# Syllabus

## Multivariable Calculus (Math23B) Fall 1999, UCSC

Lecture Sections
T TH 10-11:45 am Stevenson Acad 175 11P: M W 2-3:50 pm Applied Sci 295.
12P: M W 4-5:50 pm Applied Sci 295.

Instructor
Check the course web site periodically for revised versions of this syllabus, and to obtain homework assignments and copies of old tests and quizzes.

Teaching Assistant
Kirk Lackey

Course Description
Continuation of Calculus 23A. We will study integrals of scalar and vector valued functions of several variables, and their applications to a range of problems in geometry, physics, and engineering.

Our goal is to develop an intuitive understanding of the integrals, as well as the formal skills required to set up and evaluate them, within the context of a variety of applications. The main learning objectives of the course are as follows:

• Using double and triple integrals to set up formulas for evaluating volume, total mass, and center of mass of solid objects.
• Being able to evaluate double and triple integrals using a variety techniques including change of variables to cylindrical and spherical coordinates.
• Using path integrals to compute length of curves and total mass of thin wires; line integrals to compute total work done by moving an object in a vector field; surface integrals of scalar functions to compute the area of surfaces and total mass of thin shells; and surface integrals of vector valued functions to compute the flux of vector fields.
• To understand theorems of Green, Stokes, and Gauss, and being able to use them to evaluate line and surface integrals. Further, to apply these concepts to study the behavior of vector fields, such as those caused by forces of gravity, electromagnetism, or moving fluids.

Prerequisites
Calculus 23A--familiarity with basics of vector geometry (inner product and cross product), vector valued functions (divergence and curl), and basic rules for computing single integrals.

Textbook
The required text is Vector Calculus, by Marsden and Tromba, Fourth Edition. There is also an accompanying manual, Study Guide For Vector Calculus, by Pao and Soon, which you might find helpful.

Homework and Quizzes
There will be weekly homework assignments to be turned in every Wednesday to the teaching assistant. Late homeworks will not be accepted.

Also, there will be weekly quizzes every Monday. No calculators, textbooks, or notes will be allowed during quizzes. You should save copies of these quizzes as they are a very good source for preparing for the final and midterms.

Doing the homework problems is the most important part of this class. You may work with a group of your classmates if you are all at about the same level; however, you should definitely try to do many problems on your own. Further, try to practice doing at least some of the problems in settings which resemble that of the test and quizzes, i.e., without using your calculator or constantly referring to the textbook.

You should try to read the sections, or at least glance at them, before coming to class.
 Dates Lectures Sep 23 TH 5.1, 5.2 Intro to double integrals: Cavalieri's Principle, and Fubini's theorem 2830 TTH 5.35.4 Double integrals over general regions Changing the order of integration, The mean value theorem Oct 57 TTH 5.66.1 Triple integralsGeometry of mappings from plane to plane 1214 TTH 6.26.3 Change of variables theorem, the JacobianApplications: average values, and center of mass 1921 TTH 7.17.2 Path integralsLine integrals, total work 2528 TTH 7.37.4 Parametrized surfacesArea of Surfaces, Review Nov 24 TTH .7.5 Midterm(covers up to 7.2)Surface integrals of scalar functions 911 TTH 7.6 8.1 Surface integrals of vector functionsGreen's theorem 1618 TTH 8.28.3 Stokes theoremConservative vector fields 2325 TTH 8.4. Gauss's theoremThanksgiving (No Class) 30 T 8.5 Some more applications Dec 2 TH . Review 7 T . Final Exam(Comprehensive)

Assignments
You should plan to work on these problems over a period of several days. Getting a head start on each assignment is perhaps the most critical factor determining your success in this class.
 Homework # Due Date Problems 1 Sep 29 5.1) 1 a c, 2 a c, 4, 5, 6, 9; 5.2) 1 c, 2 c, 3, 5, 6. 2 Oct 6 5.3) 1 c, 2 a, 3, 5, 8, 11, 12; 5.4) 1 c, 2 b, 5, 7, 9. 3 Oct 13 5.6) 4, 5, 10, 12, 19, 25; 6.1) 1, 3, 7, 8. 4 Oct 20 6.2) 1, 3, 9, 13, 21, 29; 6.3) 2, 4, 9, 10, 17. 5 Oct 27 7.1) 2c, 4, 9, 12; 7.2) 1b, 2a, 3, 4, 17, 18. 6 Nov 3 7.3) 2, 5, 11, 12; 7.4) 3, 4, 6, 7, 11. 7 Nov 10 7.4 14, 16; 7.5) 2, 4, 9, 10, 11, 15. 8 Nov 17 7.6)2, 3, 4, 15, 16; 8.1) 2, 4, 12, 13, 20. 9 Nov 24 8.2) 2, 3, 8, 17, 19; 8.3) 4, 5, 7, 10, 15a. 10 Dec 1 8.4) 1, 3, 7, 8, 9, 13, 14, 18.

Tests and Exams
There will be one midterm, on Tuesday, November 2. The final exam will be on Tuesday, Dec 7, 8-11 am. No calculators, or textbooks will be allowed during the exams; however, for the midterm and the final, you may bring a single sheet of paper bearing formulas. .