Physics 320 is a course that explores some of the more advanced mathematical methods that are found in upper level physics courses. It presumes a knowledge of vector arithmetic and calculus and experience with linear algebra. The course explores the mathematics of linear systems in more detail, concentrating on methods to solve the systems of ordinary and partial differential equations normally encountered in physics. The course is very much a physicists exploration of mathematical tools that aims to develop a practical knowledge of methods. It will avoid the kind of formal rigor that would be expected in a mathematicians study of the same topics. Physicists can rely on the need for correspondance with the real world to guarantee such basic properties as continuity and existence, whose study plays so large a role in the mathematician's view of mathematics.
The course will make significant use of the text, Mathematical Methods in the Physical Sciences,3rd Edition, by Mary Boas but will also draw on occasion from other texts, including the ever useful, if terse, Schaum's Advanced Mathematics.
I anticipate a mix of lecture with some discussion and some student presentation. As always, I plan to ask a lot of questions of the class as I talk and strongly encourage you to ask questions to make sure that you understand what is happening. I cannot emphasise enough my dependence on your questions to keep the material intelligible. If you just sit and look intelligent in silence I am likely to just make the material harder to keep up the challenge. As well as questions, I also encourage comments and observations that are likely to be of general interest. I do not have a monopoly on the good ideas.
At various points during the semester you may find yourselves called upon to present to the class more or less informally. This can be as informal as taking the chalk to work through a problem in class with the whole class participating in the process or as formal as a timed presentation on a topic prepared well in advance.
Following Seth's practice developed over several years of teaching this course I plan to build the course around two different kinds of out-of-class assignment.
First, there will be a few (normally 1-3) problems assigned for each class. You are encouraged to work together/in groups on these problems are expected to bring two copies of your solutions to class. One copy will be handed in at the start of class and you will keep the other. The grade for these is based only on handing in substantial answers to all the problems. It will not be based on the quality or even the correctness of the solutions. I will return these with comments as soon as I can.
Second, there will be bi-weekly problem sets or mini-exams. These will be more substantial in length and will be due 1 week after they are set. They will normally cover the material of that two week period. These are to be treated as take-home exams and you must work on them alone, though of course you may seek help from me and I will judge what assistance I can give in each individual case. You will be able to use your notes, daily problems/solutions, and your textbook as well as Maple or Mathematica. You can must ask if you wish to use any other resource. Obviously the grade for these will be based on both the quality and the correctness of the answers.
Lastly, there will be a take-home final that will be due toward the end of finals week. It will be held under the same conditions as the mini-exams but the material will taken from the entire course.
The grade will be based mostly on the material handed in throughout the course with a small component for class participation. That portion will mostly reflect your performance as a presenter, both informally and on any formal occasions but will also reflect the general extent to which you ask useful questions or raise interesting comments.
This is only an approximate suggestion for the material to cover. Precisely how much material we can cover will depend to a significant extent on the class. If we do reasonably well then there may be opportunity to go beyond this basic list and then I will consult the interests of the class as I choose how to extend the material.
We will certainly begin with a review of of vector algebra, some linear algebra, and complex numbers.
I then expect to revisit the topic of ordinary differential equations and to explore their relationship to linear algebra. We may revisit in more depth some topics that we met in 195. We will certainly explore the ideas of solutions in terms of integral transforms, the Fourier and Laplace transforms.
From there I expect to look at the special class of differential systems called Sturm-Liouville equations and then to explore some of the classes of special function that arise from this systems. These could include Legendre polynomials, Laguerre polynomials, Hermite polynomials, and even the rarefied world of the hypergeometric functions. I expect that there will be individual projects and presentations in this topic. There might also be talk of other special functions such as the gamma function, erf, and the elliptic integrals.
Beyond ordinary differential equations lie partial differential equations and we will explore these in some depth. I hope that we will get the chance to look at both analytic and numeric methods of solving these systems. Topics here will include the wave equation, the diffusion equation, and, of course, Schrödinger's equation.
It would be nice to get to look a little at the calculus of complex functions but we will see if there is time for that. It would also be nice to introduce a little of my own particular interest in the new mathematics of geometric algebra.
One thing I can be certain of is that I will make significant use of Dirac's extremely flexible notation for vectors in Hilbert spaces of all kinds. In this we will certainly go beyond the book. I hope that the linear algebra underpinnings of almost everything that we do will be made plain throughout the course. It is often not obvious at first sight how closely related all of these topics are, and it is not neccessary to understand these connections to understand the course. However, an awareness of the relationships can help you take a much more coherent view of what can otherwise seem a rather scattered collection of materials.