MATHEMATICAL METHODS II

The course will have three related components:

• Vectors, Tensors, Differential Geometry and Topology. Covariant and contravariant vectors and tensors. Vector fields: Lie bracket and integrability. Differential forms: exterior differentiation, Poincare's theorem, integration of p-forms, Stokes' theorem. Introduction to topology: homology and cohomology.
• Groups and Group Representations. Elementary group theory: subgroups, cosets and conjugacy classes. Representations of finite groups: characters and orthogonality. Physical applications: molecular vibrations, quantum mechanics. Lie groups and Lie algebras: semisimiple and compact groups, representations, roots and weights.
• Complex Analysis. Complex differentiability. Conformal mapping and its physical applications. Cauchy, Taylor, and Laurent theorems, smooth vs. analytic functions. Applications to contour integration, solution of differential equations and asymptotics.

Below are links to various course-related documents. These files will now be in PDF format only

Homework Sets

Homework number 0, due 4pm Jan 29th. , ( Yun's solutions. )
Homework number 1, due 4pm Feb 5th. , ( Yun's solutions. )
Homework number 2, due 4pm Feb 12th, ( Yun's solutions. )
Homework number 3, due 4pm Feb 19th. ( Yun's solutions. )
Homework number 4, due 4pm Feb 26th. ( Yun's solutions. )
Homework number 5, due 4pm March 5th. ( Yun's solutions. )
Homework number 6, due 4pm March 19th. ( Yun's solutions. )
--- Spring Break ---
Homework number 7, due 4pm April 9th.
Homework number 8, due 4pm April 16th.
Homework number 9, due 4pm April 30th.
Homework number 10, due 4pm May 7th.

Lecture notes

The lecture notes are available to the general public. Please give me feedback - tell me of typos, sign errors, obscurities, or plain conceptual errors.

Textbooks

I have recommended three books: Differential Forms with applications to the Physical Sciences by Harley Flanders, Group Theory by Morton Hammermesh, and Mathematics for Physicists by Phillipe Dennery and Andre Krzywicki. All three are published by Dover Publications, and should cost no more than $15 or so each.

Grades

You may access the gradebook by using your UIUC net-id and password.

Grades for the part of the course will, as for MMA, be based on you total score on the homework (60%), and a midterm (10%) and final exam (30%).

Exams

The final exam will be held on Monday May 11th, from 1:30 to 4:30pm.

Here is an old midterm exam for you to look at.

People who figure in our story:

Elie Cartan
Cauchy
De Rham
Frobenius
H. Hopf
W. Killing
Laurent
Sophus Lie
Riemann
Issai Schur
Stokes
Hermann Weyl

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

Grader:

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Last updated 28/01/2009