Mathematical Methods of Physics I
PHYS 6124     Fall semester 2010

August 24
1. Complex variables
Reading: Chapter 6 - analytic functions, Cauchy-Riemann conditions, mappings, Riemann surfaces, conformal mappings.

August 26
2. Complex variables
Reading: Chapter 6 - EM Laplace equation with hyperbola walls, complex integrals, Cauchy residue theorem.
Problem set #1: 6.1.20 Exponent vs. log, 6.3.3 Non-analytic integration, 6.4.1 Kronecker delta, 6.4.3 Evaluate a contour integral. Optional: 6.1.15 Cute identities.

August 31
3. Calculus of residues
Reading: Chapter 6 - Laurent expansion; chapter 7 - residues

September 2
4. Integration in complex plane
Problem set #2: 7.2.4 Step function, 7.2.15 Atomic collisions integral, 7.2.21 Integral over 1/(1+x^n) . Optional: 7.2.16 An integral over ln x .
solutions

September 7
5. Branch cuts
Optional reading for this course: M. Stone and P. Goldbart, Mathematics for Physics (Cambridge University Press, Cambridge 2004), offers a very engaging, physics focused approach. A pre-publication draft can be found here. Todays example was taken from Sect. 18.1.2 Branch-cut integrals, also example 7.1.6 in A & W.

September 9
6. Asymptotic evaluation of integrals
Reading: Chapter 7, section 7.3 - Jensen's theorem (most popular Danish family names are 1. Jensen 303,089 2. Nielsen 296,850 3. Hansen 248,968, out of 5.5 million Danes), saddle point method.
Problem set #3: 7.3.2 Asymptotic expansion of Fresnel integrals, 7.3.4 Stirling formula

September 14
7. The Gamma, Airy function estimates
Reading: Bits of chapter 10, lecture notes.

September 16
8. Fourier series
Problem set #4: 14.1.1 Fourier series accuracy, 14.1.2 Phase shifted representation. Optional: 14.1.3 convergence.

September 17
For something completely different
3pm graduate seminar in L5, P. Cvitanović:
"Got turbulence? tachycardia? and what to do about it"

September 21
9. Gibbs phenomenon
Reading: Arfken and Weber 6th Edition - Section 14.5

September 23
10. Discrete Fourier transform
Reading: Arfken and Weber 6th Edition - Section 14.6
Optional reading (today's lecture): ChaosBook.org - sections H.3-H.5
Problem set #5: 14.3.4 Triangular wave, 14.3.12 (14.3.9 in the previous edition) Dirac delta function. Optional: 6.1.7 (6.1.6 in the previous edition) Finite cos or sin sums.

September 28
11. Fourier transform

September 30
12. Linear operators and matrices
Problem set #6: 15.3.4 (15.2.4 in the previous edition, formulation clearer in edition 6) Fourier transform of exp(-at), 15.3.16 (15.5.3 in the previous edition) Fourier transform of 1/k^2.
Optional: 15.3.1 (15.2.1 in the previous edition) Reality of f(x), 15.4.1 (not given in the previous edition) Neutron diffusion.

October 5
13. Eigenvalue problems: Moment of inertia tensor

October 7
14. Penrose notation for Levi-Civita tensors, determinants
Our competition: MIT 18.085 Computational Science and Engineering I
Problem set #7: NOTE - for those without friends who own the current edition, problems have links:
3.2.9 (3.2.6 in the previous edition) Jacobi identity,
3.5.1
Moment of inertia tensor,
3.5.34
(not given in the previous edition) Spectral decomposition for a Hermitian matrix.
Optional: 3.2.11 (not given in the previous edition) Quaternions,
3.5.11 (3.5.10 in the previous edition) Determinant of the moment of inertia tensor.

October 12
15. Properties of eigenvectors and eigenvalues
This is the cover of the current edition; if your cover looks like this, you got the previous edition, with all problems renumbered. Tough luck.

October 14
16. Normal modes
Problem set #8: 3.4.12 U= exp(H), 3.6.3 (3.5.20 in the previous edition) [2x2] matrix secular equation, 3.6.20 (3.5.30 in the previous edition) Normal modes, two masses beteen walls.
Optional: 3.6.17 (3.5.27 in the previous edition) eigenvalues, eigenvectors of B= exp(A), 3.6.19 (3.5.29 in the previous edition) `Rayleigh-Ritz' estimate of the 'ground state' eigenvalue.

October 16-19
Mid-term recess

October 21
17. Matrices and ODEs
Problem set #9 (sheet says #7): assignment (problem 2 is optional), solutions

October 26
18. Group theory
Reading: Sections 4.1, 4.2 up to Rotation Groups SO(2) and SO(3) subsection, and 4.7.
Lecture notes: ChaosBook.org - "World in a mirror". Skip sections 9.3 and 9.4 except for example 9.12 3-disk game of pinball.
Optional fun reading: Penrose and Tinkham

October 28
19. Group theory
Problem set #10: 4.1.5 (4.1.4 in the previous edition) Matrix reps of group actions, 4.2.4 (4.2.3 in the previous edition) Translation operator, 4.7.22 (not given in the previous edition) D_3 classes.

November 2
20. Perturbation theory for eigenvalue problems

November 4
21. WKB theory
Problem set #11: assignment, solution

November 9 - lecture by R. Grigoriev
22. Perturbation theory for differential equations

November 11
23. Separation of variables in PDEs
Reading: Sections 9.1, 9.2 and 9.3 (8.1, 8.8, 8.9, 16.1 in the previous edition)
Problem set #12: problems 1 and 2 [solutions]; optional 9.3.6 (16.2.5 in the previous edition) Heat conduction in a spherical solid.

November 16
24. Boundary value problem
Reading: Sections 10.1, 10.2 and 10.4 (9.1, 9.2 and 9.4 in the previous edition)

November 18 - lecture by R. Grigoriev
25. Sturm-Liouville problem
Problem set #13 (given by R. Grigoriev in class), due Dec 2: Compute the solution for the heat equation in an evaporating liquid wedge.

November 23
26. Rayleigh-Ritz method
Reading: Section 17.8 (Section 18.6, other bits of chapters 18 and 16 in the previous edition).
Reading: Stone and Goldbart, sect. 1.5 and "Rayleigh-Ritz and completness" pp. 130-132.
Totally optional Thanksgiving reading: Entire Stone and Goldbart chapter 1 is fun.
Problem set #14, due Dec 2: 17.8.7 (18.6.3 in the previous edition) The largest eigenvalue of a Hermitian matrix. Optional problems to play catchup for the missed earlier sets: 17.8.2 (not in the previous edition) A dumb way to find the lowest eigenvalue of harmonic oscilator. Any two problems from Stone and Goldbart section 1.7 (10 points each).

November 25
Thanksgiving

November 30
27. Green's function for ODEs
Reading: Sections 9.7 and 10.5 (16.3 in the previous edition)

December 2
28. Calculus of variations
Reading: Stone and Goldbart, Sects 1.1 - 1.3 up to 1.3.2 Noether's theorem.
Problem set #15: 9.7.5 Hemholtz `bilinear' Green's fct (16.3.5 in the previous edition), 10.5.10 1d Hemholtz Green's fct (not in the previous edition). Optional problems 9.7.16 Hemholtz Green's fct (16.3.16 in the previous edition), 10.5.1 Green's fct of linear form (not in the previous edition).

December 7
29. Quantum propagator, quantum spectrum
Lecture notes: ChaosBook.org - "Quantum mechanics: the short short introduction"