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?