Math 224a - Mathematical Methods for the Physical Sciences - Fall 2011

Announcements and Handouts

Reading for the week of 31 October-4 November: Section 7.4.

Problem Set 9 (due Thursday 17 November 2011): Problems 2, 5, 10, 11, 16, 17 and 18 on pages 345-352.

Reading for the week of 24-28 October: Sections 7.1 through 7.3.

Problem Set 8 (due Thursday 10 November 2011): Problems 1, 6, 8, 16, 20 and 23 on pages 311-318.

Reading for the week of 17-21 October: Chapter 6.

Reading for the week of 10-14 October: Chapter 5.

Problem Set 7 (due Thursday 27 October 2011): Problems 1, 3, 4, 5, 6 and 19 on pages 255-260.

Reading for the week of 3-7 October: Sections 4.1, 4.2 and 4.4.

Problem Set 6 (due Thursday 13 October 2011): Problems 2, 3, 8, 9, 13, 14 and 15 on pages 205-208. Solutions are here

Reading for the week of 26-30 September: Sections 3.5 and 3.6.

Problem Set 5 (due Thursday 6 October 2011): Problems 14, 18, 19, 20, 21 and 22 on pages 159-165. Solutions are here

Reading for the week of 19-23 September: Sections 3.3 and 3.4.

Problem Set 4 (due Thursday 29 September 2011): Problems 6, 7, 9, 12 and 13 on pages 159-165. Solutions are here

Reading for the week of 12-16 September: Sections 3.1 and 3.2.

Problem Set 3 (due Thursday 22 September 2011): Problems 1-5 on pages 159-165. For Problem 5, it helps to know that exercise 3.13 on page 122 is wrong: the formula should read i_v( \sigma \wedge \tau ) = i_v(\sigma) \wedge \tau + (-1)^p \sigma \wedge i_v(\tau) just like the formula for exterior derivative. It is also worth noting that given Problem 5, there is an easy proof of Problem 4. Solutions are here

Reading for the week of 5-9 September: Chapter 2.

Problem Set 2 (due Thursday 8 September 2011): Problems 4, 6, 9, 16, 23, 29 and 30 on pages 84-93. Solutions are here

Reading for the weeks of 25 August-2 September: Chapter 1.

Problem Set 1 (due Thursday 1 September 2011): Problems 6, 8, 11, 13, 18 and 20 on pages 31-36. Solutions are here

Lecture: Tuesday and Thursday 3:30-5:00 pm, 81 Evans

Course Home Page: http://math.berkeley.edu/~strain/224a.F11/index.html

Professor: J Strain, Office Hours Wednesday 3:00-4:00 and Thursday 12:00-1:00, 1099 Evans.

Catalog Description: Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green functions. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves.

Required Text: Michael T. Vaughn, Introduction to Mathematical Physics. Wiley-VCH, 2007.

Syllabus: The course will survey basic theory and practical methods for solving the fundamental problems of mathematical physics. It is intended for first-year graduate students in applied mathematics, physics, engineering or other mathematical sciences. The overall purpose of the course will be to develop non-numerical tools for understanding and approximating solutions of differential equations.

Convergent and asymptotic sequences, series and integrals (Chapter 1).

Linear algebra of finite-dimensional vector and inner product spaces: linear operators, adjoints, projections, spectral theory (Chapter 2).

Geometric methods: manifolds, tangent vectors, vector fields, calculus and dynamical systems on manifolds (Chapter 3).

Complex analysis: Cauchy-Riemann equations, integration, Cauchy theorem, residues (Chapter 4).

Classical differential equations of mathematical physics: Legendre, Bessel, hypergeometric (Chapter 5).

Functional analysis: Infinite-dimensional Hilbert spaces, Fourier analysis, orthogonal functions and wavelets (Chapter 6).

Linear operators on Hilbert spaces: Green functions, spectral theory and differential operators (Chapter 7).

Partial differential equations: Laplace, diffusion and wave equations. Nonlinear waves, KdV, sine-Gordon (Chapter 8).

Finite groups: symmetric groups, partitions, representations (Chapter 9).

Lie groups and their representations (Chapter 10).

Prerequisites: Math (110, 104, 185) or (121A, 121B), or equivalent background in mathematics.

Mechanics: Problem sets will be assigned each Tuesday and due the following Thursday. They will be graded and returned with solutions the following Tuesday. The course grade will be based on all but the lowest two problem sets and a final project. There will be no final exam or midterm.