Multivariable Calculus for Engineering Students

Mathematics 69.204, Section B, Winter, 1998.
This website's Internet address (URL):

Detailed Class Outline

These pages are best viewed with Netscape 2.0 or higher, and not from a Chat prompt.

Using CHAT Who's the Prof? Do I know enough? What must I know?
Classes Begin Where are the lectures? Textbook(s) Can I use a Calculator?
Will I pass? When's the Supplemental Exam? I wanna drop this! Prof's Office Hours
Tutorials FAQ on Tutorials Where's a tutor? When's the next Holiday?
Week 1 No tutorials  Fourier Series-Motivation
Fourier Series Formulae
Week 2 Tutorial 1a + answers
Tutorial 1b + answers  
Remarks on
Fourier Coefficients

Page 2 on
Fourier Coefficients

Week 3 Tutorial 2a + answers
Tutorial 2b + answers  
Remarks on
Polar Coordinates

Remarks 2 on
Polar Coordinates

Remarks 3 on
Polar Coordinates

When to use
Polar Coordinates?

Week 4 Test 1 + answers  Study guide, SOON! Homework 
Weeks 5 and 6 Geometric idea of continuity
A typical tangent plane
Another tangent plane
  Answers to
Tutorial 3
Useful Formulae in this Course Homework 
Week 7   Answers to
Tutorial 4
Useful Formulae in this Course Homework 
Week 8   Answers to
Test 2
Answers to
Tutorial 5
Week 9   Answers to
Test 3
Answers to
Tutorial 5
Week 10   Jacobians I
and all that.

Jacobians II
and all that.
Test 4 Answers
Complete Study Guide
Tutorial 6
Week 11  Sample Final Exam
Answers to B5 and B6
Complete Study Guide
Tutorial 6
Week 12   REVIEW MATERIAL: QUIZZES Complete Study Guide
Tutorial 6

Detailed class schedule

Dr. Angelo B. Mingarelli,
Herzberg Physics Office #4250
Tel/Fax: (613) 520 3534
Electronic mail:
Photograph: Just in case you don't want to come to class...
Office Hours: Wednesdays, 1:30 - 3:30 p.m.

TEXTBOOK(S): Compulsory: Edwards and Penny's Multivariable Calculus (packaged with Fourier Series).

Optional: In case you forgot your Methods of Integration, or Inverse functions, either one of my...
Integral Calculus, by A.B. Mingarelli, available in the Bookstore, around Oct. 5, 1998
The ABC's of Calculus by A.B. Mingarelli; Module on Inverse Functions, 80 pp., from the Instructor;cost $12.00 net, may come in handy, or Differential Calculus, by A.B. Mingarelli, available in the Bookstore ($25.95 + GST).

(From the Fourier Series) Sections 9.1,2,3;
From the Textbook: 10.1, 2, 3, 4, 5;  12.1, 2, 3, 4, 5, 6, 8;  13 (all);  14 (all);  15.1,2,3

The prerequisites for this course are:
69.105 or 207, and 69.114 or 117.
Students who have not passed the prerequisite courses may be automatically de-registered during the term. Those that have done poorly in the prerequisites are strongly urged to take 69.105 before attempting 69.204. Do get advice from the instructor or from the Mathematics Undergraduate Advisor Ken Small, in 4380 Herzberg Building.

Your grade will be calculated either as:
(i) Term Mark 40%;
(ii) Final Examination 60%
(iii) Final Examination 100%, whichever is better.
In any event, your final course grade is the larger of the two numbers: A and B where A=(i)+(ii) and B=(iii).

The term mark (40%) will be derived from:

Note:  The "best x of y" rule allows you to miss some of the term events for any reason (medical or otherwise).
Only under highly exceptional circumstances will a test be postponed to a later date.

No Supplemental examinations

You may use a non-programmable calculator for the examinations and tests in this course, although it is not necessary.
The Sharp 531-L is available at the Bookstore for $19.95 + Tax.

The last date for withdrawal from the course is Nov. 6. If you decide to leave the course before the end of term, it is much better, in terms of your academic career, to formally withdraw from the course than to simply ignore it and get an F.

Thursday, September 10, 1998


Mondays, 2000 Minto Building, 3:30 p.m.
Wednesdays, 2000  Minto Building, 2:30 p.m.
Thursdays, 2000  Minto Building, 4:30 p.m.


All tutorials are held on Thursdays at 2:30 p.m. and will begin on Sept. 18th (hopefully) in various locations depending on your section and the first letter of your family name (surname, cognome(n)):

FAQ on Tutorials

Where do I go? This depends on your section letter and/or on the first letter of your family name; see the preceding paragraph for details about which room you'll go to. Tutorial exercises are to be divided into individual groups of 5 students who will, as a unit,  hand in a common Exercise Set for the 2% mark. Each student in the group then receives the same mark on the given tutorial.

Do I HAVE to go? Well, if you want to have some term work to count on your behalf, then, YES, do go. If all you want to do is write the final exam, then that's OK, too; see the EVALUATION item above.

Who is my T.A.? Look above under Tutorials or go the required class and ask him/her.

STATUTORY HOLIDAY: September 7 and October 12 ...University closed. NO UNDERGRADUATE CLASSES on October 9 as this is University Day;

WINTER BREAK: Feb. 22-26, 1999

CLASSES END: Monday, December 7, 1998


Please note that the mathematics TUTORIAL CENTRE, in Herzberg Physics Building, Room 4385, will be opening around Monday, Sept. 28.
Hours for the Centre:
Monday to Thursday: 10 am to 4pm
Evening hours to be announced.
Tutors advertise frequently on the Notice Boards around the Centre.

Detailed Class Outline

Winter, 1998

Sept. 7-11  None  9.1,2,3  Fourier Series
Sep. 14-18   None   10.2-3-4 Polar Coordinates and Plane Curves 
Sep. 21-25  None  12-1-2-3-4-5 Vectors: Dot Product, Cross Product and Triple Product; Lines and Planes; Curves and Surfaces
Sep. 28 - Oct. 2  TEST 1, Oct. 1 
Tutorial room 
 12-8, 13-1 to 4 Cylindrical and Spherical Coordinates; Limits; Partial Derivatives 
Oct. 5- 9   None   13-5-6-7 Max. and Min.; Differentials, The Chain Rules;  
Oct. 12-16  None   13-8-9-10 Gradients; Directional Derivatives; Lagrange Multipliers; Second Derivative Test;
Oct. 19-23  TEST 2, Oct. 22 
Tutorial Room 
 14-1-2 Iterated Integrals; Double Integrals over Rectangles; Double Integrals over General Regions
Oct. 26-30   None   14-3-4-5-6 Using Polar Coordinates; Applications to Area and Volume; Centroids and Moments of Inertia; Triple Integrals 
Nov. 2-6  TEST 3, Nov. 5 
Tutorial Room 
14-6-7  Triple Integrals and Iterated Integrals; Use of Cylindrical and Spherical Coordinates 
10  Nov. 9-13  None   14-8-9 Surface Area, Change of Variables
11  Nov.16-20  TEST 4, Nov. 19 
Tutorial Room 
 15-1-2-3 Vector Fields; div. grad. curl; Determining Conservative Fields and Finding Potential Functions, Line integrals 
12  Nov.23-27   None    Catch-up and/or Review
13  Nov.30- Dec.4   None    Review

Using CHAT:

For information about CHAT accounts, click here
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To access the Internet from ANOTHER INTERNET SITE click here

Some Undergraduate Computer Labs

The following locations have PC's which, upon login with theword "student" will give you access to Netscape in its manyversions...
Herzberg 3393, ....Our Math. Lab: Only 5 PC 486's carry thisoption (those nearest the hall window)
Herzberg 4115, 4125, 4155, 4175.....Computer Science Labs.

More Undergraduate Computer Labs

Lanark House - Rm. 121  Campus Map
Library - Rm. 403 
Library - Rm. 411 
With two cash-card operated HP Laser printers 
Loeb - Rm. B241 Getting Help in Computer Labs 
contain's consultants' schedule
Loeb - Rm. B250 
Loeb - Rm. C460  With a cash-card operated HP Laser printer 
Loeb - Rm. C479 
Southam Hall - Rm. 507  With a cash-card operated HP Laser printer 
Southam Hall - Rm. 509 
St. Patrick's - Rm. 309A  With a cash-card operated HP Laser printer 
Please be brief in order to leave others the chance to accessthese machines.

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