Archive for heat equation

Fourier, Hamilton and Ptolemy

Posted in History, Poetry, The Universe and Stuff with tags , , , , , , , on December 17, 2018 by telescoper

As we stagger into the last week of term I find myself with just two lectures to give in my second-year module on Vector Calculus and Fourier Series. I didn’t want to present the two topics mentioned in the title as disconnected, so I linked them in a lecture in which I used the divergence theorem of vector calculus to derive the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

Fourier’s work was so influential and widely admired that it inspired a famous Irish mathematician William Rowan Hamilton to write the following poem:

Hamilton-for Fourier

The serious thing that strikes me is not the quality of the verse, but how many scientists of the 19th Century, Hamilton included, saw their scientific interrogation of Nature as a manifestation of the human condition just as the romantic poets saw their artistic contemplation and how many poets of the time were also interested in science.

Anyway I was looking for nice demonstrations of Fourier series to help my class get to grips with them when I remembered this little video recommended to me some time ago by esteemed Professor George Ellis. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

This reminds me of a point I’ve made a few times in popular talks about Astronomy. It’s a common view that Kepler’s laws of planetary motion according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as the elements used in the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed-form expression for such an orbit is considerably more compact and elegant than a Fourier representation, and also encapsulates a deeper level of physical understanding.


My Last Cardiff Lecture

Posted in Biographical, Education, The Universe and Stuff with tags , , on December 7, 2012 by telescoper

Hey ho.

This morning, as usual for a Friday, the alarm went off at 6am and I started the slow process of getting my brain in gear for a two-hour 9am lecture. As usual, by the start of the lecture I was still trying to wake up, but I at least managed to get through the performance  making only  finite number of errors.

The topic for day was Fourier series, and especially how to use them to solve interesting partial differential equations. The one I chose to illustrate the general method of separation of variables was the heat conduction equation, appropriately enough because Joseph Fourier, the man himself, developed the idea of using   trigonometric functions to represent other functions in order to solve that equation; he presented the method in his book Théorie analytique de la chaleur way back in 1822.

During the lecture I also had to distribute another bunch of questionnaires to the students to allow them to give constructive feedback vent their spleen at my incompetence and lack of organization. We already had one set of questionnaires halfway through the term, so I’m not sure why we need another one. Perhaps the students gave the wrong answers to the questions last time, so this is like a resit?

When it was all over, and I returned to my office to recover,  I suddenly realised that it was my last Cardiff lecture ever. (There is in fact another week remaining before the Christmas break, but I’m away next week and a colleague will fill in for me. ) In fact, it might have been my last undergraduate lecture ever, as I’m not sure how much time I’ll get for actual teaching when I move to my new job in the New Year. I think I’ll miss it, actually, but I’m not sure the students will!

Still, at least I get to set my alarm to a more sensible time from now on.