If you find a mistake, omission, etc., please let me know by e-mail.
The sections will meet MWF 11, 12, and/or 1,
depending on the number and schedules of students sectioning.
(In particular, we cannot know yet which TF will be teaching
which section.) The first sections will meet
Sectioning will begin Monday, Sep.15; please section by Wednesday, Sep.17 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
Unix/Linux: log into your FAS account; at the shell prompt, type ssh email@example.com and hit return.
These sites will not close at the deadline (Sep.17 at 1PM), so you should use the same procedure to section if you've missed the deadline, and we'll try to accommodate you as best we can.
If you have another problem or question about sectioning
that the sectioning
or program does not answer, please email
11-12 section: David Geraghty [geraghty], Sci Ctr 222;
office hours: Math common room (Sci Ctr 4th floor), Tuesday 4:15-5:45
12-1 section: Noam Elkies [elkies] (course head), Sci Ctr 110; office hours: Sci Ctr 335, Thursday 2:45-4:15 and by appointment
1-2 section: Wushi Goldring [wushi], Sci Ctr B10 (B=basement); office hours:
Sci Ctr 425D, Monday 3:15-4:45
Sci Ctr 425D, Wednesday 4:15-5:45
Each "[username]" means the e-mail address username@math (from outside Harvard: firstname.lastname@example.org).
11-12 section: Jessica Ranucci [jranucci],
Monday, 8-9 PM in Sci Ctr 101B;
Beatrice Liem [bliem], Wednesday, 7:30-8:30 in Winthrop Dining Hall
12-1 section: John Tan [johntan], Tuesday, 9-10 PM in Sci Ctr 111
1-2 section: Jeremy Booher [jbooher], Thursday, 8-9 PM in Sci Ctr 103B
This time each "[username]" means the e-mail address email@example.com (if you're e-mailing from a FAS account then username by itself should work).
The CA boxes are in the third floor of the Science Center, next to the sectioning office (room 308); the 21b boxes are the bottom three in the rightmost column.
The first midterm will be in the same format(*), and on the same
(*) That is, 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).
Here are solutions for the second midterm.
The topic of class on Monday, Nov.24 will be nonlinear dynamical systems, following this handout (please correct these errata in your printout).
Here is the latest version of
Students requesting to take the exam on a different date/time/location, or requesting special accommodations, should contact one of the administrative offices below:The last regular class meeting is Monday, December 15. There will be two review sessions during reading period: Wednesday and Thursday, January 7 and 8, both from 1PM to 3PM in Room 507 of the Science Center. The first will concentrate on material covered since the second midterm, and the second on material already covered on midterms 1 and 2 (which of course you're also responsible for in the final exam).
- The Accessible Education Office for requests for extended testing time or alternate exam accommodations,
- Exams Office for conflicts with religious observation,
- The student's Resident Dean of Freshmen (for freshmen), Resident Dean (for upperclass students), or the FAS Deputy Registrar (for graduate and cross-registered students) for all other requests.
Here are copies of recent final exams in Math 21b:
(with solutions at the end(*)),
and January 2006
and January 2007
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.)
Here are some more practice
exercises on the
READING PERIOD OFFICE HOURS are either by appointment or at the following dedicated times:
David Geraghty: 11:00(AM)-12:30(PM), Saturday Jan.10, and
11:00(AM)-12:30(PM), Sunday Jan.11, in the math department common room
(Sci Ctr 4th floor)
Noam Elkies: 3:00-4:30, Thursday Jan.8, Sci Ctr 335; 12:00-1:00, Monday Jan.12, Sci Ctr 335
Wushi Goldring: 6:30-7:30, Tuesday Jan.6, Sci Ctr 425D; 4:30-6:00, Wednesday Jan.7, Sci Ctr 425D