Math 20a: Calculus of several variables

Spring 2005

Instructor: Prof. Daniel Ruberman
E-mail: ruberman@brandeis.edu
Office: Goldsmith 310
Phone: 736-3074
My Homepage and Spring Schedule
Class schedule: Tuesday, Friday 12-1:30, Goldsmith 100.
Office hours: Monday 11-12, Wednesday 11-12, Thursday 10-11, and by appointment.
Textbook: Multivariable calculus (5th edition) by Stewart.

Syllabus

Chapter 11: Curves in the plane.
Chapter 13: Vectors in 3-dimensional space, surfaces.
Chapter 14: Vector functions (parametrized curves) in 3-space.
Chapter 15: Functions of several variables, derivatives, linear approximation, maxima-minima.
Chapter 16: Multiple integrals of functions of several variables, volume, surface area.
Chapter 17: Vector fields, line and surface integrals.

Exams, homework and grading

There will be 2 evening midterm exams, 3 half-hour in-class quizzes and a 3-hour final exam. first midterm exam will be February 17, 7-8:30pm and the second will be March 31, 7-8:30pm. The schedule for quizzes will be posted shortly. The class is in Block J, and the final exam will be scheduled accordingly.
Homework will be assigned and collected weekly and will be graded and returned to you.
Grading formula:
Homework 10%
Quizzes 15%
Mid-term exams 40%
Final exam 35%

Web links

Homework assignments and solutions and review sheets will be available from this page. (Note: The homework assignments and review sheets have not yet been posted. Please check back at the beginning of the semester.) Some of this will be in pdf format, for which you will need the free Acrobat Reader.
Here is a short list of trigonometric identities.
Here is a tool for integrating.
Here is some graphing software you may find useful. Just click on the link to download a compressed copy. They are freeware.
- For those with Macs, the graphing calculator (included on older models) is now available for downoading. It's good for graphing functions of two variables (surfaces in (x,y,z)-space).
- A PC version (I haven't tried it) can be found here.
- For graphing functions of one-variable (curves in the (x,y)-plane), click here (Mac only).
- For graphing functions of two variables (surfaces in (x,y,z)-space) click here for the Mac version, or here for the Windows version.

Exam schedule and solution sheets

Event	Date	Review sheet	Text sections
Quiz 1	February 11 In class	N/A	11.1 - 13.6
Quiz 2	March 22 In class		Chapter 14, 15 (through 15.5)
Quiz 3	April 19		Through Chapter 16
Exam 1	February 17 7-8:30pm Room 317 Goldsmith
Exam 2	March 31 7-8:30pm Room 317 Goldsmith		Through 15.7
Final Review	Monday, May 2 1-2:30 pm Goldsmith 317
Final	Tuesday May 3 1:30 pm - 4:30 pm Room 121 Gerstenzang		Through 17.2

Homework assignments and solutions

In addition to the even-numbered homework, which are to be handed in for grading, I have suggested (in parenthese) some odd-numbered problems for you to do for practice. The solutions to these are in the back of the book, so they won't be graded; you can write the solutions on the same paper as the even ones if you find that to be convenient. Problems given by letters A, B, C, etc. are to be accessed by clicking. Solutions will be posted after the problems are turned in, and are accessed by clicking the orange buttons.

No. Date due Sections to read Problems Solutions

1
Jan. 21 11.1-11.3; 13.1 11.1: # 2, 6, 10.
11.2: # (1), 2, 6.
11.3: # 4, 8, 14, 18, (23), 26.
13.1: # (3), 4, (13), 14, 16, (33).

2 Jan. 28 13.2, 13.3, 13.4 (start) 13.2: # 4, (5), 6, (15), 16, 18, 26, (27).
13.3: # 6, (7), (11), 12, (21), 22, 24, 36, 44, (45), 50.
13.4: 6, (7), (15), 16, (21),22.

3 Feb. 4 13.4 (end), 13.5, 13.6 13.4: (29), 30, 41.
13.5: 4, 10, (23), 24, (41), 42.
13.6: 2, (9), 10, (13), 14, (17), 20, (21-27 odd), 22, 24, 26, 28, (41), 42.

4 Feb. 11 13.7, 14.1 13.7 10, (11), (21), 22, (31, 33, 35), 32, 34, 36, 38, 44, 52, (53).
14.1: (5,13), 14, 16, (19, 21, 23), 20, 22, 24.

5 Tuesday
Feb. 15 14.2, 14.3 (through example 2, p. 900) 14.2: (5), 6, 12, (15), 16, 18, 22, 24, (31), 32, 34, (41) 44.
14.3: (3), 4, 6.

6 March 4
15.1, 15.2, 15.3 (beginning)
15.1: 6, (17), 26, (27), (28), (30), 34, (43), 44, (53-58), 60.
15.2: (1), 6, (7), 10, 18, (27), 28, 32, (33).
15.3: (5,6), 8, (15), 16, 24, (29), 32.

7 March 11 15.3 (end), 15.4, 15.5 (beginning) 15.3: (41), 42, (53), 56, (67), 68, (69, 71).
15.4: 4, 12, (13), (19), (23), 24, 30.
15.5: (3), 4, 6, 8, 12.

8 March 18 15.5 (end), 15.6 15.5: (21), 22, 24, (27), (31), 34, 38,(39), 40, (45), 46, (47)
15.6: (7), 8, 16, 22, (23), (27), 28, 32, (37), 38, 42, (55), (61)

9
Tuesday
Mar. 22 15.7 (through p. 994) 15.7: (1), 2, 4, (7), 8, 12, 18, 38, (43), 44.

10
April 1
16.1, 16.2
16.1: (11), 12, (13), 14.
16.2: 2, (11), 12, 16, 18, 24, (25), 28.
In #18, you may recall that the integral of ln(z) dz is best done by parts (set u=ln(z), dv=dz).

11 April 8 16.3, 16.4, 16.5 (first section only, to p. 1046). 16.3: (1), 2, 4, 6, (11), 14, 20, (25), 28, (43), 44.
16.4: (7), 8, 10, (15), 22, 26, 28.
16.5: (1), 2.

12
April 15 16.7, 16.8 16.7: 4, (7), 8, 10, (19), 20, (31), 32 (Just do two of the other 5: dxdydz and dzdydx).
16.8: 2, (3), (5), 6, (9), 10, (17), 18, 30.

13
April 22***
See note below. 17.1, 17.2 17.1: (3), 6, 8, (11), 12, (13), 14, (15), 16, (17), 18, (21), 24, (29), 30. (These are easy, so don't worry about the large number of problems.)
17.2: 2, (3), 4, 8, 10, 14, (17), 18, (19), 20.

No.	Date due	Sections to read	Problems
1	Jan. 21	11.1-11.3; 13.1	11.1: # 2, 6, 10. 11.2: # (1), 2, 6. 11.3: # 4, 8, 14, 18, (23), 26. 13.1: # (3), 4, (13), 14, 16, (33).
2	Jan. 28	13.2, 13.3, 13.4 (start)	13.2: # 4, (5), 6, (15), 16, 18, 26, (27). 13.3: # 6, (7), (11), 12, (21), 22, 24, 36, 44, (45), 50. 13.4: 6, (7), (15), 16, (21),22.
3	Feb. 4	13.4 (end), 13.5, 13.6	13.4: (29), 30, 41. 13.5: 4, 10, (23), 24, (41), 42. 13.6: 2, (9), 10, (13), 14, (17), 20, (21-27 odd), 22, 24, 26, 28, (41), 42.
4	Feb. 11	13.7, 14.1	13.7 10, (11), (21), 22, (31, 33, 35), 32, 34, 36, 38, 44, 52, (53). 14.1: (5,13), 14, 16, (19, 21, 23), 20, 22, 24.
5	Tuesday Feb. 15	14.2, 14.3 (through example 2, p. 900)	14.2: (5), 6, 12, (15), 16, 18, 22, 24, (31), 32, 34, (41) 44. 14.3: (3), 4, 6.
6	March 4	15.1, 15.2, 15.3 (beginning)	15.1: 6, (17), 26, (27), (28), (30), 34, (43), 44, (53-58), 60. 15.2: (1), 6, (7), 10, 18, (27), 28, 32, (33). 15.3: (5,6), 8, (15), 16, 24, (29), 32.
7	March 11	15.3 (end), 15.4, 15.5 (beginning)	15.3: (41), 42, (53), 56, (67), 68, (69, 71). 15.4: 4, 12, (13), (19), (23), 24, 30. 15.5: (3), 4, 6, 8, 12.
8	March 18	15.5 (end), 15.6	15.5: (21), 22, 24, (27), (31), 34, 38,(39), 40, (45), 46, (47) 15.6: (7), 8, 16, 22, (23), (27), 28, 32, (37), 38, 42, (55), (61)
9	Tuesday Mar. 22	15.7 (through p. 994)	15.7: (1), 2, 4, (7), 8, 12, 18, 38, (43), 44.
10	April 1	16.1, 16.2	16.1: (11), 12, (13), 14. 16.2: 2, (11), 12, 16, 18, 24, (25), 28. In #18, you may recall that the integral of ln(z) dz is best done by parts (set u=ln(z), dv=dz).
11	April 8	16.3, 16.4, 16.5 (first section only, to p. 1046).	16.3: (1), 2, 4, 6, (11), 14, 20, (25), 28, (43), 44. 16.4: (7), 8, 10, (15), 22, 26, 28. 16.5: (1), 2.
12	April 15	16.7, 16.8	16.7: 4, (7), 8, 10, (19), 20, (31), 32 (Just do two of the other 5: dxdydz and dzdydx). 16.8: 2, (3), (5), 6, (9), 10, (17), 18, 30.
13	April 22*** See note below.	17.1, 17.2	17.1: (3), 6, 8, (11), 12, (13), 14, (15), 16, (17), 18, (21), 24, (29), 30. (These are easy, so don't worry about the large number of problems.) 17.2: 2, (3), 4, 8, 10, 14, (17), 18, (19), 20.

*** Note about final homework: The official due date for these problems is the last class, on Friday April 22. However, recognizing that some people are leaving early, I will accept this assignment at the review session to be held on Monday May 2. There will be no further extensions for this assignment!

Academic Honesty

You are expected to be honest in all of your academic work. The University policy on academic honesty is distributed annually as a section of the Rights and Responsibilities handbook. Instances of alleged dishonesty will be forwarded to the Office of Campus Life for possible referral to the Student Judicial System. Potential sanctions include failure in the course and suspension from the University. If you have any questions about my expectations, please ask me.