# 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.

• 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.
Homework 10%
Quizzes 15%
Mid-term exams 40%
Final exam 35%

• 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
Review
solutions
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.

*** 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!