Math 335, Accelerated (Honors) Advanced Calculus, Winter, 2008

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. (3/12/08) The review session will be at 4:00 pm, Friday, March 14, in Padelford C401. C401 is also reserved for your use at 10 am on Sunday, March 15.
  2. (3/11/08) Sample problems for the final.
  3. (3/07/08) An entertaining classic article: The Marquis and the Land-Agent.
  4. (3/06/08) A recent article about Euler (including a video).
  5. (3/06/08) 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.
  6. (3/06/08) A brief summary of facts about Fourier analysis. I will probably add some items to this list.
  7. (2/27/08) 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.
  8. (2/27/08) In 1926 Kolmogorov gave an example of an L1 function whose Fourier series diverges everywhere. An exposition is in Bari, "Treatise on Trigonometric Series".
  9. (2/23/08) Here's the fix for Yonatan's question.
  10. (2/21/08) Abstract for Jerry Folland's course on the hydrogen atom.
  11. (2/21/08) I answered a question of Yonatan incorrectly and there were some other errors in my proof of Dirichlet's theorem. I'll fix them on Monday.
  12. (2/21/08) Talk today:

    We invite you to join us for our first lecture + pizza of Winter 2008. Emily Kirkman will be giving a talk entitled "Efficient Circular Planarity Testing" on Tuesday, Feb. 21st at 5:00 pm in PDL C-36. (Abstract below).

    ABSTRACT: Efficient Circular Planarity Testing In studying inverse problems of electrical networks at the UW REU, it has been determined that the underlying electrical network is recoverable if the graph is circular planar. Given a graph with designated boundary, it is circular planar if it can be drawn in a disc with the boundary nodes on the disc boundary, and all interior nodes inside the disc with no edge crossings. We present an algorithmic approach to determining in linear time whether a graph with boundary is circular planar, which will be demonstrated with an implementation in Sage. We will start with basic definitions so it should be accessible to any undergraduate student. This is a great talk to attend if you are interested in working on Sage or participating in the summer UW REU.

  13. (2/21/08) There is a talk Tuesday, Feb 26 in 102 Smith Hall that should appeal to you. Here is an abstract:

    Hilbert's 17th problem asked whether every non-negative polynomial over the reals can be written as a sum of squares of rational functions. This was proved in the affirmative by Artin in the 1930's initiating the field of real algebraic geometry which is concerned with finding real solutions to polynomial equations and inequalities. One of the cornerstones of this theory is the Positivstellensatz due to Stengle in 1974, which is the analog of Hilbert's Nullstellensatz over the reals. The most efficient current version of the positivstellensatz is due to Mihai Putinar who arrived at it via the theory of moments. This theorem has allowed all sorts of new techniques for polynomial optimization leading to many unexpected applications.

    This talk is an exposition of the history of this fascinating subject initiated by Hilbert that weaves analysis and algebra together and has then lead into applications. Mihai is a fantastic speaker and the talk itself is a repeat of an AMS plenary address.

  14. (2/21/08) An article on the Gamma function.
  15. (2/21/08) Harold Edwards' book is Riemann's Zeta Zunction.
  16. (2/21/08) William Dunham's book is Euler: The Master of Us All.
  17. (2/20/08) Here's the link to our mcm papers.
  18. (2/20/08) 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.
  19. (2/20/08) Here's a suggestion for next quarter's term paper: Nineteen proofs of Euler's formula: V_E+F=2.
  20. (2/20/08) The review session for the second midterm will be in Smith 309 at 5:00 pm on Thursday.
  21. (2/18/08) Sample problems for the second midterm.
  22. (2/12/08) Jeff will hold an office hour on Monday, February 18, in his office in Padelford at 4:30 pm.
  23. (2/12/08) Linear constant coefficient odes.
  24. (2/12/08) Sage Seminar: An Introduction to Symbolic Manipulation and Calculus in Sage; Padelford C-401; 4:00pm Thursday Feb. 21.
  25. (2/12/08) Jeff's office hour on Friday, February 15, will be at 5:00, this week only.
  26. (2/06/08) Homework from section 7.3 will be postponed to February 19.
  27. (2/04/08) The rearranged alternating harmonic series.
  28. (2/1/08) A note on Euler's constant. A link to an article with a more sophisticated computation.
  29. (1/30/08) Dave Duncan's thesis on the Kakeya Problem.
  30. (1/30/08) The review for the midterm will be on Thursday at 5:00 pm in Padelford C-36.
  31. (1/28/08) Sample problems for the first midterm.
  32. (1/23/08) I will not cover Raabe's test and I will not ask you to work any problems using it.
  33. (1/23/08) From now on, Jeff's Monday office hour will be held in Lowe 115 from 4:30 to 5:30.
  34. (1/18/08) Jeff's office hours are Mondays at 4:30 and Fridays at 3:30 and he is looking a bigger room for Monday, but the January 21 (even though it is a holiday) office hour will be in his office.
  35. (1/15/08) Here is a proof of the Poincare Lemma.
  36. (1/11/08) 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.
  37. (1/09/08) 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.
  38. (1/04/08) I will be out of town January 6-8. Bob Phelps, an internationally recognized analyst will give a lecture in the regular classroom on Monday, January 7, on Hilbert Space. Here is your chance to learn from an expert. Jeff will hold quiz section as usual on January 8.
  39. (1/04/08) Lord Kelvin and the Age of the Earth by Joe Burchfield is a terrific book on how math gets used in science.
  40. (1/04/08) I need lots of Mathday volunteers. Please consider helping. You can see the program at Mathday.
  41. (1/04/08) The book The Pleasures of Counting by Thomas Korner is very entertaining.
  42. (1/04/08) A proof of the Riemann-Lebesgue Lemma.
  43. (1/04/08) Here is a summary of interesting facts about sine series.
  44. (1/04/08) 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.
  45. (1/04/08) 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
  46. (1/04/08) The Banach-Tarski paradox
  47. (1/04/08) 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)
  48. (1/04/08) For problem number 3 in section 5.8, assume that Laplacian(f)=div(H) has a solution. You don't need to justify this.
  49. (1/04/08)On the Convergence of Fourier Series is an article with an alternate (and pretty) discussion of some of the results we will discuss.
  50. (1/04/08) An article on Fourier Series of Polygons
  51. (1/04/08) The AMS has two popular links, Math in the Media and a monthly Feature Column.
  52. (1/04/08) An article on Cantor's ternary function. It gives a brief introduction to some ideas of measure theory.
  53. (1/04/08) Rearranging Conditionally Convergent Series
  54. (1/04/08) Creating More Convergent Series, an article about rearranging terms in a series.
  55. (1/04/08) An interesting article on gravity.
  56. (1/04/08) 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.
  57. (1/04/08) Make sure you check Jerry Folland's website for misprints.
  58. (1/04/08) Syllabus(pdf)

morrow@math.washington.edu