# Math 12 (Fall 2011): Calculus+

Quick links on this page: News | Introduction | Class coordinates | Instructor information | Exam Schedule | Grading | Homework Policy | Textbook | Course Description | Weekly Schedule

Quick links to other Math 12 webpages: Homework Assignments | Class Notes | Exam review page

## News

• 12/1/2011: Office hours for the exam period are as follows: Monday 1-4pm, Tuesday 1-4pm, Wednesday 1-3pm.
• 11/28/2011: The final exam location has been updated - it will take place in Kemeny 007.
• 10/14/2011: An exam review section of the page is available, and contains links to lots of practice problems. There will soon be a review sheet there as well.
• 10/12/2011: Midterm times are now listed under the exam section. The exam will cover everything through chapter 14.6 (gradient vectors), excluding 14.5, on the chain rule.
• 9/29/2011: Tutorial room changed to 312 Silsby, effective immediately!
• 9/20/2011: First version of this webpage goes up. If you are taking this class, please read this webpage carefully, especially the sections on grading and homework.

## Introduction

Math 12 is a calculus class intended for incoming first-year students who have taken the equivalent of Calculus BC and scored a 5 on the AP exam. We cover the standard curriculum of multivariable calculus; doing this in a trimester versus the usual semester at most universities means the pace of this class will be rather quick.

Math 11 and math 12 more or less cover the same material, but as math 12 is intended for students who would like more of a challenge, certain topics will be covered more in-depth (for example, the intepretation of the derivative as a linear transformation) and homework assignments may be more difficult. All this being said, transferring between the two classes should not be too much of a problem.

## Class Coordinates

Room: Kemeny 007
Time: 1:45pm - 2:50pm, Monday, Wednesday, Friday
X-hour: Thursday, 1:00pm - 1:50pm. We may use the X-hour a few times for mandatory classes (if we fall behind schedule or I need to reschedule a class), and some of the X-hour classes will be optional, like review sessions for exams or talks about topics which are related but not essential parts of the course curriculum.

Tutorial room: 312 Silsby
Tutorial times: 7:00pm - 9:00pm, Thursday, Sunday
TA: Sarah Wolff

## Instructor information

Name: Andrew Yang
Office: Kemeny 316
Office Hours: Tuesday 2:30pm - 4:00pm, Thursday 1:00pm - 2:20pm.

## Examination Schedule

There will be two midterms and a final exam. All exams are closed book and no calculators or computational assistants of any kind.

• Midterm 1: Tuesday, October 18 2011, 5:00pm - 7:00pm, Rockefeller 001
• Midterm 2: Tuesday, November 15 2011, 5:00pm - 7:00pm, Rockefeller 001
• Final exam: December 4, 3pm-6pm, Kemeny 007.

If you are unable to be at any of these exams, please contact me as soon as possible so we can setup alternate test-taking arrangements.

Your grade in this class will be determined by homework and exams.

There will be two types of homework assignments: Webwork and written assignments. Webwork is an automated computer-based grading system where you get slightly randomized problems, and try to solve them until the computer tells you your answer is correct. We will generally use Webwork for easier questions which are more suited for computer automated grading. These will usually be given out three times a week, and shouldn't take too long to complete (not much more than 60 to 90 minutes, say.)

Written assignments will consist of questions which will usually be more difficult than Webwork assignments. You will need to write down solutions which justify your answers and turn them in.

There will be two midterm exams and a final examination. All exams will be at a specified location and will be closed book.

Each of the above contributes to your final grade in the following fashion:

• Homework, Webwork: 15%
• Homework, Written: 15%
• Midterms: 20% each (40% total)
• Final exam: 30%
Your final letter grade is computed using a curve. Exactly what the distribution of letter grades cannot be determined in advance, although it should not stray too far from the way grades were distributed in previous terms.

## Homework Policy

Written homework assignments will be posted on this website and will be usually due about a week after they are posted. Late assignments will only be accepted when granted an extension, which must be requested from the instructor several days in advance. In general, extensions will only be granted for health-related reasons or family emergencies. Exceptions may be made for school-related travel.

The homework collaboration policy for this class is more or less in line with other Dartmouth math classes. You are allowed to collaborate with others on homework, but must write your own solutions. A good rule of thumb is that you should never be copying phrases or sentences from anyone else or any source. You may use theorems, lemmas, etc. that we have covered from the textbook, but in general you should not use theorems, lemmas, etc. from sections of the book we have not covered or from external sources. Also, please write down the people you collaborated with and outside sources (namely, anything besides the required textbook) you consulted on your homework assignments.

## Textbook

The required book for this class is Calculus, 7th edition, by James Stewart, ISBN 978-0538497817.

There are many books about calculus. The following are a few books which might be worth consulting:

• Div, grad, curl, and all that, 4th Edition, by H.M. Schey, ISBN 978-0393925166.
This book takes a more informal and physics-inspired approach to the material in the second half of the class. It is well-written, contains many exercises, and is not particularly long, so this book might be a useful supplement to the main text.
• Calculus, Volume II, 2nd edition, by Tom Apostol, ISBN 978-0471000075.
This book contains a lot of the material we will be covering in this class, in a slightly more abstract way. This book also contains linear algebra, and so is able to present multivariable calculus in a more generalized and uniform setting.
• Calculus on Manifolds, by Michael Spivak, ISBN 978-0805390216.
This book is for more ambitious students, who are interested in understanding the material in the second half of the class in a more general context. Students who intend to major in mathematics or theoretical physics will be the ones who find this book relevant. If you choose to read this book you will probably need outside assistance to do so.

## Course description

Our primary goal in this class is to cover chapter 13 through 16 of the textbook. We begin by quicking thinking about lines and planes in three dimensional space. We then study differential calculus of functions of several variables, which includes topics like vector-valued functions, partial derivatives, directional derivatives, tangent planes, the second derivative test, and Lagrange multipliers. After that we study integral calculus of functions of several variables, which includes topics like multiple integration and coordinate changes. Finally, we apply everything learned earlier to the study of line and surface integrals, and conclude with the three great theorems of the subject, Green's Theorem, the Divergence Theorem, and Stokes' Theorem.

## Weekly schedule

This schedule is preliminary and will almost certainly be adjusted over the course of the term.

Week 1: Geometry in $$\mathbb{R}^{3}$$

• Vectors
• Dot product
• Reading: Chapter 12.1, 12.2, 12.3

Week 2: More geometry in $$\mathbb{R}^{3}$$, vector-valued functions

• Cross product
• Lines and planes in $$\mathbb{R}^{3}$$
• Vector-valued functions and their derivatives and integrals
• Reading: Chapter 12.4, 12.5, 13.1, 13.2

Week 3: Vector-valued functions, derivatives of multivariable functions

• Arc length
• Limits of multivariable functions
• Partial derivatives
• Reading: Chapter 13.3, 14.1, 14.2, 14.3

Week 4: Applications of partial derivatives

• Tangent planes
• Directional derivatives
• The second derivative test
• Reading: Chapter 14.4, 14.6, 14.7

Week 5: Lagrange multipliers, the chain rule

• The method of Lagrange multipliers
• Derivatives as linear transformations
• The chain rule

Week 6: Double integration

• Double integrals over rectangular regions
• Double integrals over arbitrary regions
• Double integration using polar coordinates
• Reading: Chapter 15.1, 15.2, 15.3, 15.4

Week 7: Triple integration

• Triple integrals over arbitrary regions
• Spherical and cylindrical coordinates
• The Jacobian: arbitrary coordinate changes
• Reading: Chapter 15.7, 15.8, 15.9, 15.10

Week 8: Line integrals

• Vector fields
• Line integration
• Fundamental theorem of line integrals
• Reading: Chapter 16.1, 16.2, 16.3

Week 9: Green's Theorem, curl and divergence, parameteric surfaces

• Green's Theorem
• Curl and divergence
• Parametric surfaces
• Reading: Chapter 16.4, 16.5, 16.6

Week 10/11: Surface integrals

• Surface integration
• The Divergence Theorem
• Stokes' Theorem
• Chapter 16.7, 16.8, 16.9