If you find a mistake, omission, etc., please let me know by e-mail.

The

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
**Monday, September 22**.

**Sectioning**
will begin Monday, Sep.15;
please section by Wednesday, Sep.17 at 1PM.
To section by computer:

Windows --> SecureCRT application --> Quickconnect, protocolssh2, hostnameulam.fas.harvard.edu, usernamesection

or:

Mac OS X --> Applications --> Utilities --> Terminal: at the prompt, typessh section@ulam.fas.harvard.eduand hit return

or:

Unix/Linux: log into your FAS account; at the shell prompt, typessh section@ulam.fas.harvard.eduand 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
webpage
or program does not answer, please email
Susan Milano

Here's the course syllabus, including the provisional homework assignments through the second midterm, just before Thanksgiving.

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: `username@math.harvard.edu`).

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
`username@fas.harvard.edu`
(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.

Both

The first midterm will be in the same format(*), and on the same
material (Sections

(*) 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's another way to see that the row rank of any matrix

Here's what you have to know about

Some practice problems from the second midterm: this exam from the Fall 2006 edition of 21b, and more practice problems [now the correct link -- sorry it pointed to the 1st-midterm problems before!]. Here are the answers to problems 1, 2, 3, 4, 5 of the 2006 midterm, and solutions to practice roblem 1 and the T/F questions and practice problems 2,3, by 2006 TF's L.Kang and N.Sun respectively. As with the first midterm, there are plenty of other problems in the textbook that you can use for further practice.

Here are solutions for the second midterm.

Here is the schedule of homework assignments past the second midterm date.

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
**(updated Dec.3)** in your printout.

The

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.

- 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 theFAS Deputy Registrar(for graduate and cross-registered students) for all other requests.

Here are copies of recent final exams in Math 21b:
May 2004,
May 2005
(with solutions at the end(*)),
and January 2006
and January 2007
(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.)

Here are some more practice
exercises on the **+**2f''+2f'=0,**-**2f''+2f'=0.

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