lived from 1793 until 1841, and who left a substantial legacy for modern theoretical physicists, in Green’s theorems and Green’s functions. George Green is also credited as being the first person to use the word “potential” in electrostatics. Green was the son of a Nottingham miller who, amazingly, taught himself mathematics and did most of his best work, especially his remarkable Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism (1828) before starting his studies as an undergraduate at the University of Cambridge (which he did at the age of 30, after his father died, and he leased out the mill he consequently inherited, to pay for his studies).

Extremely unusually for British mathematicians of his time, Green taught himself from books that were published in France. This gave him a huge advantage over his national contemporaries in that he learned the form of differential calculus that originated with Leibniz, which was far more elegant than that devised by Isaac Newton (which was called the method of fluxions).

Great scientist though he was, Newton’s influence on the development of physics in Britain was not entirely positive. Newton was held in such awe, especially in Cambridge, that his inferior mathematical approach was deemed to be the “right” way to do calculus and generations of scholars were forced to use it. This held back British science until the use of fluxions was phased out. Green himself was forced to learn fluxions when he went as an undergraduate to Cambridge despite having already learned the better method.

Unfortunately, Green’s great pre-Cambridge work on mathematical physics didn’t reach wide circulation in the United Kingdom until after his death. William Thomson, later Lord Kelvin, found a copy of Green’s Essay in 1845 and promoted it widely as a work of fundamental importance. This contributed to the eventual emergence of British theoretical physics from the shadow cast by Isaac Newton which reached one of its heights just a few years later with the publication a fully unified theory of electricity and magnetism by James Clerk Maxwell.

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Advice for Prospective Physics Students

Posted in Education with tags , , , , , , , on February 25, 2013 by telescoper

Just got time for a quickie this morning before I head up to the Big Smoke for the first meeting of the Astronomy Grants Panel of the Science and Technology Facilities Council (STFC). I had thought that the last round would be my last, but I must have misbehaved somehow and my sentence has been extended accordingly.

Anyway, I thought I’d follow up Saturday’s post with a response to a question that a few prospective Physics students asked me during our Admissions Day. Those attending these days have already applied to this University but, at this stage of the annual undergraduate admissions cycle, the applicants are still deciding which University to put as their first or firm choice. I see our job in the Admissions Days simply to make available as much information as possible to the applicants to help them make a choice. Quite a few people I spoke to on Saturday, however, said that they had already made up their minds and just wanted some advice about what to do during the summer after they have finished their A-levels to help them prepare for their undergraduate studies in Physics (and/or Astronomy).

The answer I gave was pretty simple: practice your mathematics, especially your calculus! 

The justification for exhortation is that the big difference between Physics at A-level and Physics at undergraduate University level is that the latter is taught in a much more mathematical way than the former. This is because the physical laws that underpin our understanding of the natural world are expressed in a mathematical language; the more fluent you are in this language the easier you will find it to assimilate the physical concepts. To put it another way, you will find it difficult to understand the physical meaning of what is being taught if you are struggling with the mathematical meaning of the symbols being used or the manipulations needed to obtain useful solutions to the relevant equations.

Newton’s Second Law, for example,  relates the rate of change of momentum of a body to the force exerted upon it. If you’re comfortable with calculus you don’t think twice about writing d(mv)/dt for the rate of change of momentum and then constructing a differential equation which you can (hopefully) solve. You won’t absorb the importance of laws like this unless you become so familiar with the mathematics that it ceases to occupy the part of your mind that’s needed to really think.

I think that learning to do Physics is a bit like learning to play a musical instrument. Practicing such basic mathematical procedures as integration and differentiation is analogous to the five-finger exercises you have to do when learning to play the piano. The more you practice them, the greater the extent to which they become hard-wired. Your brain can therefore concentrate on the more interesting conceptual stuff – that’s really the hard part of learning Physics. We do of course do as much as we can to help with this once you’ve got to University, but doing some preparation on your own beforehand would greatly smooth the transition.

So I’d tell any prospective physics student wondering what to do this summer to get hold of as many basic calculus exercises as they can and do them whenever they get the chance. It may not be the most exciting way to spend your post A-level holiday, but it is the single thing you can do that will best prepare you for life as a Physics student.

On the other hand, the advice I’d give to physicists rather later in their careers is to think very carefully before agreeing to be on committees or panels…