MATHEMATICAL METHODS I

Course Page

PHYS 508, Fall 2009

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 3rd.
Solutions 0

Homework number 1, due Thursday Sept 10th.
Solutions 1

Homework number 2, due Thursday Sept 17th.
Solutions 2

Homework number 3, due Thursday Sept 24th.
Solutions 3

Homework number 4, due Thursday Oct 8th.
Solutions 4

Homework number 5, due Thursday Oct 15th.
Solutions 5

Homework number 6, due Thursday Oct 22rd.
Solutions 6

Homework number 7, due Thursday Nov 12th.
Solutions 7

Homework number 8, due Thursday Nov 19th.
Solutions 8

Homework number 9, due Thursday Dec 3rd.
Solutions 9

Homework number 10, due Thursday Dec 10th.
Solutions 10

Homework number 11, optional problems.

Exams

There will be a midterm exam in class sometime in November. It will be closed book, and will cover everything up to (but not including) partial differential equations.

Ihe final exam for the fall 2009 semseter is scheduled for 8-11AM monday Dec 14th. It will be a closed book exam, and will be held in the usual lecture room: 144LLP

Old Exams:

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

Textbook

I recommend (but do not require) that you purchase Mathematics for Physics: A guided tour for graduate students by myself and Paul Goldbart. (Cambridge University Press 2009). The list price is $90, but Amazon has it for $72 (+shipping). I do not yet know what the UI bookstore is selling it for. This beautifully produced book is an expanded version of the lecture notes for both PHYS 508 and PHYS 509.

As a cheaper alternative you can buy Mathematics for Physicists by Phillipe Dennery and Andre Krzywicki (Dover Pulications, $12.95) as an alternative textbook. It covers a fair bit of the material in this course, and will be useful for the complex-variable part of PHYS 509.

A set of online lecture notes is still available for download, but, now that the book is published, I am no longer maintaining them, so typos are not being corrected.

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, outside 2117 ESB.

Graders:

Min Yu
Office: 212D MSEB
Phone: 3-3053
minyu2@illinois.edu
Office Hour: Wednesday 5:00-6:00pm in 212D.

Xianhoa Xin
Office: 390U (Loomis-Seitz-Interpass)
Phone: 3-3053
xin2@uiuc.edu
Office Hour: Monday 2:30-3:30pm in 390U.

Back to Mike's Home Page

Last updated 16/09/08