# Math 22b, Spring 2006

Course Description: This course covers multivariable calculus, up through the classic theorems of vector calculus: Stokes', Green's and Gauss's theorems. We will use the book Vector Calculus (Fifth Edition) by Marsden and Tromba, and cover most of Chapters 2-8.

Grading: There will be one final (40%), two midterms (20% each), quizzes and homework.

Office Hours: For the first two weeks, office hours will be Tuesday 11-12, Wednesday 2-3, and Thursday 11-12, and by appointment.

Homework Assignments as well as review sheets for exams can be found on WebCT.

Contact information: The easiest way to reach me is via email: ruberman @brandeis.edu. My office is 310 Goldsmith, and my phone is x3074.

First Midterm Exam: In class on Thursday March 13. The exam will cover through section 6.4.
Quiz 1: In class on Wednesday March 26, covering through section 7.1.

Final Exam: Monday, May 8, time and room TBA.

Course Outline: We will cover most of the topics below; the ones in parentheses will be covered depending on the pace of the class.

2. DIFFERENTIATION
2.1 The Geometry of Real-Valued Functions
2.2 Limits and Continuity
2.3 Differentiation
2.4 Introduction to Paths
2.5 Properties of the Derivative

3. HIGHER-ORDER DERIVATIVES: MAXIMA AND MINIMA
3.1 Iterated Partial Derivatives
3.2 Taylor's Theorem
3.3 Extrema of Real-Valued Functions
3.4 Constrained Extrema and Lagrange Multipliers
(3.5 The Implicit Function Theorem)

4. VECTOR-VALUED FUNCTIONS
4.1 Acceleration and Newton's Second Law
4.2 Arc Length
4.3 Vector Fields
4.4 Divergence and Curl

5. DOUBLE AND TRIPLE INTEGRALS
5.1 Introduction
5.2 The Double Integral Over a Rectangle
5.3 The Double Integral Over More General Regions
5.4 Changing the Order of Integration
5.5 The Triple Integral

6. THE CHANGE OF VARIABLES FORMULA AND APPLICATIONS OF INTEGRATION
6.1 The Geometry of Maps from R2 to R2
6.2 The Change of Variables Theorem
6.3 Applications of Double and Triple Integrals
(6.4 Improper Integrals)

7. INTEGRALS OVER PATHS AND SURFACES
7.1 The Path Integral
7.2 Line Integrals
7.3 Parametrized Surfaces
7.4 Area of a Surface
7.5 Integrals of Scalar Functions Over Surfaces
7.6 Surface Integrals of Vector Functions

8. THE INTEGRAL THEOREMS OF VECTOR ANALYSIS
8.1 Green's Theorem
8.2 Stokes' Theorem
8.3 Conservative Fields
8.4 Gauss' Theorem
(8.5 Applications to Physics, Engineering, and Differential Equations)

### General Course Policies

You may discuss the homework problems with other students in the class; however, if you do, you should write on your homework submission the students with whom you discussed the assignment. (You do not need to mention any help you received from the TA's or instructor.) You may not copy the written work of another student or allow another student to copy your written work. What you submit should be your own work.

If you are a student with a documented disability on record at Brandeis University and wish to have a reasonable accommodation made for you in this class, please see me immediately. Please keep in mind that reasonable accomodations are not provided retroactively.

You are expected to be honest in all of your academic work. The University policy on academic honesty is distributed annually as section 5 of the Rights and Responsibilities Handbook. Instances of alleged dishonesty will be forwarded to the Office of Campus Life for possible referral to the Student Judicial System. Potential sanctions include failure in the course and suspension from the University. If you have any questions about my expectations, please ask.

Last updated: January 18, 2006