Physics | Techniques in Theoretical Physics
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

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.