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syllabus - Fall 2010 Mathematical Physics Physics(PHZ 3113...

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R esearch in theoretical physics often proceeds in two stages. The first sharpens a vague idea into a specific question whose answer can be found by solving a specific mathematical problem. The second, much easier stage is to solve that problem. The balance of courses in the undergraduate physics curriculum re- flects the dichotomy of physics research. Most courses are con- cerned primarily with fundamental theoretical concepts, their mathematical expression, and relation to the material world. These courses involve solving problems, of course, but they are about the natural phenomena we observe and how we describe them mathe- matically. The present course, in contrast, is about solving problems. Mathematics is a tool for a physicist, a means to an end. It is not an end in itself. The objective is not to prove mathematical theorems, important though they may be. Rather, the first goal for a physicist is to learn how to use those theorems and associated mathe- matical structures to calculate results. A second goal, which often becomes more important as one matures scientifically, is to see why the definitions of mathematical structures are as they are. Mathematical definitions often have roots in models of physical phenom- ena, and understanding they are interesting often yields insight both into how the mathematics works and into the conceptual content of the associated physical models. Course Objectives The first objective of this course is to develop students’ facility with certain mathematical tech- niques. By its end, each student should be familiar enough with these that he or she can waste little time wondering how to proceed when solving problems in future courses. It may some- times be difficult in practice to get a numerical result, but it should be clear in principle how to go about it. The primary responsibility for this objective is the students’. As with any tool, there is no substitute for practice when learning to use the mathematical methods we discuss. Students should therefore commit to devoting substantial time to the assigned problems. The second objective is to highlight applications of mathematical methods to physical systems. As discussed above, such applications shed light on both the mathematics and the logic under- pinning physical models. This allows students to develop valuable intuition for subsequent courses where these same methods will be applied. We will accomplish this objective primarily in the lectures. These will tend to focus in roughly equal parts on what the mathematical struc-
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