Math 556: Methods of Mathematical Physics, I

Text

Principles of Applied Mathematics: Transformation and Approximation Revised Updated edition (2000), by J. P. Keener. We cover Chapters 1-5, and 7.
The course aims to show what advanced mathematics can tell us about fundamental problems in physics and engineering

Course outline

Review of finite dimensional linear algebra, including minimax principle for eigenvalues
Metric spaces and orthogonal polynomial bases
Integral equations/compact operators, and their spectral theory
Differential/unbounded operators, and Green functions and eigenfunctions
Calculus of variations
Transform and spectral theory, including Fourier, Laplace and Z transforms

Prerequisites

Advanced calculus and ordinary differential equations (undergraduate).

Homework: 7 or 8 overall. Extensions are given by prior arrangement only. Your lowest homework score will be dropped.
I encourage students to work together on the homework. But you must write up your own solutions in your own words, so that you really learn the material before the tests. Please write on the first page of your homework the names of all people with whom you discussed it (this is an ethical matter: "give credit where credit is due").

In-class Test: Wednesday 5 October.

Take-home Test: Thursday 17 November - Friday 18 November.

Final exam: 1:30-4:30pm, Friday 16 December
You may not use books, notes, calculators or computers during the in-class test or the final exam.

Grading policy: There will be no makeup tests except for medical reasons or family emergencies. Vacation or leisure plans are never a valid excuse for missing a test or exam.
Your letter grade for the course will be based on your total score out of 500 points:

Final exam: 200 pts (40% of grade)
Test 1: 100 pts (20% of grade)
Test 2: 100 pts (20% of grade)
Homework: 100 pts (20% of grade)

How to Succeed: Always re-work your lecture notes before the next class, checking every step, filling in missing details and keeping a running list of definitions. Read the textbook as you re-work your notes. We cover only the most essential points in class, and the text has helpful additional comments; reading them will help you get the big picture. Before each class, ask me about the points you didn't follow in the last class.
Go over your HW solutions the day you get them back.

If you have any inquiries or concerns, please contact me by e-mail at laugesen@illinois.edu.

- Richard Laugesen