MATHEMATICAL METHODS I

Course Page

PHYS 508, Fall 2008

10:30 to 11:50 Monday, Wednesday, Room 144 Loomis Laboratory.

This web-page contains links to documents such as handouts and other useful stuff. These files are now only in PDF (.pdf) format.

Syllabus Outline

The course covers four related areas:

Calculus of Variations. Equations of mathematical physics as variational problems, conservation laws, Lagrange multipliers, origin of eigenproblems, variational approximation schemes.
Ordinary differential equations. Linear equations: Solution space, linear independence, Wronskians, normal forms. Eigenvalue problems: importance of boundary conditions, formal and true self-adjointness, completeness of eigenfunctions, Fourier series, continuous spectra and Fourier integrals. Green Functions: Range-nullspace theorem, Fredholm alternative, constructing Green functions via jump conditions.
Partial Differential equations. Classification of PDE's. Hyperbolic equations: wave equation, method of characteristics, shocks and weak solutions. Heat equation: solution by integral transforms. Elliptic equations: Dirichlet and Neumann problems, Poisson's equation, Legendre functions, spherical harmonics, Bessel and spherical Bessel functions, examples from electrostatics.
Integral Equations. Type I and type II Fredholm and Volterra equations, solution via Fourier and Laplace transforms, Abel's equation. Separable Kernels: compact and Hilbert-Schmidt operators, Fredholm alternative again. Perturbation methods: Neumann and Fredholm series.

A more detailed, but approximate, week-by-week syllabus can be found here.

Homework Sets

There are weekly homework sets. Your solutions must be deposited in the Phys 508 box before 4pm on the due date, which will always be a Thursday. This time has been chosen to encourage you to go to the Physics Department Colloquia which are at 4pm each Thursday. Attending the Colloquium is an important part of your broader physics education.

Homework number 0, due Thursday Sept 4th.
Solutions 0

Homework number 1, due Thursday Sept 11th.
Solutions 1

Homework number 2, due Thursday Sept 18th.
Solutions 2

Homework number 3, due Thursday Sept 25th.
Solutions 3

Homework number 4, due Thursday Oct 2nd.
Solutions 4

Homework number 5, due Thursday Oct 16th.
Solutions 5

Homework number 6, due Thursday Oct 23rd.
Solutions 6

Homework number 7, due Thursday Oct 30th.
Solutions 7

Homework number 8, due Thursday Nov 6th.
Solutions 8

Homework number 9, due Thursday Nov 13th.
Solutions 9

Homework number 10, due Thursday Nov 20th.
Solutions 10

Homework number 11, optional problems.

Exams

There will be a midterm exam sometime in early November.

The final exam will be on the morning of Friday Dec 19th --- 8am to 11am. This is late in the exam week, so make sure to take this into account in your travel plans.

Old Exams:

Midterm Exam, Fall 2002
Midterm Exam, Fall 2003
Final Exam, Fall 2003

Lecture notes

The lecture notes are a fairly accurate representation of the course as given in the classroom. They are no longer password protected, and so are available to the general public.

Loyal readers will note that this new edition of the online "notes" now has Paul Goldbart listed as a co-author. Paul has in fact been a secret co-author for several years. When I first taught this class I based my course outline on a set of notes that Paul had prepared for his own version of 598MMA. My previous online notes retained this outline, but substantially expanded the content. For this new edition, Paul has gone over my added text in detail, improving the pedagogy and clarity of its frequently all-too-terse exposition. The fruit of our joint labours is to be published by Cambridge University Press.

We would like to thank our past Math Methods I students for all their help in fixing typos and warning us of obscurities. There must still be some undiscovered. Please feel free to contact me if you find errors.

Textbook

Although the lecture notes should be self contained, I recommend Mathematics for Physicists by Phillipe Dennery and Andre Krzywicki (Dover Pulications, $12.95) as an alternative textbook. It covers much of the material in this course, and will be useful for the complex-variable part of PHYS 509.

Grades and Gradebook

Registered students may access the on-line gradebook by using your university username and password. You will need to accept cookies, and have JavaScript turned on for the gradebook to work.

Your grade in the course will be determined as from your total scores weighted as follows: Homework 60%, Midterm exam 10%, Final Exam 30%.

Cultural Enrichment Links

Some of the material in the course is supposed to introduce you to the wider culture of mathematical physics and its applications in the real world. Here are links relating to some of the topics discussed:

People A Short Biography of George Green can be found here. This site also contains the biographies of many other mathematicians and mathematical physicists.
Wave Phenomena Images of Hydraulic Jumps and Bores can be found here. Images and description of the Kelvin ship wake can be found here and here
Solitons A nonlinear pulse obeying the KdV equation does not form a shock, but instead decays into solitions. Here is a movie of this phenomenon. It is taken from the ``soliton lab'' created by Kanehisa Takasaki at the univerity of Kyoto.

Staff

Finding me:
Office: 2117 ESB.
Phone: 3-2891.
e-mail: m-stone5@uiuc.edu
My office hour is Thursday 8-9am, in 2117 ESB.

Grader:

Xianhoa Xin
Office: 390U (Loomis-Seitz-Interpass)
Phone: 3-3053
xin2@uiuc.edu
Office Hour: Monday 1-2pm in 390U.

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Last updated 16/09/08