University of Illinois at Urbana-Champaign

Math 242 (Calculus of Several Variables) Fall 2002

Course Description
This course is an introduction to multi-variable calculus. In previous courses, you have learned how to apply the techniques of calculus to functions of a single real variable. However, virtually all "real-world" applications of calculus which you will encounter involve more than one variable. In this course, you will study differential and integral calculus for functions of several variables.

We will begin with a study of vectors and three-dimensional geometry. This foundational material will give us language and notation to describe functions of several variables. We will then study how to formulate calculus for such functions. Differential calculus involves the notion of partial derivatives. We will learn how to solve maxima and minima problems in this setting using Lagrange multipliers. Integral calculus involves multiple integrals over different regions in the plane and in space. We will learn how to set up and evaluate such integrals.

The culmination of our study is (or rather, should be) a version of the Fundamental Theorem of Calculus relating these two notions: partial differentiation and multiple integration. I say ``should be'' because time constraints make it impossible for us to thoroughly treat this important topic. However, I will definitely allow time during the final week of class to discuss results such as Green's Theorem and Stokes' Theorem, which are multi-dimensional versions of the Fundamental Theorem of Calculus.

The prerequisite for this course is the Math 120-130 sequence or equivalent (single-variable differential and integral calculus).

  • Tutoring Room: this room is staffed by advanced students who are available to answer questions about the course material. The tutoring room for this semester is 447 Altgeld Hall (Mon-Thu 7-9pm). If you feel that the assistance you receive is in any way unsatisfactory, let me know the date and time as well as the problem you encountered.

    Schedule of Topics

    (Note: This schedule is tentative and may be revised at a later date.)

    Week of Topics
    Aug. 26 12.1: Three-Dimensional Coordinate Systems
    Sept. 2 12.2: Vectors
    12.3: The Dot Product
    12.4: The Cross Product
    Sept. 9 12.5: Equations of Lines and Planes
    12.6: Cylinders and Quadric Surfaces
    12.7: Cylindrical and Spherical Coordinates
    Sept. 16 13.1: Vector Functions and Space Curves
    13.2: Derivatives and Integrals of Vector Functions
    13.3: Arc Length and Curvature
    Sept. 23
    First midterm exam on 9/24
    covering 12.1-12.7     		13.3
    Sept. 30 13.4: Motion in Space
    14.1: Functions of Several Variables
    Oct. 7 14.2: Limits and Continuity
    14.3: Partial Derivatives
    Oct. 14 14.4: Tangent Planes and Linear Approximations
    14.5: The Chain Rule
    Oct. 21
    Second midterm exam on
    10/24 covering 13.1-14.5     		14.6: Directional Derivatives and the Gradient Vector
    14.7: Maxima and Minima
    Oct. 28 14.7
    14.8: Lagrange Multipliers
    Nov. 4 15.1: Double Integrals over Rectangles
    15.2: Iterated Integrals
    15.3: Double Integrals over General Regions
    Nov. 11 15.4: Double Integrals in Polar Coordinates
    15.5: Applications of Integration
    Nov. 18
    Third midterm exam on
    11/29 covering 14.6-15.3     		15.5
    15.6: Surface Area
    Dec. 2 15.7: Triple Integrals
    15.8: Triple Integrals in Cylindrical and Spherical Coordinates
    Dec. 9 15.9: Change of variables in Multiple Integrals
    16:

    Final Exam on 12/17 in 161 Noyes Lab from 8:00-11:00. Cumulative