There is also a web page on Courseworks, with most of the same information as this one, and a wiki about using computer software with the course.

Time: TTh 1:10-2:25 p.m.
Place: Math 207
Textbook: Calculus: Early Transcendentals (sixth edition) by James Stewart.
Office hours: TTh 2:30-3:30 p.m. in Math 424.
Help room hours: M 2:00-5:00 p.m.
Undergraduate teaching assistants: Soo Youn Lim, Elliot Smalling
Help room hours: F 11:00-1:00, 3:00-5:00
Final exam date: December 22, 1:10-4:00 p.m.

Syllabus Problem sets Handouts Policies Advice

## Prerequisites.

Calculus I or equivalent is required, though Calculus II or equivalent is recommended. See this web page for more details about which calculus class is appropriate for you.

## Description and goals.

Math V1201 extends techniques from differential calculus to functions of several variables. The main goals are:

• To develop techniques for solving higher-dimensional problems occuring throughout science and engineering. There is a particular focus on maximization / optimisation of functions subject to constraints.
• To develop a geometric language for talking about higher-dimensional spaces, with a particular emphasis on two- and three-dimensions.
• To develop a geometric intuition about vectors in two- and three-dimensions necessary for the more general study of vector spaces in linear algebra (Math V2010).
• To review a few other fundamental mathematical tools (notably, the complex numbers) central to the sciences and engineering.

In the process, the course covers a substantial amount of beautiful mathematics. Many simple phenomena in one dimension (e.g., continuity) become much more subtle -- and interesting -- in higher dimensions. Other phenomena (e.g., linear approximation) become clearer in the context of functions of several variables.

## Policies.

 Homework 30% Midterm 1 20% Midterm 2 20% Final exam 30%

The lowest two homework scores will be dropped. Because of the size of the class, late homework will not be accepted.

### Homework

Problem sets are due on Tuesdays at the beginning of class, except as noted below. If you can't make it to class, put it in the drop box for our section, across from Math 409, on the 4th floor.

You're welcome to work on the homework together. However, you must write up your final answers by yourself. I consider writing them up together cheating.

You are also generally welcome to use any resources you like to solve the problems. However, any resource you use other than the textbook (Stewart) must be cited in your homework. This includes electronic resources (including Wikipedia and Google) and human resources (including the help room and your classmates). Failure to cite sources constitutes academic misconduct.

### Students with disabilities

Students with disabilities requiring special accommodation should contact Office of Disability Services (ODS) promptly to discuss appropriate arrangements.

### Missed exams

If you have a conflict with any of the exam dates, you must contact me ahead of time so we can make arrangements. (At least a week ahead is preferable.) If you are unable to take the exam because of a medical problem, you must go to the health center and get a note from them -- and contact me as soon as you can.

## Syllabus and schedule.

Note: "+" indicates material not discussed in the textbook. Material in parentheses will probably be omitted from class (but discussed in problem sets). "-" indicates we only discuss parts of those sections not involving integrals.

Date Material Textbook Announcements
09/08 Brief overview. Vectors in Rn. Coordinate systems. §12.1, 12.2, pp. 1001, 1005-1006 Welcome to V1201.
09/10 Dot product. Correlation. §12.3, +
09/15

Cross product.

§12.4

Problem set 1 due.
09/17 Equations for lines and planes (parametric, implicit). §12.5
09/22

More on lines and planes in space.

§12.5

Problem set 2 due.
09/24

Some more surfaces in space.

§12.6, +

09/29 Review.   Problem set 3 due.
10/01 Midterm 1.
10/06

Parametric curves.

§13.1, 13.2

10/08

Velocity, acceleration.

§13.4

10/13

Complex numbers. The complex exponential function.

Appendix H

Problem set 4 due.

10/15

More complex numbers. Some differential equations.

§17.1

10/20

Review of continuity and limits in one dimension.

§2.2, 2.4, 2.5

Problem set 5 due.
10/22

Continuity and limits of functions of several variables.

§14.1, 14.2

10/27

Partial derivatives.

§14.3

Problem set 6 due.
10/29

Tangent planes and linear approximation. The chain rule, version 1.

§14.4, 14.5

11/03

Election day: go vote!

11/05

Directional derivatives. More linear approximation. Gradient.

§14.6 Problem set 7 due.
11/10

Review.

Problem set 8 due.
11/12 Midterm 2.
11/17 Critical points and maximization. §14.7
11/19 Lagrange multipliers. §14.8
11/24

More Lagrange multipliers.

§14.8

Problem set 9 due.
11/26

No class: be thankful.

12/1 Linear transformations and matrices. +
12/3 Linear approximation. The Jacobian of a map Rn->Rm. + Problem set 9 part b due.
12/8 Composition of linear transformations, matrix multiplication, the chain rule. §14.5, + Problem set 10 due.
12/10 Review.

## Problem sets.

• Problem set 1 (PDF). Due September 15.
• Solutions have been posted in Courseworks, in the Class Files section.
• Problem set 2 (PDF). Due September 22.
• Solutions have been posted in Courseworks, in the Class Files section. An alternative solution to the proof problem (12.4.46) is also posted there.
• Problem set 3 (PDF). Due September 29.
• Solutions have been posted on Courseworks, in the Class Files section.
• Problem set 4 (PDF). Due October 13.
• Solutions have been posted on Courseworks, in the Class Files section.
• Problem set 5 (PDF). Due October 20.
• Problem 2 corrected in this version.
• Solutions have been posted on Courseworks, in the Class Files section.
• Solutions corrected (several mistakes in original).
• Problem set 6 (PDF). Due October 27.
• Revised: a few problems deleted (but will appear next week).
• Solutions have been posted on Courseworks, in the Class Files section.
• Problem set 7 (PDF). Due November 5.
• Solutions have been posted on Courseworks, in the Class Files section.
• Problem set 8 (PDF). Due November 10.
• Solutions have been posted on Courseworks, in the Class Files section.
• Problem set 9 (PDF). Due November 24.
• Revised: some problems now due December 3.
• Solutions to part 1 posted on Courseworks, in the Class Files section.
• Problem set 10 (PDF). Due December 8.
• Solutions have been posted on Courseworks, in the Class Files section.

## Handouts.

• Poicies handout (PDF). All of the information is contained on this web page.
• Week 1 Review Sheet (PDF). Summary of most important material covered in week 1.
• Printable version of Mathematica "quadric surfaces" worksheet used in class on 9/24. The Mathematica worksheet itself, for use with Mathematica, is here. The "flooding" (traces) movie is here. (Warning: the printable version and traces movie are both several megabytes.) (The movie was made with PoVRay.)
• Here's the Sage worksheet about Newton's method and the Newton fractal I used on 10/15.
• Linearization Handout (PDF). Notes on which the last three lectures are based.
• (This is the 5th revision since it was first posted.)