Math 335, Accelerated (Honors) Advanced Calculus, Winter, 2011

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.

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The following are links to current course information.

1. (3/9/11) Sample problems for the final.
2. (3/3/11) Abel's theorem on Fourier series.
3. (2/25/11) Proof of the Riemann-Lebesgue Lemma.
4. (2/17/11) Sample problems for the second midterm. I've changed this version from the one posted earlier.
5. (2/14/11) Uniform version of Abel's test
6. (2/10/11) A summary of facts about power series.
7. (2/8/11) A short note on compactness.
8. (2/6/11) A note on double series.
9. (2/9/11) A note on Abel's test on series (corrected on 2/9/11).
10. (2/1/11) Don't forget about Mathday, March 21. I still need lots of help.
11. (1/31/11) Dirichlet's test, a better discussion than I gave in class.
12. (1/20/11) Sample problems for the first midterm.
13. (12/31/10) A general point-wise convergence theorem.
14. (12/31/10) 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.
15. (12/31/10) Binomial series.
16. (12/31/10) 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.
17. (1/30/10) Infinite sums, some notes on summing possible infinite sets of numbers.
18. (12/31/10) A simple discussion of the fundamental solution of a constant coefficient linear differential equation.
19. (12/31/10) A primer on differential equations.
20. (12/31/10) A direct, simple, proof of a convergence theorem for improper integrals.
21. (12/31/10) 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.
22. (12/31/10) A simple, quick, introduction to differential forms is Differential Forms by Harley Flanders which is available as a Dover reprint.
23. (12/31/10) Change of variables formula for spherical coordinates.
24. (12/31/10) Spherical Coordinates
25. (12/31/10) 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.
26. (12/31/10) Smith's Prize exams. Look at 1854, #8.
27. (12/31/10) A proof of the Poincare Lemma. (Corrected on 1/18/11)
28. (12/31/10) The Marquis and the Land Agent by G. N. Watson, The Mathematical Gazette, Vol. 17, No. 222 (Feb., 1933), pp. 5-17
29. (12/31/10) I will not cover Raabe's test and I will not ask you to work any problems using it.
30. (12/31/10) 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.
31. (12/31/10) 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.
32. (12/31/10) 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)
33. (12/31/10) For problem number 3 in section 5.8, assume that Laplacian(f)=div(H) has a solution. You don't need to justify this.
34. (12/31/10) 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.
35. (12/31/010) Make sure you check Jerry Folland's website for misprints.
36. (12/31/10) Syllabus(pdf)