# Math 32B: General Course Outline

## Course Description

**32B. Calculus of Several Variables. (4) **Lecture, three hours; discussion, one hour. Requisite: courses 31B & 32A with a grade of C- or better. Introduction to integral calculus of several variables, line and surface integrals. 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) The section on polar coordinates should be used to emphasize areas inside polar curves, as a preview of polar double integrals and cylindrical coordinates, and not arcane polar coordinate curves.

2) The sections on Green's Theorem, Stokes' Theorem, and the Divergence Theorem are extremely important. Time must be left to cover these sections in detail.

Outline update: R. Brown, 8/12

## Schedule of Lectures

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

1 |
16.1 |
Integration in two Variables |

2 |
16.1 |
Integration in two Variables |

3 |
16.2 |
More General Regions |

4 |
16.3 |
Triple Integrals |

5 |
12.3 |
Polar Coordinates |

6 |
16.4 |
Integration in Polar Coordinates |

7 |
16.4 |
Integration in Polar Coordinates |

8 |
16.5 |
Applications of Multiple Integrals |

9 |
16.6 |
Change of Variables |

10 |
16.6 |
Change of Variables |

11 |
17.1 |
Vector Fields |

12 |
17.1 |
Vector Fields |

13 |
17.2 |
Line Integrals |

14 |
17.2 |
Line Integrals |

15 |
17.3 |
Conservative Vector Fields |

16 |
17.3 |
Conservative Vector Fields |

17 |
17.4 |
Parametrized Surface |

18 |
17.4 |
Parametrized Surface |

19 |
17.5 |
Surface Integrals |

20 |
18.1 |
Green's Theorem |

21 |
18.1 |
Green's Theorem |

22 |
18.2 |
Strokes' Theorem |

23 |
18.2 |
Stokes' Theorem |

24 |
18.2,3 |
Stokes' Theorem, Divergence Theorem |

25 |
18.3 |
The Divergence Theorem |

26 |
18.3 |
The Divergence Theorem |