Mathematical Methods I

29:171

Fall 2006

Class Information

• Room: 301, Van Allen Hall
• Time: 10:30AM - 11:20AM
• Days: Monday, Wednesday, Friday
• Text: Mathematics for Physicists, Philippe Dennery and Andre Krzywicki, Dover, 1995(1967)
  

Instructor Information

• Instructor: Wayne Polyzou
• Office: 306 Van Allen Hall
• Office Hours:M 2:30-3:30, Tu 1:30-2:30, W 1:30-2:30
• Grader: Yuzhi Liu - e-mail - yuzhi-liu@uiowa.edu, office 658A, phone: 5-2920
• Instructor schedule.
• E-mail: polyzou@uiowa.edu
• Phone: 319-335-1856

Grading Policy

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 (20%), hour exam scores (25% x2), and the final exam (30%). Homework assignments and important announcements will appear on the web version of this syllabus (http://www.physics.uiowa.edu/~wpolyzou/phys171/). Homework solutions, exam solutions, and lecture notes will be posted on the library's electronic reserves and here. Homework will be due on Wednesdays. 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.

General Information

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 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 historical references.
• 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 references.
• 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,
  K. Yoshida, Springer, 1980.
  Has useful material on semigroups of operators and material that is important in quantum mechanics.
• Trace Ideals and Their Applications,
  Barry Simon, 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 Application, V1,2,
  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 systems.
• Ordinary Differential Equations,
  V. I. Arnold, MIT Press, 1981.
  Clear and concise treatment of differential equations from a modern point of view.
• 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 spaces.
• Complex Variables,
  R. Redheffer and N. Levinson, Holden Day, 1970.
  Clear elementary reference directed at physicists, mathematicians and engineers
  
• The Theory of Functions,
  E. C. Titchmarch, Oxford, 1932.
  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 excellent mathematicians.
• Algebra
  S. Lang, Springer, (1965).
  Standard graduate text on algebra.
  
• Algebra
  T. Hungerford, Springer, 1974.
  Graduate text on algebra, clear presentation of many topics.
  
• 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 groups.
  
• 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.
• Lie Algebras in Particle Physics,
  H .Georgi, Benjamin Cummings, 1982.
  Nice treatment of Lie Algebras relevant to symmetries of particle physics.
  
• 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 freedom.
  
• 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 field theory.
  
• 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 text.
  
• 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
  Steven Weinberg, Cambridge, 1995.
  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

• Week 1
• Monday, August 21
• Lectures:
  
• Reading: Chapter 1 D&K
• Wednesday, August 23
• Homework #1: First Assignment, due 8/30
  
• Friday, August 25

• Week 2
• Monday, August 28
  
• Wednesday, August 30 - Homework set #1 due.
• Homework #2: Second Assignment, due 9/6
• Reading: Chapter 1 D&K,
  
• Friday, September 1

• Week 3
• Monday, September 4, Labor Day, no class
• 
  
• Wednesday, September 6 - Homework set #2 due.
• Homework #3: Third Assignment, due 9/13
• Reading:
  
• Friday, September 8

• Week 4
• Monday, September 11
  
• Wednesday, September 13, Homework Set #3 due.
• Homework #4: Fourth Assignment, due 9/20
• Reading:
  
• Friday, September 15

• Week 5
• Monday, September 18
  
• Wednesday, September 20 - Homework set #4 due.
• Homework #5: Fifth Assignment, due 9/27
• Reading:
  
• Friday, September 22

• Week 6
• Monday, September 25
  
• Wednesday, September 27 - Homework set #5 due.
• Homework #6: Sixth Assignment, due 10/4
• Reading:
  
• Friday, September 29

• Week 7
• Monday, October 2
  
• Wednesday, October 4 - Homework set #6 due.
• Homework #7: Seventh Assignment, due 10/11 - Note that on problem 5 the lower limit of integration should - infinity not 0 [the original integral can only be done in terms of special functions]
• Reading:
• Homework #7:
• Reading:
  
• Friday, October 6

• Week 8
• Monday, October 9
  
• Wednesday, October 11 - Homework set #7 due.
• Homework #8:
• Reading:
  
• Friday, October 13

• Week 9
• Monday, October 16
• Exam 1
  
• Wednesday, October 18
• Homework #8: Eighth Assignment [Note - it is OK to hand the homework to me by 5:00 on the day that it is due( you may slide it under my door), however homework will not be accepted after 5, and all homework should be given directly to me.]
• Reading:
  
• Friday, October 20 in room 301

• Week 10
• Monday, October 23
  
• Wednesday, October 25 - Homework set #9 due.
• Homework #9: Ninth Assignment
• Reading:
  
• Friday, October 27

• Week 11
• Monday, October 30
  
• Homework #10: Tenth Assignment
• Wednesday, November 1 - Homework set #9 due.
• Homework #11:
• Reading:
  
• Friday, November 3

• Week 12
• Monday, November 6
  
• Wednesday, November 8
• Homework #11: Eleventh Assignment
• Reading:
  
• Friday, November 10

• Week 13
• Monday, November 13 - Exam 2; Professor Meurice will lecture this week
  
• Wednesday, November 15
• Reading:
  
• Friday, November 17

• Thanksgiving Recess

• Week 14
• Monday, November 27
  
• Wednesday, November 29 - Homework set #11 due.
• Homework #112: Assignment 12
• Reading:
  
• Friday, December 1

• Week 15
• Monday, December 4
  
• Wednesday, December 6 - Homework set #13 due.
• Reading:
  
• Friday, December 8, Last Day of Classes.

• Final Exam 9:45 A.M. Wednesday, December 13, 301 Van

College Information

• Information for students who need accommodations due to disabilities

Students with Disabilities

Any student who has a disability which may require some modification of seating, testing, or other class requirements, should contact me that appropriate arrangements may be made. Students with disabilities should also contact the Office of Student Disabilities Services (335-1462).

Student Complaints:

A student who has a complaint related to this course should follow the procedures summarized below. The full policy on student complaints is on-line in the College's Student Academic Handbook (http://www.clas.uiowa.edu/students/academic_handbook/ix.shtml)

Ordinarily, the student should attempt to resolve the matter with the instructor first. Students may talk first to someone other than the instructor (the departmental executive officer, or the University Ombudsperson) if they do not feel, for whatever reason, that they can directly approach the instructor.

If the complaint is not resolved to the student's satisfaction, the student should go to the departmental executive officer.

If the matter remains unresolved, the student may submit a written complaint to the associate dean for academic programs. The associate dean will attempt to resolve the complaint and, if necessary, may convene a special committee to recommend appropriate action. In any event, the associate dean will respond to the student in writing regarding the disposition of the complaint.

For any complaint that cannot be resolved through the mechanisms described above, please refer to the College's Student Academic Handbook for further information.