MATH 25a Honors Multivariable Calculus and Linear Algebra Course Announcements Date Time Announcement 16th Jan 4:55pm From noon on Tues 17th onwards you can pick up HW 13 from the Math 25 mailboxes on the third floor of Science Center. Also, the final lot of practice problems has been uploaded. 13th Jan 9.45am Some of the HW solutions have been uploaded. (Note that HW13 part 1 is there, even though the document has the title HW12 solutions.) 10th Jan 6.00pm More detailed exam information added - see below. More practice questions posted - more to come. 9th Jan 6.20pm Practice questions posted - more to come. 9th Jan 5.20pm Final exam info and office hours updated. 14th Dec 10.00am Office Hours on Thursday 15th Dec 3-4pm. No office hours on Friday 16th Dec. Office hours on Monday 19th December from 2-4pm (so one hour later than usual). 16th Nov 2.40pm Webpage for course updated - old announcements removed etc, etc. 14th Nov 5.30pm Midterm exam has been returned. See me for solutions. Final Exam information When/where: 23rd January at 9:15am in 102 Sever Hall - try to arrive 15mins early. Office hours: Tues 10th Jan 3-5pm, Wed 11th Jan 3-5pm, Thurs 12th Jan 3-5pm Mon 16th Jan 1-3pn, Wed 18th Jan 1-3pm, Fri 20th Jan 1-3pm and Sunday 22nd Jan 1-3pm and by appointment What to bring: Pens, pencils, etc. This is a closed book exam. Don't bring any notes or textbooks or a calculator. What's covered: The exam will be based around the second part of the course - on linear algebra and vector spaces. However, you'll also be expect to know the parts of the first half of the course that are relevant. Some examples of what I mean: 1) vector spaces are over a field and we studied these in the beginning of semester, 2) recall that we discussed inner products, norm and how you can get a metric from a norm, we studied metrics early on, 3) many maps are injective or surjective and we looked at these early in the semester, etc, etc. We went over some subjects quickly at the end of semester. I expect you to know the statement of the Spectral Theorem, how to prove it and also how to use it. Know the statement of the Jordan Basis Theorem and how you might use it. However, you don't need to worry about the proof. (In particular nilpotent operators or generalised eigenspaces.) However, do make sure you understand determinants and traces of both operators and matrices! So Ch 7 pages 127 - 137 could be on the exam, as could Ch 10 pages 213 - 236. You don't need to worry about the end of chapter 7, chapters 8 and 9 EXCEPT to understand the statement of the Jordan Basis Theorem and how to use it. Having said this, at least one of the exam questions will ask you to apply the things you know to new situations. There are so many different things that I could ask here and some of the potential topics are in the omitted parts of ch 7, 8 and 9. Know that whatever "unseen" material I'll ask, you'll have plenty of guidance in the way I ask the question. Also note that you'll all be in exactly the same situation! What's it like: The format of the exam will be similar to that of the midterm. There will be five questions, however there won't be any choice. You will have to do all the questions. I expect that the exam will take you about 2 and a half hours, so you should have some spare time to think about things. You can expect that some questions will have parts to them. The layout will be similar to the midterm as well: you'll get a packet with questions, then blank space to work on, then another question. I'll add some extra blank paper to the end of the exam. We will also have lots of scratch paper if you need it. I think the exam as a whole will be the same level of difficulty or perhaps a bit more challenging than the midterm exam. The exam questions will be of varying levels of difficulty. Some of them will be proofs, some will be calculations and I might even include some true/false quesitons. Some questions will be directly based on class or HW material. Other questions will ask you to apply what you know to new situations. Practice Problems: Practice problems in a .pdf file Also any of the questions in Axler or your HW assignments are really good to look at. Essential information Classes: MWF 10-11 in 309a Science Center Course webpage: math25a.html Instructor: Elizabeth Denne Office hours: Mondays 1-3pm, Fridays 2-3pm and by appointment Office: 535 Science Center Email: denne@math.harvard.edu Phone: 617 495 2210 Cool Stuff for Math 25'ers To Do Physics Talk: Lisa Randall gives a public lecture on her new book "Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions" in Science Center Lecture Hall D on Wednesday 14th December at 8pm. Math Table: See Ivan talk at the Math Table on Tuesday 13th December at Mather House. Dinner at 5.30pm and talk at 6pm. Kuumba Singers Concert: See Dawn sing in the Kuumba Singers Christmas Concert. Held in Memorial Church on both Friday 9th December at 7pm and Saturday 10th December at 8pm. Concert is free. Bach Society Concert: See Heather and Elizabeth perform in the Bach Society Concert on Saturday 10th December at 8pm in Paine Hall. Tickets are $6 for students/seniors and $10 for general admission. Glee Club Concert: Sam sings with the Harvard Glee Club and Radcliffe Choral Society at "Christmas On the Common," Saturday December 10, 8pm, First Church Cambridge, congregational, 11 Garden Street. Christmas carols, sacred music, and other fun stuff. Putnam Math Competition: The competition is this weekend Saturday Dec 3, 2005. The exam is in two sessions, both held in Science Center hall C. The session times are 10am-1pm and then 3pm-6pm. Alison and Ivan have both taken the Putnam exam in previous years and are happy to answer questions or give you tips! Best of luck! Fallen Angels Concert: Zoe's a cappella group is singing in a concert from 1-2pm this Saturday 3rd December at the Ice Rink outside the Charles Hotel. They'll be singing their usual pop repetoire plus some fun carols. 2006 Harvard-MIT Math Tournament: HMMT is an annual math competition for high school students run by Harvard and MIT students. Intersted in writing up problems or helping out on the day (Feb 25, 2006)? See http://web.mit.edu/hmmt or email Elizabeth Goodman esqgoodm@fas.harvard.edu for more information. Math Table: Each Tuesday undergraduates interested in mathematics eat dinner together in Mather house and hear a 30min math talk. Talk to Alison or Ivan about this or see http://www.math.harvard.edu/mathtable/index.html. Numb3rs: TV show on CBS each Friday 10pm about a math professor from ''CalPsi'' help his brother in the FBI fight crimes using mathematics. (See also http://www.cbs.com/primetime/numb3rs/.) I'm personally very happy to see a mathematician on TV who is a) young b) not insane c) not wearing glasses and d) very cool. Textbooks Principles of Mathematical Analysis by Walter Rudin, third edition, published by McGraw-Hill. Linear Algebra Done Right by Sheldon Axler, Second Edition, published by Springer. Both of these are available from Harvard COOP, or from on-line bookstores. The following texts are on reserve in Cabot library: Principles of Mathematical Analysis by Walter Rudin, third edition, published by McGraw-Hill. Linear Algebra Done Right by Sheldon Axler, Second Edition, published by Springer. Linear Algebra with Applications by Otto Bretscher, third edition published by Prentice Hall. Finite-dimensional vector spaces By Paul. R. Halmos, published by Springer-Verlag. A useful secondary source but be warned some of the notation used is non standard! Course Assistants Who: Alison Miller Ivan Corwin Email: miller5@fas.harvard.edu corwin@fas.harvard.edu Office hours: Loker 8-10pm Loker 8-10pm Problem Sessions: Sun 7-8pm room SC 411 Mon 5-6pm room SC 411 Course Summary and Course Outline Summary: This course contains a rigorous treatment of linear algebra, point-set and metric topology, and the calculus of functions of several variables. Emphasis is placed on careful reasoning, and on learning to understand and construct proofs. You should take this class if you are very interested in mathematics and want a thorough proof-based review of the topics before moving on to other mathematics. You will be required to work hard (at least 10 hours each week) during this course! Outline: CLICK HERE for a day to day calendar of material covered in class. This page also contains the readings and references for each class. Confused about which course to take? CLICK HERE for an excellent description of the differences between Math 21, 23, 25 and 55. In the end, you should take the course which challenges, but not overwhelms you. You should aim to find the classroom environment where you will produce your best work. Grades, Attendance and other matters Exams: 1 midterm and 1 final exam. Midterm to be held Wednesday 9th November. Grading Policy: Homework 2/5; midterm 1/5; final exam 2/5. The class will not be graded "on a curve": if everyone deserves an A, everyone will get an A. Attendance: Attendance will not be taken at each class. However, it is much harder to learn the material on your own, so you are strongly encouraged to attend each class. You must attend the midterm and final exams. Make-up exams will only be given in special circumstances. Problem Sessions: You should attend, each week, at least one of the problem sessions held by the course assistants. Drop date: The drop date for the course is Monday 3rd October. However note that fees will be waived up to the 5th Monday of semester (17th Oct) for students who change between any of the following courses: Math 21a, Math 23a, Math 25a and Math 55a. Final exam: Exam group 3: date, time etc TBA Midterm Exam When: 7-9pm Wednesday 9th November 2005 Where: Room 507 Science Center What: Material from weeks 1 through 6. That is Rudin Chapters 1-3, plus anything from the HW. What to bring: Your brain and a pen or pencil. This is a closed book exam so notes or textbooks are not allowed. What to expect: The exam has 6 questions in total. However you will be expected to do just 5 of them. You will have to do the first two questions, then you will do 3 of the 4 remaining questions. (So you'll have some choice.) I think you will probably need all 2 hours to do the exam. Expect an exam that will challenge you, but that is not impossible to do. If you've done your HW and thought about the material in class, you will be able to do a good chunk of the exam with thought. However, expect some more challenging questions as well! Note that anything from the class or the text or from the HW is fair game. You should expect some questions asking for definitons, proofs and maybe calculations. You should also expect to see some questions that are on material that is not identical to the material we've covered in class. However, note that I am quite aware that we only have 2 hours in which to do the exam. So for example, it would be unreasonable to ask you to construct the reals from the rationals. But it would not be unreasonable for me to ask you to define what a dedekind cut is and to define addition for the set of cuts and show it obeys some of the field axioms. So for the long proofs we've done in class, make sure you understand what the key steps are and how they work. Handouts Date: What it's about: Where to fnd it: 14th Nov Solutions for the Midterm exam See me for extra copies 4th Nov Practice questions for the Midterm exam See me for extra copies 3rd Oct Mathematical notation used in class pdf file here 3rd Oct Academic Integrity - acknowledging sources of help pdf file here 26th Sept Summary of set notation See me for extra copies 23rd Sept Summary of field axioms and LUB property from Rudin See me for extra copies Homework Assignments Assignment Date due Handed in? Returned? Solutions HW 1 (pdf) Tuesday 27th September Yes Yes Complete solutions (pdf) HW 2 fixed (pdf) Tuesday 4th October Yes Yes Complete solutions (pdf) HW 3 (pdf) Wednesday 11th October Yes Yes Complete solutions (pdf) HW 4 (pdf) Tuesday 18th October Yes Yes Complete solutions (pdf) HW 5 (pdf) Tuesday 25th October Yes Yes Complete solutions (pdf) HW 6 (pdf) Tuesday 1st November Yes Yes Part 1 solns (pdf) and Part 2 solns (pdf) HW 7 (pdf) Tuesday 8th November Yes Yes Complete solutions (pdf) HW 8 (pdf) Tuesday 15th November Yes Yes Part 1 solns (pdf) and Part 2 solns (pdf) HW 9 (pdf) and HW 9 (tex) Tuesday 22nd November Yes Yes Part 1 solns (pdf) and Part 2 solns (pdf) HW 10 (pdf) and HW 10 (tex) Tuesday 29th November Yes Yes Part 1 solns (pdf) and Part 2 solns (pdf) HW 11 (pdf) and HW 11 (tex) Tuesday 6th December Yes Yes Part 1 solns (pdf) and Part 2 solns (pdf) HW 12 (pdf) and HW 12 (tex) Tuesday 13th December Yes Yes Part 1 solns (pdf) and Part 2 solns (pdf) HW 13 (pdf) and HW 13 (tex) Tuesday 20th December Yes Yes Part 1 solns (pdf) and Part 2 solns (pdf) Homework policy Homework: Assignments are handed out each week. They will form an essential part of the course. The assignments and the dates they are due will be posted on the course website. Submitting HW: Homework should be turned in to the course mailbox (outside 325 Science Center) by noon on the day that the assignment is due. Late HW: Late homework will be accepted only in exceptional circumstances and only with prior approval. HW grade: Your lowest homework score will be dropped at the end of the semester. Working together: You are strongly encouraged to discuss the homework problems both with your fellow students and with the course assistants. However, you must write up your solutions by yourself. (Copying someone else's homework is unacceptable.) Collaborating on exams is not permitted. Keeping the graders happy: To make the job of grading easier, could you please follow the following guidelines: Write your name on your HW. Neat, legible handwriting. We will not grade anything we cannot read! Write on one side of the paper only. The problems should be in the order assigned. Staple (or paper-clip) all pages together.