Math 240-002, Spring 2007

Announcements:

· May 5. The final exam was checked. I think that you, Shuvra and me worked fine in this course: though the final's average dropped a little bit with respect to the midterms (probably because there were no partial credits in the multiple choice questions), our performance was the best one! So, we have the best (relative) quotas of A's and B's! Four students received 100 in all exams (2 midterms and final), and I decided to give them A+. The results and the final grades are posted now on the gradebook. You can see your exams at my office on Monday, 2-4 pm, and I'll submit the results to the secretary on Monday evening. Have a great summer!

· April 27. The final exam will take place on Thursday, May 3, 12-2pm at Logan Hall, room 17.

· April 26. I'll hold a review session on Tuesday, May 1, 5-7 pm, at DRL 4C4. My plan is to solve few relatively difficult problems from past final exams and to answer all your questions.

· April 19. This Sunday there will be a Sunday review, DRL A4, 7-9pm. It covers the whole course. Please prepare your questions.

· April 13. I checked the midterm and uploaded the grades (currently not cut to 100) onto the gradebook. Also, I posted a handwritten solution on the bottom of this page. The results are very good! Keep working similarly until the final exam!

· April 9. A review session before the midterm will be held on Tuesday, April 10, 6:30-7:30 pm at DRL A7.

· April 4. Few announcements:

I requested a Sunday review on this Sunday.
I will be replaced by Professor Tong Liu on Friday.
I posted the new homework (there is no quiz next week).
The midterm on Wednesday will be in the same format as the first midterm (3 free response and 5 multiple choice). It covers the following material: Stokes and divergence theorems; simple substitutions in ODE's; solutions of homogeneous linear second order ODE's of the following types: linear (I can ask to find the general solution or to solve an initial/boundary value problem), Cauchy-Euler; the only non-homogeneous case I can ask is the example with a forced motion where f(t)=Acos(at)+Bsin(at), especially the resonance; solutions by power series about ordinary and regular singular points; systems of linear first order ODE's (two next lectures).

· April 3. I posted a review of the material of chapters 2,3 and 5 (the first four sections in the review are introductory).

· March 30. I replaced problem 5.2/8 by 5.2/6 in HW11.

· March 26. Finally, the second midterm will be held on Wednesday, April 11 !!

· March 25. I posted a brief review of the vector calculus part of the course. Let me know if you have any comments (typos, missing material, etc.).

· March 23. The second midterm is scheduled for Friday, April 13. I heard an opinion that this may be a bad date for a midterm because of a certain campus activity. I consider a possibility to move the midterm to Wednesday, April 11, but before doing so I would like to know your opinion on this issue. If you would like to move the date please send me an e-mail.

· March 6. I posted the remaining part of HW8. It contains few problems on surface integrals and Stokes theorem and will be used in the next quiz (the first week after the break). I'll be replaced by professors Rimmer and van Erp on Monday, 3.12 and Wednesday, 3.14. They will start the ODE part of the course. I prefer to postpone the divergence theorem (section 9.16) and the summary of the vector calculus part of the course until my return on Friday, 3.16.

· March 2. The results of the midterm were uploaded on the blackboard. Five grades above 100 are cut to 100, the average is 76.09 and the standard deviation is 19.16.

· February 27. A solution of the sample midterm was posted.

· February 23. Few announcements regarding midterm, reviews, etc:

· (i) The midterm is in class, it covers all linear algebra and sections 9.7-9.12,9.15,9.17. You can bring one A4 handwritten (both sides) list of paper, its size is about 11.5" x 8" (a strange size 5" x 8" which was written earlier is a typo).

· (ii) I decided to cancel the next week quiz, because the last homework will be reflected in the midterm rather intensively.

· (iii) There will be a Sunday review on chapters 8 and 9. Prepare your questions and come.

· (iv) On Tuesday, February 27, 6-8pm at DRL A5, I'll hold a review of the vector calculus part of the course. If the time permits, it'll be possible to answer your questions on linear algebra as well.

· (v) I was asked to prepare a sample vector calculus midterm. I'll try to do it during the weekend -- check the homework part of the webpage.

· February 19. On this Wednesday, February 21, 16:30 at DRLA6, there will be an undergraduate colloquium lecture by professor Stephen S. Shatz. The title is "Analysis when 10^n is small (even when n is positive). Combining this with usual analysis". It is going to be an exciting lecture for students interested in pure mathematics.

· February 19. I would like to recall that math240 workshops are hold regularly on Wednesdays, 4:30-6:00 pm at DRL 4E19.

· February 7. I changed the time of my Wednesday's office hour. The new time is 6-7pm. I hope it will allow to avoid time conflicts with your classes.

· February 2. The final date for the first midterm is Wednesday, February 28 !!

· January 31. There will be a Sunday review this Sunday, 7-9pm, DRL A4. In principle, the second half of chapter 8 will be reviewed: eigenvalues and eigenvectors of a matrix/symmetric matrix, characteristic polynomials and Cayley-Hamilton theorem, diagonalization and orthogonal diagonalization. Please say me if you have any other suggestion.

· January 16. It was finally decided that the TA's grade will be based only on quizzes, so you have not to submit the homework. Nevertheless, I would strongly recommend to do the homework and write down a detailed solution. To give you an additional motivation to work on the homework (except the fact that the quizzes will be based on homework problems!) I will post answers by next Fridays.

General Info:

Instructor:

· Michael Temkin

· E-mail: temkinmath.upenn.edu

· Office: DRL 4N61 (215 - 573 - 1029)

· Office Hours: Mon 2-4pm, Wed 6-7pm, or by appointment

· Lectures: MWF 1-2pm DRLB A7

Teaching Assistant:

· Shuvra Gupta

· E-mail: shuvramath.upenn.edu

· Office: DRL 3E2 (215 - 898-8175)

· Office Hours: Mon 6-7pm, Wed 6-7pm

· Recitations: DRLB 3C4

115-211: T 8:30-9:30
115-212: T 9:30-10:30
115-213: R 8:30-9:30
115-214: R 9:30-10:30

Texts:

· Advanced Engineering Mathematics, 3rd Ed., Zill and Cullen.

Evaluation:

· TA's grade (homework & quizzes) 20%

· Midterm I; Friday, February 23 20% The final date is Wednesday, February 28 !!

· Midterm II; Friday, April 13 20% The final date is Wednesday, April 11 !!

· Final Exam; Thursday, May 3rd, 12-2 pm 40% The final exam will take place at Logan Hall, room 17 !!

All grades will be posted on the Blackboard. The midterm exams will be held at class, their dates are tentative.

Course description:

· Here is math 240 homepage of Math Department. It contains syllabus, core problems, old final exams, etc.

The course consists of three parts: (1) Linear algebra and (2) Vector calculus (slightly more than half a course), (3) ODE (ordinary differential equations). Maple is an optional tool which can turn out to be helpful in your studies. It can also be used in lecture demonstrations.

Exams:

All exams are without books and calculators. You may bring one handwritten cheating sheet of size 11.5"x8" (both sides).

Homework:

The exercises are assigned on Wednesdays on this webpage. The homework is not obligatory, but I strongly recommend to do it in full details.

Homework Assignments:

More difficult problems are marked with *.

Homework 1. Ex. 8.1/12,18,40; 8.2/8,10,12; 8.3/4,6,12. The answers are here.
Homework 2. Ex. 8.4/5,7,13,25,29; 8.5/4,7,8,9,22,23,31,33. The answers are here.
Homework 3. Ex. 8.6/5,9,17,19,27,35,36; 8.7/1,7; 8.8/4,11,15,18,19*. The answers are here.
Homework 4. Ex. 8.9/1,16; 8.10/2,8,9,21,25; 8.12/2,13,18,24,26. The answers are here.
Homework 5. This homework is prepared in format of a midterm, so it gives you an opportunity to practice the linear algebra part of exams. There are ten free response questions and the recommended time is one hour (both midterm and final exam will contain less linear algebra questions, and some questions will be in the multiple choice format). I strongly recommend to work on the homework as follows: repeat the material and prepare your cheating sheet, print the homework and work on it during one hour and using the cheating sheet only, stop your work as the time is over, then you can solve it again on another paper and using any material to check yourself and complete the homework Handwritten solution will be available on this webpage on the next Friday. Good luck, the homework is here.
In problem 1 (i) of the sample midterm one should replace "roots of A" by "eigenvalues of A" (or "roots of the characteristic polynomial of A").
A handwritten and scanned solution of HW5 is here. Two pages were scanned oppositely, so you'll have to rotate them by the pdf viewer.
Homework 6. Ex. 9.7/13,30; 9.8/3,16,19,29; 9.9/1,3,6,12,15,20,24; 9.10/40; 9.11/28. The answers are here.
Homework 7. Ex. 9.17/8,15,19,27,28; 9.15/13,23; 9.12/5,7,16,17,23,25,29. The answers to even problems are here.
Homework 8. A first part of the homework is here, it is prepared in format of a sample midterm. There are 6 free response questions and the suggested time is half an hour. Check your knowledge and your cheating sheet! It will be possible to discuss these problems on the Tuesday's review. The remaining part of the homework will be posted later.
A corrected handwritten and scanned solution of the sample midterm is here.
Homework 8, continuation. 9.13/5,15,23,31,33; 9.14/5,9*,13. The answers to even problems are here.
Homework 9. Ex. 9.16/3,4,5,6,8,14; 2.5/2,8,18; 3.6/4,8,12. The answers to even problems are here.
Homework 10. Ex 3.7/1,7,12,15 (just find the coefficients in the last problem); 3.8/27,38; 3.9/10,15,18. The answers to even problems are here.
Homework 11. Ex. 5.1/18,23,24,29; 5.2/6,17,18,19 (in all power series problems you should find the first three non-zero terms). The answers to even problems are here.
Homework 12. Ex 10.2/1,5,9,13,19,21,25,29. There will be no quiz on this homework, but it practices the part of chapter 10 that is covered by the midterm.
To avoid overlaps with the midterm and the final, I prefer not to post an original practice midterm this time. But I recommend to solve the following problems from old final exams. Spring 2004: 3,11; Spring 2005: 3,4,6,7,11; Fall 2005: multiple choice 6,7,13,15 d (the only reason to exclude 8 is that it is given with a mistake -- a coefficient of y'' is missing), and free response 2,4; Spring 2006: 3,7,8,9,10,13,20. Certain problems from this list have been discussed/solved in class. The suggested time is 5-10 minutes per problem (10 minutes can be spent only for two types of problems: power series solutions and systems of ode's). A reasonable set for half an hour is Spring 2005: 3,11 and Spring 2006: 7,9,20.
The last homework will posted on Monday. It will cover the last theme we are going to study -- the Laplace transform. There is no quiz next week.
Homework 13. Ex. 4.1/11,15,25,29;4.2/11,15,25,32,33,34,35,38. The homework practices the last type of problems we study in the course. There is no quiz this week, but I do recommend to work on the homework.

Lectures:

Each lecture's title is followed by relevant core problems. Make sure you can solve them (you can skip the details if the problem is easy for you).

January 8. Vector spaces and n-spaces, 7.6. Matrix algebra, 8.1. Ex. 8.1/4,12,13,18,23,33,36,39,40
January 10. Systems of linear equations, 8.2. Ex. 5,7,9,11,23.
January 12. Rank of a matrix, 8.3. Ex. 1,5,9,13,17.
January 17. Determinants, 8.4. Ex. 3,7,21,29.
January 19. Properties of determinants, 8.5. Ex. 4,7,12,15,21,31,33,35,39.
January 22. Inverse matrices, 8.6. Ex. 1,5,9,11,21,49.
January 24. Inverse matrices, 8.6. Cramer's rule, 8.7. The eigenvalue problem, 8.8. Ex. 8.6/27,51; 8.7/1,9,11; 8.8/1.
January 26. The eigenvalue problem, 8.8. Ex. 7,15,21.
January 29. Powers of matrices, 8.9. Symmetric matrices, 8.10. Ex. 8.9/3,9,13;8.10/1.
January 31. Orthogonal matrices, 8.10. Diagonalization, 8.12. Ex. 8.10/7,15,19; Ex. 8.12/5,15,37.
February 2. Diagonalization and orthogonal diagonalization, 8.12. Ex. 21,25,27.
February 5. Review of linear algebra, 8/1,2,3,4,5,6,7,8,9,10,12. Brief review notes which were shared at class can also be found here. Repeat the material using the text, the review notes and your lecture notes. Make sure that you remember how to solve all previous homework. Prepare your cheating sheet with basic formulas. Make sure that you know how to solve the following review exercises using the cheating sheet only: pp. 447-449;2,3,4,6,7,8,9,11,13,15,17,18,19,20,23,25,30,40,41,43. Exploit my or TA's office hours if you experience difficulties with the exercises.
February 7. Method of least squares, 8.15. Vector functions, 9.1. Ex. 8.15/3,5,7; 9.1/13,15,17,25,29,33,39. Read sections 8.15 and 9.1, and look through section 9.7.
February 9. Vector fields, curl and divergence, 9.7. Ex. 9,13,27,33,39. Read section 9.7 and look through section 9.8.
February 12. Line integrals, 9.8. Ex. 3,5,9,15,19,25,33. Read section 9.8 and look through section 9.9, solve exercises 9.8/11-14 as an illustration to section 9.9.
February 14. Independence of path, 9.9. Ex. 3,13,17,23,27. Read section 9.9 and look through sections 9.10 and 9.11.
February 16. Review of line integrals, sections 9.8 and 9.9. Review of double integrals, sections 9.10 and 9.11. Ex. 9.10/9,11,17,23,37,45; 9.11/3,5,9,10,13,27,31,35. Read sections 9.10 and 9.11 and look through section 9.17.
February 19. Change of variables in double integrals, 9.17. Ex. 1,3,5,11,17,25. Read section 9.17 and look through section 9.15.
February 21. Triple integrals, 9.15. Ex. 1,5,11,17,21,53,77. Read section 9.15 and look through section 9.12.
February 23. Green's theorem, 9.12. Ex. 1,3,9,13,15,18,23,25. Read section 9.12 and look through section 9.13.
February 26. Surface integrals, 9.13. Ex. 1,5,11,19,27,33,37. Read section 9.13 and look through section 9.14.
February 28. Midterm 1. The results are available through blackboard. You can find the solutions here.
March 2. Stokes theorem, 9.14. Ex. 1,3,5,11,15,17.
March 12. Solutions by substitutions, 2.5. Ex 5,10,13,19,25,33. Read section 2.5 and look through section 3.6.
March 14. Cauchy-Euler equation, 3.6. Ex. 1,9,11,15,27. Read section 3.6 and look through section 9.16. In addition, repeat the material about surface integrals, we will need it on the next lecture.
March 16. Divergence theorem, 9.16. Ex. 3,7,11,15,17. Read section 9.16. Repeat the whole vector calculus part of the course (chapter 9).
March 19. Non-linear second order differential equations, 3.7. Ex. 1,5,17. Read section 3.7 and look through section 3.8.
March 21. Free undamped and damped motions, 3.8.1 and 3.8.2. Ex. 3.7/13; 3.8/9,22,27.
March 23. Driven undamped and damped motion and resonance, 3.8.3. Ex. 3.8/31. Read sections 3.8.1-3 and look through sections 3.9,3.10 and 3.11 (we will discuss them very briefly in class).
March 26. Boundary-value problems, 3.9; non-linear models, 3.10; elimination in systems of linear ODE's, 3.11. Ex. 3.9/9,15,27; 3.10/3,9,13 (optionally); 3.11/1,7,19. Repeat chapters 2 and 3, especially homogeneous ODE of second order (linear and Cauchy-Euler). Look through section 5.1.
March 28. Solutions about ordinary points, 5.1. Ex. 3,9,11,15,21,31. Read section 5.1 and look through section 5.2.
March 30. Solutions about regular singular points, 5.2. Ex. 5,11. Read about the method of Frobenius in section 5.2.
April 2. Solutions about regular singular points and indicial equation 5.2. Ex. 15,21. Look through sections 5.2 and 5.3.
April 4. Special functions, 5.3. Repeat the material of chapter 5. Practice solving linear ODE by power series. First practice equations with ordinary points, then equations with regular singular ones.
April 6. Systems of linear ode's. Preliminary theory, 10.1; homogeneous systems with distinct real roots, 10.2.1.
April 9. Homogeneous systems with real roots (distinct and repeated), 10.2.1 and 10.2.2. Ex. 10.2/5,11.
April 11. Midterm 2.
April 13. Homogeneous systems with complex roots, 10.2.3; Solution by diagonalization 10.3; Solution by matrix exponential, 10.5. The material was sketched. Do not solve practical exercises, but look through the text if you are interested in the theory. The general formula (1) on page 600 is especially nice. We will study Laplace transform (chapter 4) on the next week.
April 16. Laplace transform, 4.1; Inverse Laplace transform 4.2. Ex. 4.1/1,7,15,19,25,31,37; 4.2/5,11,17,23. Read sections 4.1 and 4.2.
April 18. Laplace transform of derivatives and applications to nonhomogeneous linear ode's, 4.2.2. Ex. 4.2/33,39.
In the last lecture I'll summarize briefly the material studied in the course: linear algebra, vector calculus and ode's. It will be the best time to ask any questions about the material. (Surely, I'll be available for your questions till the exam, but my answers on Friday can help other students too.) So, prepare the questions!

Review notes:

February 5. A review of the linear algebra part of the course is here.
March 6. You can find here brief notes where a chemical reaction is balanced by solving a system of linear equations by Gauss-Jordan method.
March 25. A review of the vector calculus part of the course is here.
April 3. A review of the material about differential equations is here.

Solutions:

March 6. A handwritten solution of the sample midterm on linear algebra is here.
March 6. A handwritten solution of the sample midterm on vector calculus (without surface integrals, and Stokes and divergence theorems) is here.
March 6. A handwritten solution of the first midterm is here.
April 13. A handwritten solution of the second midterm is here.