Friday, June 24, 2016

Taylor expansions are a really basic skill and concept that undergraduate physics majors need to master

This past semester I taught part of two undergraduate courses: thermodynamics for second year, and solid state physics for fourth years. I was particularly struck by two related things.

1. Taylor expansions kept coming up in many contexts: approximate forms for the Gibbs free energy (e.g. G vs. pressure is approximately a straight line with slope equal to the volume), Ginzburg-Landau theory, Sommerfeld expansion, linear response theory, perturbation theory, and solving many specific problems (e.g. where one dimensionless parameter is very small).

2. Many students really struggled with the idea and/or its application. They have all done mathematics courses where they have covered the topic but understanding and using it in a physics course eludes them.

Physics is all about approximations, both in model building and in applying specific theories to specific problems. Taylor expansions is one of the most useful and powerful methods for doing this. But, it is not just about a mathematical technique but also concepts: continuity, smoothness, perturbations, and error estimation.

Does anyone have similar experience?
Can anyone recommend helpful resources for students?

1 comment:

  1. Hello Ross,
    I really enjoy reading your blog and appreciate not only your comments on condensed matter but also on our educational duties.

    I agree 100% with you about the importance of Taylor series expansion and the underlying physical aspects related to it. As for the problem, does it really concerns only Taylor expansion? Indeed I always find striking that the students have a serious difficulty when they try to apply what they had learned in Math classes to Physics classes.

    This semester, teaching Classical Mechanics (5th semester), I could not anticipate at all that some students would have problems trying to state a difference between total and partial derivatives, and I was discussing Lagrangian dynamics!

    As for Taylor series, experience told me that I should expect these difficulties. I looked for resources on books and on the web. But seriously, do the students lack resources these days? It was in a conversation with colleagues from the ecology department (theoretical ecologists, great enthusiasts of condensed matter themselves) that I found a partial solution: you have to give students small activities before classes, so they can work these key mathematical concepts for themselves. I usually write something informal on the subject, and connect it to some physical stuff. Anyway, you have to dedicate 10 minutes of the lecture to discuss that, summing up the ideas. Usually, I give 24-26 lectures in a semester and I prepare 10-12 of these notes (1-2 pages each) and is really worth it.

    Will all the students be prepared? Well, I think this is another topic...