# Math 32A: General Course Outline

## Course Description

**32A. Calculus of Several Variables. (4) **Lecture, three hours; discussion, one hour. Requisite: course 31A with a grade of C- or better. Introduction to differential calculus of several variables, vector field theory. 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.

## Textbook(s)

J. Rogawski, *Multivariable Calculus*, (4th Edition), W.H. Freeman & CO.

1) Some problems may refer to polar coordinates. One only need inform the students that x = r cos q and y = r sin q. Polar coordinates are done in detail in 32B in order to help with areas, double integrals, etc.

2) The first two of Kepler's Laws should be done if at all possible.

3) There are two lectures on limits and continuity, in order to introduce the concepts of open, closed sets, etc.

Outline update: R. Brown, 9/14

## Schedule of Lectures

Lecture | Section | Topics |
---|---|---|

1 |
13.1 |
Vectors in the Plane |

2 |
13.2 |
Vectors in Three Dimensions |

3 |
13.3 |
Dot Product |

4 |
13.4 |
Cross Product |

5 |
13.5 |
Planes in Three-Space |

6 |
12.1 |
Parametric equations |

7 |
14.1 |
Vector-Valued Functions |

8 |
14.2 |
Calculus of Vector-Valued Functions |

9 |
14.3,4 |
Arc-Length and Speed; Curvature |

10 |
14.5,6 |
Motion in Three-Space; Planetary Motion |

11 |
15.1 |
Functions of Two or More Variables |

12 |
13.6 |
A Survey of Quadric Surfaces |

13 |
15.2 |
Limits and Continuity |

14 |
15.2 |
Limits and Continuity |

15 |
15.3 |
Partial Derivatives |

16 |
15.3 |
Partial Derivatives |

17 |
15.4 |
Differentiability and Tangent Planes |

18 |
15.4 |
Differentiability and Tangent Planes |

19 |
15.5 |
Gradient and Directional Derivatives |

20 |
15.5 |
Gradient and Directional Derivatives |

21 |
15.6 |
Chain Rule |

22 |
15.6 |
Chain Rule |

23 |
15.7 |
Optimization |

24 |
15.7 |
Optimization |

25 |
15.8 |
Lagrange Multipliers |

26 |
15.8 |
Lagrange Multipliers |