# Math 32AH: General Course Outline

## Course Description

32A. Calculus of Several Variables (Honors). (4) Lecture, three hours; discussion, one hour. Enforced requisite for course 32AH: course 31A with grade of B or better. Honors sequence parallel to courses 32A. 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

1.1

Notation, Functions, Vectors

2

1.1

Dot and Cross Products

3

1.2

Open and closed subsets of Rn

4

1.3

Limits of functions, Continuity

5

2.1

Derivative in one variable as linear approximation

6

2.1

Vector valued functions and their derivative

7

2.2

Differentiating functions Rn R: partial derivates

8

2.2

Partial derivatives, continuously differentiable functions

9

A.2

Matrices as linear transformations

10

2.10

Derivatives Rn Rm; Jacobian

11

2.3/2.10

Chain rule

12

2.6

Higher differentials, Schwarz Lemma

13

2.7

Taylor polynomial

14

1.4

Limits of sequences

15

1.5

Suprema, Completeness, Bolzano-Weierstrass

16

1.6

Compactness, Extreme value Theorem

17

2.1

Mean Value Theorem

18

A.8

Crashcourse Eigenvalues

19

2.8

Critical points, Hessian

20

2.9

Lagrange Multipliers

21

A.4/A.7

Invertible Matrices and Determinants

22

3.4

Inverse function Theorem

23

3.1

Implicit function Theorem

24

3.2

Curves

25

3.3

Surfaces

26

3.3

Immersions, Submersions, Submanifolds