P321 | 3308 | Schwandt

The primary goal of this course is to expose you to some of the more advanced analytical and numerical techniques which we expect you to master, and which you will apply repeatedly in solving scientific problems, both in your upcoming physics classes, and in your scientific career. Some of these techniques can be illustrated by reviewing the key physics ideas presented in your introductory classes (P201/202/301), but in a mathematically more organized and sophisticated manner than you may have encountered previously. In this sense, this class is intended to serve as a “bridge”, between the introductory physics courses where the emphasis is more on the concepts and less sophisticated analytical techniques, and on the higher—level courses which are generally more rigorous. It may be easiest to describe this course by emphasizing what it is not. This course is not a math class. The syllabus is organized around topics which feature certain mathematical tools and concepts. However, at every stage, we will emphasize the physical applications of these concepts, and give detailed examples of how these concepts arise in physics, and how to solve realistic physics problems using these tools. Similary, this course is not a remedial physics course. Even though many of the examples will re—visit topics you covered in the first three introductory classes (P201/202/301), the emphasis will be on how either new concepts, or concepts which you have been exposed to in math classes but in a more abstract fashion, can be applied to “real—life” physics problems. We will start by reviewing vectors and vector spaces. In particular, we will discuss how the use of curvilinear coordinate systems can simplify physical situations which exhibit cylindrical or spherical symmetry. Having discussed vector spaces, we will move on to vector differential operators. We will define the divergence, curl and gradient operators, and discuss their applications primarily in the field of electricity and magnetism (E & M). We will then discuss line and surface integrals, giving numerous examples in various physical systems. We will then examine classical fields and their properties. Again, the emphasis will be on their applications in E & M. Next, we will discuss systems of linear equations. We will derive matrix methods to solve such systems. We will discuss properties of matrices and determinants, the eigenvalues of matrices, and “normal modes” of a physical system. Finally, we will discuss differential equations. We will begin with ordinary differential equations, with special emphasis on their applications in classical mechanics and electrical circuits. Then, we will discuss partial differential equations, in particular their applications in quantum mechanics, and in particular their applications in quantum mechanics, focussing on Schroedinger's equation. The class will not assume any mathematics other than the 2—semester calculus sequence. On the other hand, for students who may have taken such classes as multivariable calculus, linear algebra and/or differential equations, we want to keep the level interesting and challenging. Consequently, the precise level of the class will depend on the math background of the students as determined at the beginning of the semester. There may be times where we will discuss topics, such as multivariable calculus, which are familiar to some but not all of the students. At those times, we will provide optional (and informal) supplementary lectures for interested students.