Physics 503
Methods of Mathematical Physics
Fall 2001
Tuesday & Thursday, 11:20am-12:40pm.
Physics P128
Instructor: Dr. Alexander (Sasha) Abanov,
Assistant Professor
Office: Physics B102
Phone: (631)632-8174
E-mail:
alexandre.abanov@sunysb.edu
Web page:
http://felix.physics.sunysb.edu/~abanov/
Introduction
It is impossible to neither study nor do research in physics
without the use of mathematical methods. This course is
designed to give an introduction (or review) to some of
those methods. It is not a "mathematical course" in a sense
of rigorousness and completeness. In fact, any of the
"topics" mentioned below can easily take one or two
semesters in the form of a regular mathematical course.
Instead, I am going to (i) give a motivation for some of
the methods showing its necessity to solve problems, (ii)
give an introduction to the method, and (iii) apply the
method to solving more problems. The course, therefore, will
consist of related but somewhat independent topics devoted
to different methods with the emphasis made not on
foundations but on applications of the methods to solving
diverse problems.
The main prerequisite for the course is the knowledge of
standard (real variable) mathematical calculus. Also, you
have to either know something about complex numbers or to be
a quick learner to be able to follow the course.
The final grade for this course will be based on grades for
homeworks (70%), which will be given every
two weeks and on the grade for the final exam (30%).
Topics to be covered
Here is a list of topics I am planning to cover
in the course.
- Functions of a complex variable
(1/2 of the course)
- Complex numbers
- Analytic functions
- Contour integration
- Conformal mapping
- Asymptotic methods (1/6 of the course)
- Asymptotic series
- Laplace method
- Method of steepest descent
- Method of stationary phase
- Special functions (1/6 of the course)
- Gamma function
- Bessel functions
- Other special functions
- Basics of topology (1/6 of the course)
- Topological spaces
- Homotopy theory
- Classification of textures and defects of ordered media
Recommended Books
These books are all recommended but not required. They are
available from the University bookstore.
-
G. F. Carrier, M. Krook, and C. E. Pearson,
Functions of a complex variable: Theorie and Technique,
McGraw-Hill book company, New York, 1983.
This is a very good textbook which contains most of the
topics I am going to present.
-
G. B. Arfken, H. J. Weber, Mathematical methods
for physicists,
Academic press, London, 2001.
This book is probably the bestseller on the subject. It is
not a good textbook but contains all of the topics (except
for topology) and is nice to have as a reference.
Additional Books
-
C. M. Bender and S. A. Orszag,
Advanced Mathematical
Methods for Scientists and Engineers,
McGraw-Hill book company, New York, 1978.
This is a very good textbook on asymptotic methods.
-
B. A. Fuchs and B. V. Shabat,
Functions of a complex variable and
some of their applications,v. I,
Pergamon press, 1964.
This book contains a lot of examples, especially on conformal mappings
and residue calculus.
-
B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov,
Modern Geometry-Methods and Applications : Part II,
the Geometry and Topology of Manifolds,
(Graduate Texts in Mathematics, Vol 104)
Springer Verlag, 1985, ISBN: 0387961623.
This is a very nice books with the basics of homotopy theory and a lot of
examples.
-
Mikio Nakahara,
Geometry, Topology, and Physics,3rd edition,
Cambridge, Massachusetts, MIT press, 1987, ISBN: 0852740956.
Many topics relating geometry and physics are presented. In particular,
chapter 4 is dealing with homotopy groups and their use for physics.
-
N. D. Mermin,
The topological theory of defects in ordered media, Rev. Mod.
Phys., 51, 591 (1979).
This is a review article which explains how to use the homotopy theory
for the classification of topological defects in ordered media.
Homeworks
-
Homework #1, due Thursday, September 13, 2001
(PDF )
-
Homework #2, due Tuesday, September 25, 2001
(PDF )
-
Homework #3, due Thursday, October 4, 2001
(PDF )
-
Homework #4, due Thursday, October 18, 2001
(PDF )
-
Homework #5, due Thursday, November 1, 2001
(PDF )
-
Homework #6, due Thursday, November 15, 2001
(PDF )
-
Homework #7, due Thursday, November 29, 2001
(PDF )
-
Homework #8, due Thursday, December 13, 2001
(PDF )
-
Homework #9, not for credit
(PDF )
Some of the homotopy groups used in physics
(PDF )
Last updated December 28, 2001