Mathematical Methods 29:4761
Mathematical Methods I
- Room: 618, Van Allen Hall
- Time: 9:30AM - 10:45AM
- Days: Tuesday, Thursday
- Text: Mathematics for Physicists,
Philippe Dennery and Andre Krzywicki, Dover, 1995(1967)
- Instructor: Wayne Polyzou
- Office: 306 Van Allen Hall
- Office Hours:W 1:00-2:00, Tu 1:00-2:00, F 1:00-2:00 or by appointment
- Grader: Calvin Mera
- E-mail: firstname.lastname@example.org
- Office: 302 Van Allen Hall
Possible final grades are A+,A,B,C,D,F. The grade of A+ is for
performance that is a full grade above an A. Grades are based on
homework scores (15%), hour exam scores (25% x2), and the final exam
(35%). Exam dates will be determined by the instructor after
consultation with the students.
Homework assignments and important announcements will appear
on the web version of this syllabus
solutions, exam solutions, and lecture notes will be posted on
the class website
here. Homework will be due on Thursdays. The
lecture notes are for your benefit, but they are no substitute for
taking your own good notes during lectures. My lecture notes are
normally written during the evening before each lecture and posted on
the morning of the lecture. I do not have time to proofread the notes
so be warned that they may have errors. If you do not understand
something in the posted lecture notes, check with me before or after
class. I will try to correct errors as I go so expect changes in the
latter parts of the notes.
This is the first half of a two semester course on mathematical
methods in physics. The purpose of this course is to expose students
to the type of mathematics that is used in intermediate and advanced
physics classes. The main focus for the first semester will be
on complex analysis, linear algebra and analysis. These are topics from
the first two chapters and part of the third chapter of the text.
These are both used extensively in the core graduate courses.
The text for this course is, "Mathematics for Physicists",
Philippe Dennery and Andre Krzywicki. I will also lecture on
supplementary material that not covered by the text.
In addition to the text there are a number of excellent
references on specific areas of mathematics that are used in physics.
The references listed below go deeper in many of the subjects that I
will cover in this class and cover some relevant areas of mathematics
that will not be covered in this class; I have chosen them because
they are the books that I have found to be useful both as a
student, teacher, and researcher.
- Functional Analysis,
Frigyes Reisz and Bela Sz.-Nagy, Dover, 1990(1950).
Readable treatment of functional analysis.
- Functional Analysis and Semigroups,
Einar Hill, Ralph S. Phillips, AMS Colloquium Publications,
Vol. XXXI, 1957.
The best reference on analytic properties of resolvent and semigroups.
- Methods of Modern Mathematical Physics, Vol 1-IV,
Michael Reed and Barry Simon, Academic Press, (1972,1975,1978,1979).
This is a four volume set of books that cover almost all aspects
of functional analysis that are relevant for physics. Contains excellent
- Real and Complex Analysis,
Walter Rudin, McGraw Hill, 1972.
Standard first year graduate reference on analysis.
- Real Analysis,
H. L. Royden, Mac Millan, 1968.
Main competitor to Rudin.
- Generalized Functions, V1-5,
I. M. Gelfand, G. E. Shilov (V1-3), I. M. Gelfand and N. Ya. Vilenkin (Vol 4),
I. M. Gelfand, M. I. Graev, N, Ya. Vilenkin (Vol 5),
Academic Press (1964,68,67,64,66).
Readable and well written - the definitive reference on distribution theory,
harmonic analysis, infinite dimensional integration. One of my favorite
- Linear Operators (Parts I,II and III),
N. Dunford and J. Schwartz, Wiley, (1957,1963,1971).
Comprehensive three volume work on linear operators.
- Methods of Mathematical Physics, Vol I and II,
R. Courant and D. Hilbert,
Wiley, 1989(1937)(v1) , 1962 (v2).
Comprehensive reference written by leading mathematical physicists.
- Functional Analysis,
Has useful material on semigroups of operators and material that is
important in quantum mechanics.
- Trace Ideals and Their Applications,
AMS Mathematical Surveys and Monographs, V120, 2005.
Has unique material that in important in statistical physics,
quantum mechanics, and quantum field theory.
- An Introduction to Probability Theory and its
W. Feller, Wiley, 1950.
Standard reference on probability theory, proof of the law of large numbers
and central limit theorem.
- Differential Equations, Dynamical Systems, and Linear Algebra,
M. Hisrch and S. Smale, Academic Press, 1974.
Modern treatment of linear algebra and differential equations -
nice emphasis on qualitative methods that are important for dynamical
- Ordinary Differential Equations,
V. I. Arnold, MIT Press, 1981.
Clear and concise treatment of differential equations from a modern point of
- Transversal Mappings and Flows,
R. Abraham and J. Robbin, Benjamin Cummings, 1967.
Nice treatment of generic properties of dynamical systems that is
hard to find elsewhere.
- Singular Integral Equations,
N. I. Muskhelishvili, Dover, 1992 (1953).
One of the earliest references in treating a class of integral equation
that are important in scattering theory and imaging.
- Perturbation Theory for Linear Operators,
T. Kato, Springer, 1966.
Contains important material on time dependent scattering theory,
also illustrates many important concepts using finite-dimensional examples.
- Scattering Theory by the Enss Method,
P. Perry, Harwood, 1983.
The first section gives a beautiful introduction to functional analysis, and
uses Weiner's theorem to give a neat geometrical characterization
of spectral properties of linear operators that have applications in scattering.
- Foundations of Modern Analysis,
J. Dieudonne, Academic Press, 1969.
Elegant work on analysis by one of the Bourbaki
contributors; formulates many theorems of elementary calculus in a
geometric manner that applies equally to infinite and finite dimensional
- Complex Variables,
R. Redheffer and N. Levinson,
Holden Day, 1970.
Clear elementary reference directed at physicists, mathematicians and
- The Theory of Functions,
E. C. Titchmarch,
Classic reference, contains much of what is now called
mathematical physics. Easy to read.
- Applied Analysis,
C. Lanczos, Dover, 1988(1956).
Contains practical material relevant to mathematical physics.
- Mathematical Physics,
Robert Geroch, University of Chicago Press, 1985.
A unique abstract treatment of mathematical physics that starts from
category theory. Good job of motivating why certain abstract mathematical
structures are important.
- A Survey of Modern Algebra,
G. Birkhoff and S. Maclaine,
Mac Millan, 1965(1941).
Standard introductory reference on algebra. Written by two
S. Lang, Springer, (1965).
Standard graduate text on algebra.
T. Hungerford, Springer, 1974.
Graduate text on algebra, clear presentation of many topics.
- Lie Algebras in Particle Physics,
H .Georgi, Benjamin Cummings, 1982.
Excellent and readable treatment of group theory
- The Theory of Groups,
H. J. Zassenhaus, Dover, 1999(1958).
Clear and compact reference that focuses on group theory.
- Theory of Group Representations,
M. A. Naimark and A. I. Stern, Springer, 1982.
Nice treatment of group representation theory.
- The Theory of Lie Groups,
C. Chevalley, Princeton, 1999(1946).
This is a classic and well written reference. It deals
with some of the fundamental properties of Lie Groups.
- The Classical Groups - Their Invariants and Representations,
H. Weyl, Princeton, 1939.
A classic reference which includes material on the classification of
- Representation Theory of Semisimple Groups,
A. Knapp, Princeton, 1986.
Readable and useful.
- Group Theory and its Application to Physical Problems,
M. Hammermesh, Dover, 1989(1962).
One of the earlier references on group theory written by a physicist.
- Gian-Carlo Rota on Combinatorics,
G. C. Rota, Birkhauser, 1995 .
Collection of Rota's papers.
Has useful material on Mobius functions, Zeta functions and partial
orderings that is important in theories involving many degrees of
- General Topology,
J. L. Kelly, D. Van Nostrand, 1955.
Clear treatment of the branch
mathematics that is used to properly formulate convergence.
- Topological Groups,
L. S. Pontryagin, Gordon and Breach, 1966,
Stands out as one of the most readable book on advanced mathematics -
contains excellent treatment of foundation material for topics that are
important in Lie groups.
- Differential Geometry, Lie Groups, and Symmetric Spaces,
S. Helgason, Academic Press, 1978.
Best treatment of symmetric spaces, pretty good for differential geometry
and Lie groups as well.
- A Comprehensive Introduction to Differential Geometry (V1-5),
M. Spivak, Publish or Perish, .
Readable treatment of differential geometry
that is relevant for general relativity and gauge theories - contains
some fun historical material.
- The Topology of Fiber Bundles,
N. Steenrod, Princeton University Press, 1951.
One of two treatments of the formal mathematics behind gauge symmetries
- Fiber Bundles,
Dale Husemoller, Springer, 1966.
Covers same material as Steenrod.
- Measure Theory
P. Halmos, Springer, 1974(1950).
The subject is becoming more important in physics, especially for
problems in dynamical systems, statistical physics, and quantum
- Foundations of Differentiable Manifolds and Lie Groups,
M. Warner, Scott Foresman, 1970.
Nice compact treatment of Differentiable Manifolds and
Lie groups. Now available in a Dover edition.
- PCT Spin Statistics and All That,
R. F. Streater and A. S. Wightman, Benjamin Cummings, 1964.
Clear treatment of SL(2,C), finite dimensional representations of the
Lorentz group, representation analytic functions of several variables,
distribution theory with applications to quantum field theory.
- Algebraic Topology,
E. H. Spanier, Springer, 1989(1966).
The first comprehensive textbook on the subject. Most
contemporary mathematicians learned the subject from this
- Gravitation and Cosmology,
Steven Weinberg, Wiley, 1971.
Nice treatment of Reimannian Geometry in Chapter 6, nice treatment of
the Weyl Tensor.
- The Quantum Theory of Fields, VI
Chapter 2 contains a nice treatment of the
representation theory of the Poincare group, projective representation
and central extensions of groups,
- Operator Algebras and Quantum Statistical Mechanics V1,2,
O. Bratelli and D. Robinson, Springer 1979,1981.
Mathematics of systems of and infinite number of degrees of freedom.
Homework Assignments and Calendar
- I have added some typed notes - they are a work in progress - they have some errors but may be more readable than the hand written notes - note that
they do not exactly overlap what I have covered in lectures.
. Feel free to point out any errors that need to be corrected.
- Week 1
- Tuesday, August 27
- Reading: Chapter 1 D&K
- Wednesday: August 28 - no office hours today - Lobbying for the American
Physical Society in Des Moines
- Thursday, August 29
- Reading: Chapter 1 D&K
- Homework #1:
Assignment 1 (solutions) , due, Thursday, September 5
- Week 3 - This week and next week Professor Reno will teach this class.
- Tuesday, September 10
- Lecture 5:
- Thursday, September 12
- Homework #3 (solutions):
Assignment 3, due, Thursday, September 19
- Week 4 - This week and next week Professor Reno will teach the class.
- Tuesday, September 17
- Lecture 7:
- Thursday, September 19
- Homework #4 (solutions):
Assignment 4, due, Thursday, September 26
- Week 5
- Tuesday, September 24
- Thursday, September 26
- Homework #5 (solutions):
Assignment 5, due, Thursday, October 3
- I will not be available for office hours on Friday.
- Week 9
- Tuesday, October 22
- Thursday, October 24
- Homework #9:
Assignment 9, due, Thursday, October 31 (correction on homework 9
- for problem 6 the norm of A should be less than 1, not infinity).
- Friday, October 25 - No office hours today - out of town
- Week 10
- Tuesday, October 29
- Thursday, October 31
- Homework #10:
Assignment 10, due, Thursday, November 7 - correction - problem 4 should read ''let H be'' rather than ''let A be''.
Tuesday, November 12
Thursday, November 14
Assignment 12, due, Thursday, November 21
- Thanksgiving Recess Nove 25-Nov 29
Tuesday, December 3
Thursday, December 5
Assignment 14, due, Thursday, December 12
- Friday, December 13, Last Day of Classes.
- Final Exam: Monday 12/16, 5:30-7:30 PM, 618 Van
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