# Math 32BH: General Course Outline

## Catalog Description

32B. Calculus of Several Variables (Honors). (4) Lecture, three hours; discussion, one hour. Enforced requisite for 32BH: courses 31B and 32A, with grades of B or better. Honors sequence parallel to courses 32B. P/NP or letter grading.

## Course Information:

The following schedule, with textbook sections and topics, is based on 26 lectures. The remaining classroom meetings are for leeway, reviews, and two midterm exams. These are scheduled by the individual instructor.

Math 32AB is a traditional multivariable calculus course sequence for mathematicians, engineers, and physical scientists.

The course 32A treats topics related to differential calculus in several variables, including curves in the plane, curves and surfaces in space, various coordinate systems, partial differentiation, tangent planes to surfaces, and directional derivatives. The course culminates with the solution of optimization problems by the method of Lagrange multipliers.

The course 32B treats topics related to integration in several variables, culminating in the theorems of Green, Gauss and Stokes. Each of these theorems asserts that an integral over some domain is equal to an integral over the boundary of the domain. In the case of Green's theorem the domain is an area in the plane, in the case of Gauss's theorem the domain is a volume in three-dimensional space, and in the case of Stokes' theorem the domain is a surface in three-dimensional space. These theorems are generalizations of the fundamental theorem of calculus, which corresponds to the case where the domain is an interval on the real line. The theorems play an important role in electrostatics, fluid mechanics, and other areas in engineering and physics where conservative vector fields play a role.

## Schedule of Lectures

Lecture Section Topics

1

4.1

Integration on the Line

2

4.2

Integration in Higher Dimensions

3

4.2

Integration in Higher Dimensions

4

4.3

Multiple Integrals and Iterated Integrals

5

4.3

Multiple Integrals and Iterated Integrals

6

4.4

Change of Variables for Multiple Integrals

7

4.4

Change of Variables for Multiple Integrals

8

4.5

Functions Defined by Integrals

9

4.8

Lebesgue Measure and the Lebesgue Integral

10

5.1

Arc Length and Line Integrals

11

5.1

Arc Length and Line Integrals

12

5.2

Greens Theorem

13

5.3

Surface Area and Surface Integrals

14

5.4

Vector Derivatives

15

5.5

Divergence Theorem

16

5.6

Some Applications to Physics

17

5.6

Some Applications to Physics

18

5.6

Some Applications to Physics

19

5.7

Stokes Theorem

20

5.8

Integrating Vector Derivatives

21

5.8

Integrating Vector Derivatives

22

5.9

Higher Dimensions and Differential Forms

23

5.9

Higher Dimensions and Differential Forms

24

5.9

Higher Dimensions and Differential Forms

25

5.9

Higher Dimensions and Differential Forms

26

5.9

Higher Dimensions and Differential Forms