If you find a mistake, omission, etc., please let me know by e-mail.
The sections will meet MWF 10 and 11, and also MWF 12 with sufficient enrollment. The first sections will meet Monday, September 25.
Sectioning will begin Monday, Sep.18; please section by Wednesday, Sep.20 at 1PM. To section by computer:
Windows --> SecureCRT application --> Quickconnect, protocol ssh2, hostname ulam.fas.harvard.edu, username section
Mac OS X --> Applications --> Utilities --> Terminal: at the prompt, type ssh firstname.lastname@example.org and hit return.
NB The homework from 7.6 is now due Wednesday Nov.22 (since Nov.20 is the midterm date). The next few homeworks are:
Afterwards we'll follow a combination of some of the remaining sections of the textbook and ``chapter 10'', Prof. Taubes' notes on linear differential equations written over the last year or two for the end of Math 21b. Please also note these errata and comments in your printout.
Here are some practice exercises
on this material. Please ignore problems 11, 19, 20, 21, and 24,
which concern material we won't cover.
Please also the following typographical errors in the solutions:
in solution 29, the exponents
[Added Jan.13 and 21: E. Beh notes that Problem 31 is incorrect as stated, and that the printed solution indicates that the intended differential equation was
The remaining homework assignments for the semester are:
10-11 section: Elizabeth Denne [denne], Sci Ctr 507;
office hours: Sci Ctr 535, Monday 2-3, Tuesday 3-4, and Friday 1-2
11-12 section: Noam Elkies [elkies] (course head), Sci Ctr 309A; office hours: Sci Ctr 335, Thursday 2:45-4:15 and by appointment
12-1 section: Maksym Fedorchuk [fedorchuk], Sci Ctr 310; office hours: Sci Ctr 242B, Monday 11-12 and Thursday 5-6
Each "[username]" means the e-mail address username@math (from outside Harvard: email@example.com).
10-11 section: Heather Carmichael [carmich], Friday 3:30-5 PM, Sci Ctr 101B
11-12 section: Louis Kang [lkang], Monday 7:30-9 PM, Sci Ctr 216;
11-12 section: (William) Lin Cong [lincong], Thursday 6-7:30 PM, Sci Ctr 116;
12-1 section: Nike Sun [nsun], moved to Thursday 4-5:30 PM, Sci Ctr 103B.
Each "[username]" means the e-mail address username@fas (from outside Harvard: firstname.lastname@example.org).
If you've been to the Math Question Center (MQC), we'd like to hear what you think about it!
Please take a few minutes to fill out a quick survey so we can make the MQC even better!
Some practice problems for the first midterm (taken from last year) are here. The actual exam will have 3 multi-part problems, each worth 10 points for a complete solution (including showing all work), and 10 True/False questions, worth one point each (with no requirement for justifying answers).
Thanks to the CA's for quickly putting together answers for the practice problems (1,2, 3,4,5)!
Here are the answers to problems 1, 2, 3, 4 of the first midterm exam.
The actual exam will have 4 multi-part problems (but with roughly the same total number of parts as on the first midterm), each worth 10 points for a complete solution (including showing all work), and 10 True/False questions, worth one point each (with no requirement for justifying answers).
Here are the answers to problems 1, 2, 3, 4, 5 of the second midterm exam.
The review sessions for the final exam will take place during the second week of reading period in Science Center Room 507: Monday January 8 from 11AM to 1PM, and Friday January 12 from 10AM to 12 Noon.
Here are copies of recent final exams in Math 21b:
(with solutions at the end(*)),
and January 2006 (problems only).
Note that there are a handful of problems on material
(usually from 10.5 and 10.6) that is not on the syllabus for
this year's exam.
(*) Corrections: in the 2004 exam, T/F question 7b, the bottom two entries of A were evidently switched, so the problem as stated is FALSE rather than the intended TRUE [thanks to Eugene Beh]; in the 2004 solutions, Question 6, parts b and c should have factors of sin(2x) and cos(2x) instead of sin(x) and cos(x) [thanks to Eric Petersen]; in the 2005 solutions, Question 6, part b should have cos(2t)+sin(t) instead of cos(t)+sin(t) as the last basis element [thanks again to Eugene Beh]. (In question 6c, if you listed the natural basis elements of 6a in a different order, then your answer to 6c will look different though the two 4*4 matrices must be similar.)
Brief solutions for the homework from the ``Chapter 10'' handout are here (remember that you're not responsible for 8.3 on the final); thanks to the CA's for sending these.