Math 335, Accelerated (Honors) Advanced Calculus, Winter, 2010

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. (3/10/10) Science and Music by James Jeans is a good introduction to the subject.
  2. (3/10/10) Sample problems for the final.
  3. (3/8/10) Now is the time to look at: (1/15/10) A primer on differential equations.
  4. (3/8/10) A general point-wise convergence theorem.
  5. (3/4/10) I answered Harry's question incorrectly. Here is a correct response.
  6. (3/3/10) Cauchy and limits.
  7. (3/3/10) More Fourier facts
  8. (3/3/10) Facts about sine and cosine series.
  9. (2/19/10) My statement of Stirling's formula for n! was correct. I misstated the formula for the gamma function. It should be
    xΓ(x)/((2 π x)1/2(x/e)x) → 1 as x → ∞.
  10. (2/18/10) How Newton made his famous discoveries: By always thinking about them. I keep the subject constantly before me and wait til the first dawnings open little by little into full light.
  11. (2/16/10) Sample problems for the second midterm
  12. (2/10/10) Binomial series.
  13. (2/5/10) A sentence from a letter by C. G. J. Jacobi to A. von Humboldt, If Gauss says he has proved something, it seems very probable to me; if Cauchy says so, it is about as likely as not; if Dirichlet says so, it is certain.
  14. (1/30/10) Infinite sums, some notes on summing possible infinite sets of numbers.
  15. (1/26/10) Thomson 331 is reserved Friday, January 30, from 2:30 to 5:00, and Padelford C-401 is reserved Saturday, January 31, from 10:00 to 3:00 for review. Chad plans to be at both of these.
  16. (1/26/10) Sample problems for the first midterm.
  17. (1/26/10) Rearranging alternating harmonic series.
  18. (1/20/10) MAC talk:"Capitalist Investment and Political Liberalization" by Roger Myerson; Friday, January 22, Kane 210. Math Across Campus link.
  19. (1/20/10) Touring the Calculus Gallery by William Dunham.
  20. (1/15/10) A simple discussion of the fundamental solution of a constant coefficient linear differential equation.
  21. (1/15/10) A primer on differential equations.
  22. (1/14/10) A direct, simple, proof of a convergence theorem for improper integrals.
  23. (1/06/09) Problem 5.8.4b is not precisely stated. Assume that what is meant is that C is a piecewise smooth simple closed curve that is the boundary of an open connected set that contains the origin.
  24. (1/05/09) A simple, quick, introduction to differential forms is Differential Forms by Harley Flanders which is available as a Dover reprint.
  25. (12/31/09) Change of variables formula for spherical coordinates.
  26. (12/31/09) Spherical Coordinates
  27. (12/31/09) Benford's Law. An application of Benford's law.
  28. (12/31/09) The book A=B.
  29. (12/31/09) Here's a proof of the Cauchy-Binet formula, which has a nice application to give a formula for the measure of a parameterized manifold in Rn.
  30. (12/31/09) Smith's Prize exams. Look at 1854, #8.
  31. (12/31/09) A proof of the Poincare Lemma.
  32. (12/31/09) 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.
  33. (1/20/10) The Marquis and the Land Agent by G. N. Watson, The Mathematical Gazette, Vol. 17, No. 222 (Feb., 1933), pp. 5-17
  34. (12/31/09) 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.
  35. (12/31/09) 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.
  36. (12/31/09) In 1926 Kolmogorov gave an example of an L1 function whose Fourier series diverges everywhere. An exposition is in Bari, "Treatise on Trigonometric Series".
  37. (12/31/09) An article on the Gamma function.
  38. (12/31/09) Harold Edwards' book is Riemann's Zeta Function.
  39. (12/31/09) William Dunham's book is Euler: The Master of Us All.
  40. (12/31/09) 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.
  41. (12/31/09) Dave Duncan's thesis on the Kakeya Problem.
  42. (12/31/09) I will not cover Raabe's test and I will not ask you to work any problems using it.
  43. (12/31/09) 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.
  44. (12/31/09) 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.
  45. (12/31/09) Lord Kelvin and the Age of the Earth by Joe Burchfield is a terrific book on how math gets used in science.
  46. (12/31/09) I need lots of Mathday volunteers. Please consider helping. You can see the program at Mathday.
  47. (12/31/09) The book The Pleasures of Counting by Thomas Korner is very entertaining.
  48. (12/31/09) 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.
  49. (12/31/09) 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
  50. (1/20/10) The Banach-Tarski paradox.
  51. (12/31/09) 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)
  52. (12/31/09) For problem number 3 in section 5.8, assume that Laplacian(f)=div(H) has a solution. You don't need to justify this.
  53. (1/20/10) On the Convergence of Fourier Series is an article with an alternate (and pretty) discussion of some of the results we will discuss.
  54. (1/19/10) Fourier Series of Polygons
  55. (12/31/09) The AMS has two popular links, Math in the Media and a monthly Feature Column.
  56. (1/19/10)Rearranging Conditionally Convergent Series.
  57. (1/19/10) Creating more convergent series, an article about rearranging terms in a series.
  58. (12/31/09) 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.
  59. (12/31/09) Make sure you check Jerry Folland's website for misprints.
  60. (1/02/10) Syllabus(pdf)

morrow@math.washington.edu