University of British Columbia
Department of Mathematics

# Multivariable Calculus - MATH 253 section 104 - 2011W1

## Time and Place

Tuesdays and Thursdays, 2:00-3:20pm at Chemistry D300.

## Office Hours

Tuesdays and Thursdays, 1-2pm at WMAX 109 or by appointment. Feel free to send me emails with questions about the material.

If you need help, you should also consider the drop in tutoring.

## Course Outline

This is an introduction to the calculus of several variables. The main topics are partial derivatives and multiple integrals - basically adding extra dimensions to first year calculus. Hence an appreciation of three dimensional geometry is essential.

Sections numbers refer to Stewart, Multivariable Calculus, 6e (for Early Transcendentals, subtract 1 from the section numbers):

1. Vectors, quadratic surfaces (Sections 13.1-13.6, 11.1, 11.3, 11.5)
2. Partial derivatives, increments, chain rule (Sections 15.1-15.5)
3. Taylor's Polynomials in 1 and multiple dimensions (Sections 12.10, 12.11 and notes provided here)
4. Directional derivative and Gradients (Section 15.6)
5. Max/min, Lagrange multipliers (Sections 15.7-15.8)
6. Double integrals (Sections 16.1-16.5)
7. Triple integrals (Sections 16.7-16.8)

Here is a list of Objectives and Summary of Topics.

## Textbook and other material

Early Transcendentals, or Multivariable Calculus by James Stewart Edition 6e.

## Final Exam

The final exam is scheduled for Friday, Dec 16, 8:30am, at MATH 100. If you think you may have a conflict or a hardship (3 exams *within* 24 hours with Math being the middle exam) then you should contact Mar Ness (ness at math.ubc.ca) by email with your name, student number, and an explanation of the issue. You are also welcome to stop by the Math Office and speak to Mar.

## Midterms and Quizzes

There will be 2 midterms, 80 minutes long each, on Tuesday at regular class time and location. Each will count for 17.5 points of the final grade.

There will be 3 quizzes, 20 minutes long each, on Tuesdays at the beginning of class. Each will count for 5 points of the final grade.

Additionally, a mark will be given for participation in clicker polls in class. This mark will be used instead of the lowest quiz mark, if it is for the benefit of the student.

The material for midterm and quizzes is everything we studied until that point. The quizzes questions will be very similar to questions from your homework assignments.

The dates are:

No books, notes, calculators, other human beings or similar aids will be allowed in any of the exams.

There is a Vista section for, where you can see your marks and register your clicker.

## Material covered and Homework

Sections numbers refer to Stewart, Multivariable Calculus, 6e (for Early Transcendentals, substract 1 from the section numbers).
• Sep 8 - 2D and 3D space, coordinate systems, Cartesian coordinates, distance between 2 points.
Suggested exercises: section 13.1:  1-8,10,11,13,15,17,21,22,27-32,33,35,38,39. Extra question: invent your own coordinate system!
• Sep 13 - Vectors: geometric and algebraic representation, addition, length.
Suggested exercises: section 13.2:  1-5,7,9,11,17,19,21-25,27,31,37,38,41.
• Sep 15 - The dot product, algebraic definition, properties, geometric meaning, projections and components, direction angles and cosines.
Suggested exercises: section 13.3:  1,2,3,5,7,9,11,12,15,17,19,21,23,24,25,26,27,28,35,37,39,41,42,43,44,50,51,52,53,55,56,57,58,60. Extra question: what does it mean when the dot product of is positive or negative?
• Sep 20 - Quiz (20min) Lines and Planes (13.5).
Suggested exercises: section 13.5:  1,2,4,5,6,9,11,13,17,19,21.
• Sep 22 - More Line and Planes (13.5), Cross product (part of section 13.4).
Suggested exercises:section 13.5: 23,24,26,27,29,30,33,34,36,37,43,45,46,49,50,55,60,62,63,65,66,69,71,73; section 13.4: 3,5,7,13,17,19,22.
• Sep 27 - Conic sections (11.5) and Quadratic surfaces (13.6).
More exercises regarding lines and planes: HW1 and Solutions. Question no. 5 is particularly interesting.
• Sep 29 - Quadratic surfaces (13.6) and Functions of several variables (15.1).
Suggested exercises:section 13.6:  1,2,4,7,9,11,15,19,21-28,29,36   ; section 15.1:  1,3,8,9,11,13,17,18,27,28,30,31,32,33,34,35-38,41,43,45,46,55-60,61-64,65,66.
• Oct 4 - Quiz (20min). Continuity of functions of several variables (15.2). Suggested exercises: section 15.2:  1,25,26,29-38.
• Oct 6 - Partial Derivatives (15.3) (excluding higher order derivatives and partial differential equations), Tangent Planes (15.4) (excluding differentials).
Suggested exercises:section 15.3:  1,9,10,15,17,19,21,25,31,33,35,39,41. section 15.4:  1,3,5,11,13,15,19,21.
• Oct 11 - Higher order partial derivatives (15.3), Higher order approximations / Taylor Polynomials (not in textbook, see these notes.).
Suggested exercises:section 15.3:  5-8,49,50,51,53,55,57,59,63,67,87,91. Also, HW2 and Solutions.
• Oct 13 - Local minimum and maximum (not absolute) (15.7).
Suggested exercises:section 15.7:  1,3,4,5,7,9,11,13,15,17,19,37.
• Oct 18 - Midterm at SCRF 100.
• Oct 20 - Absolute minimum and maximum (15.7).
Suggested exercises:section 15.7:  29,31,33,34,35,43,44,46,47,49,51,53,56.
• Oct 25 - The chain rule (15.5).
Suggested exercises:section 15.5:  1,3,5,7,9,11,13,15,21,23,25,27,29,31,33,39,58.
• Oct 27 - Directional derivative and the Gradient (15.6).
Suggested exercises:section 15.6:  1,3,5,7,9,11,15,19,21,25,27,28,29,31,38.
• Nov 1 - Quiz (20min). Tangent planes to level surfaces (15.6), Lagrange Multipliers (15.8).
• Suggested exercises:section 15.8:  1,3,5,7,9,11,13,18,19,25,26,31,32,34,35,37,39.
• Nov 3 - Lagrange Multipliers (15.8), Partial Differential Equations (15.3).
Suggested exercises:section 15.8:  40,41,42,45. section 15.3:  71-77,81,82,83,85,87. section 15.5:  45-50,52,53.
• Nov 8 - Double integrals (16.1), Iterated integrals (16.2).
Suggested exercises: section 16.2:  3,5,7,9,11,13,15,17,19,21,25,27,31,35.
• Nov 10 - Double integrals over general domains (16.3).
Suggested exercises: section 16.3:  1,3,5,7,9,11,13,15,17,19,21,25,31,33,39,41,43,45,47,51,59.
• Nov 15 - Midterm at SCRF 100.
• Nov 17 - Double integrals in polar coordinates (16.4).
Suggested exercises: section 16.4:  5,9,11,13,15,17,19,21,23,25,28,29,31,33,35.
• Nov 22 - Application of double integrals (16.5).
Question 1 here: HW4. Make sure you understand double integrals well before we start with triple integrals.
Also, some review questions: HW3 with Solutions.
• Nov 24 - Application of double integrals (16.5). Triple integrals over general domains (16.6).
Suggested exercises: section 16.5:  1,3,5,7,9,11,13,15; section 16.6:  1,3,5,7,9,11,13,15,17,21,27,29,31,33,35,37,39,51,52,53. Also ,question 2 here: HW4.
• Nov 29 - Triple integrals in cylindrical coordinates (16.7).
Suggested exercises: section 16.7:  5,6,7,8,9,11,15,17,19,21,23,25,26,27; section 16.8:  1,3,5,6,7,8,9,11,13,15,17,19,21,25,27,29,31,35,39,44. Also ,question 3 here: HW4 (solutions).
• Dec 1 - Triple integrals in spherical coordinates (16.8).

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## Resources

• Brian Seymour's Math 200 Resources include a lot of useful explanations, exercises with solutions, etc.
• Past exams.
• I use Mathematica to make some graphs (specifically 3D ones). It's a great software if you're interested in math. Many of its features are also available for free at WolframAlpha.