Math 335, Accelerated (Honors) Advanced Calculus, Winter, 2009

This is the Math 335 homepage. Consult it from time to time to find useful information for the course. I will include links to the syllabus and other course information.

The following are links to current course information.

  1. (3/12/09) Paul Halmos's expostion of the Spectral Theorem for bounded self-adjoint operators on a Hilbert space. (Not good enough for applications to quantum mechanics.) Here's an Unbounded Operator version.
  2. (3/12/09) The review sessions will be in Padelford C-401 from 10:00--1:00 on both Saturday and Sunday. Nate will be there on Saturday.
  3. (3/11/09) The Heat kernel.
  4. (3/10/09) Sample problems for the final exam.
  5. (3/09/09) Facts about certain sine and cosine series.
  6. (3/04/09) English translation of Fourier's book Analytical Theory of Heat. Highly recommeded. Very well written; explains Fourier's motivation, lots of detailed examples, and shows how he eventually realized that the coefficients are found by integrating: p. 185, "we arrive at a very remarkable result ..." and he proceeds to write down the integral form for the coefficients.
  7. (3/03/09) Nate's office hours this week will be 4:00--5:00 Wednesday, 4:30--5:30 Thursday.
  8. (3/03/09) Gerver's paper on Riemann's "differentiable nowhere" function; and more; and more.
  9. (2/25/09) One of Dirichlet's theorems on convergence of Fourier series applies to functions that are bounded and monotonic on a finite number of intervals (functions that can be drawn). The arcsine is such a function. I am not going to prove this convergence theorem in class.
  10. (2/25/09) The product of two Riemann integrable functions is Riemann integrable.
  11. (2/25/09) Proof of the Riemann-Lebesgue Lemma.
  12. (2/19/09) The review sessions will be in Padelford C401, Saturday and Sunday, 11:00-2:00. Nate will be there on Saturday.
  13. (2/19/09) Abel's lemmas
  14. (2/18/09) A proof of the Bohr-Mollerup theorem can be found in Theory of Functions by C. Caratheodory (it is not in Ahlfors).
  15. (2/18/09) Sample problems for the second midterm.
  16. (2/11/09) Differential equations
  17. (2/05/09) Absolute and Uniform Covergence. Uniform Convergence and absolute convergence does not imply that the absolute values converge uniformly.
  18. (1/30/08) Summing the alternating harmonic series.
  19. (1/29/09) Review sessions for the midterm will be held in C401 Saturday and Sunday from 11:00--2:00. Nate will be there on Saturday.
  20. (1/27/09) Sample problems for the first midterm.
  21. (1/23/09) Cauchy and calculus by Judith Grabiner.
  22. (1/23/09) Nate's Thursday office hour will now be from 5-6 pm.
  23. (1/21/09) Mathematics Digital Library
  24. (1/21/09) Math Across Campus lecture at 3:30 on January 22 in Kane 220. Combinatorial Optimization in Action; Martin Grotschel.
  25. (1/20/09) Change of variables formula for spherical coordinates.
  26. (1/15/09) Spherical Coordinates
  27. (1/08/09) Benford's Law. An application of Benford's law (corrected 1/12/09).
  28. (1/07/09) Math is Best.
  29. (1/07/09) The book A=B.
  30. (1/06/09) Here's a proof of the Cauchy-Binet formula, which has a nice application to give a formula for the measure of a parametrized manifold in Rn.
  31. (1/06/09) Nate's office hours will be: M 3:30--4:30, W 5:00--6:00, Th 1:30--2:30.
  32. (1/06/09) Thanks to Dylan I've corrected an error in the note on the Poincare lemma. I should have stated that on any domain an exact form is closed.
  33. (1/05/09) Smith's Prize exams. Look at 1854, #8.
  34. (1/05/09) A proof of the Poincare Lemma.
  35. (1/02/09) I fixed a misprint in the syllabus. The class meets TTh in Padelford C36.
  36. (12/23/08) Special issue of the Notices of the AMS on formal proof.
  37. (12/17/09) If you send a message to mathlib@u.washington.edu you can get on the mailing list to get the weekly newsletter from the math library. It has all of the new acquisitions of the week with links to them. It's an excellent way to keep up with the latest publications.
  38. (12/17/08) An entertaining classic article: The Marquis and the Land-Agent.
  39. (12/17/08) A recent article about Euler (including a video).
  40. (12/17/08) There is a scattered discussion of Weierstrass's non-differentiable function in Fourier Analysis : An Introduction by Elias M. Stein & Rami Shakarchi. Hardy's discussion of Weierstrass's non-differentiable function.
  41. (12/17/08) In 1966, Lennart Carleson proved that the Fourier series of an L2 function converges almost everywhere. At that time it was the outstanding problem in Fourier analysis. He won the 2006 Abel Prize for this theorem. His proof has not been greatly simplified. The best current version is in Michael Lacey's paper.
  42. (12/17/08) In 1926 Kolmogorov gave an example of an L1 function whose Fourier series diverges everywhere. An exposition is in Bari, "Treatise on Trigonometric Series".
  43. (12/17/08) An article on the Gamma function.
  44. (12/17/08) Harold Edwards' book is Riemann's Zeta Zunction.
  45. (12/17/08) William Dunham's book is Euler: The Master of Us All.
  46. (12/17/08) I think it is correct that the notation for the Gamma function is due to Legendre and Pi(x) is Gauss's notation for Gamma(x+1). A reference is Gamma Function by Askey and Roy. Askey is THE expert on special functions.
  47. (12/17/08) Interesting article: Nineteen proofs of Euler's formula: V_E+F=2.
  48. (12/17/08) Dave Duncan's thesis on the Kakeya Problem.
  49. (12/17/08) I will not cover Raabe's test and I will not ask you to work any problems using it.
  50. (12/17/08) Starter books on manifolds and Stokes's theorem: Loomis and Sternberg, Advanced Calculus; Hubbard and Hubbard, Vector Calculus, Linear Algebra, and Differential Forms; Flanders, Differential Forms.
  51. (12/17/08) In problem #3, section 5.7, the curve should be oriented in the counter-clockwise direction when viewed from high above the x-y plane.
  52. (12/17/08) Lord Kelvin and the Age of the Earth by Joe Burchfield is a terrific book on how math gets used in science.
  53. (12/17/08) I need lots of Mathday volunteers. Please consider helping. You can see the program at Mathday.
  54. (12/17/08) The book The Pleasures of Counting by Thomas Korner is very entertaining.
  55. (12/17/08) Trigonometric Series by A. Zygmund and Fourier Analysis by T. Korner are superb references. Zygmund's book is a nearly complete reference for theoretical results. Korner's book has a broad collection of uses of Fourier analysis. Korner's book would be a good place to start to find material for your term paper for 336. It is readable and written for students that are at your level.
  56. (12/17/08) Two interesting books are Inequalities, by G. H. Hardy, J. E. Littlewood,and G. Pólya, and Pi And The AGM : A Study In Analytic Number Theory And Computational Complexity by Jonathan M. Borwein and Peter B. Borwein
  57. (12/17/08) The Banach-Tarski paradox
  58. (12/17/08) There is an error in the answer to problem 2b in section 5.8. The answer should be (xz2/2, -xyz-z2/2-x2/2, 0)+grad(f)
  59. (12/17/08) For problem number 3 in section 5.8, assume that Laplacian(f)=div(H) has a solution. You don't need to justify this.
  60. (12/17/08)On the Convergence of Fourier Series is an article with an alternate (and pretty) discussion of some of the results we will discuss.
  61. (12/17/08) An article on Fourier Series of Polygons
  62. (12/17/08) The AMS has two popular links, Math in the Media and a monthly Feature Column.
  63. (12/17/08) An article on Cantor's ternary function. It gives a brief introduction to some ideas of measure theory.
  64. (12/17/08) Rearranging Conditionally Convergent Series
  65. (12/17/08) Creating More Convergent Series, an article about rearranging terms in a series.
  66. (12/17/08) The 1854 Smith Prize Exam at Cambridge University that Stokes wrote can be found in the Michigan online library. The Smith Exams are in the last volume and this exam is on page 320. Apparently William Thomson (Lord Kelvin) stated the result to Stokes in a letter in 1850. James Clerk Maxwell won the Smith Prize in 1854 and Gabriel Stokes himself won it in 1841 and Thomson in 1845. Other winners are Arthur Cayley (1842), G.H. Hardy (1901), Arthur Eddington (1907), Alan Turing (1936). A history of the prize.
  67. (12/17/08) Make sure you check Jerry Folland's website for misprints.
  68. (12/17/08) Syllabus(pdf) (coming soon)

morrow@math.washington.edu