Multivariable Calculus:



Collaborative Learning Workbook

David B. Damiano
Margaret N. Freije















Copyright © by Brooks/Cole Publishing Company

Contents

Maple for Multivariable Calculus: A Summary of Useful Commands

xvii

Chapter 1: Euclidean Space and Vectors

Discussion 1: Functions of Several Variables

Discussion 2: The Coordinate System in ${\bf R}^{3}$

Discussion 3: Using the Language of Vectors

Discussion 4: The Geometry of Lines and Planes

Discussion 5: Introduction to Optics

Chapter 2: Parametric Curves and Vector Fields

Discussion 6: Parametric Curves

Discussion 7: Plotting Parametric Curves Using Maple

Discussion 8: Further Exploration of Parametric Curves

Discussion 9: The Derivative of a Parametrization

Discussion 10: Applications of Parametric Curves

Discussion 11: Vector Fields

Discussion 12: Critical Points of Vector Fields

Discussion 13: Vector Fields as Models

Discussion 13A: Physical Models

Discussion 13B: Models of Interacting Populations

Discussion 13C: Epidemic Models

Chapter 3: Differentiation of Real-Valued Functions

Discussion 14: Contour Plots

Discussion 15: Plotting Functions of Two Variables in Maple

Discussion 16: Rates of Change

Discussion 17: Local Linearity of Functions of Two Variables

Discussion 18: Solving Systems of Equations

Chapter 4: Critical Points and Optimization

Discussion 19: Critical Points

Discussion 20: Using Maple to Find Critical Points

Discussion 21: The Second Derivative Test

Discussion 22: Constrained Optimization

Discussion 23: Lagrange Multipliers

Chapter 5: Integration

Discussion 24: Total Accumulation

Discussion 25: Iterated Integrals in the Plane

Discussion 26: Integration in Polar Coordinates

Discussion 27: Iterated Integrals in Space

Chapter 6: Integration on Curves

Discussion 28: Riemann Sums on Paths in Space

Discussion 29: Path Independence

Discussion 30: Green's Theorem

startsection section10mm-.5 Chapter 1: Euclidean Space and Vectors 

This chapter begins by developing the Euclidean coordinate system in the plane and in space. The emphasis is on using the formal language of mathematics, set notation, to describe subsets of the plane and space as well as giving prose descriptions of these subsets. Sections 1.2 and 1.3 introduce vectors and vector algebra. In these sections the language of vectors is used as another way to describe subsets of the plane and space and to express other geometric concepts. Both vectors themselves and the vector operations are discussed symbolically and graphically. Thus a vector can be represented by an ordered pair or triple, or by an arrow in the plane or in space. The dot product is defined symbolically by the standard formula and then interpreted geometrically in terms of the angle between two vectors. Finally in Section 1.4, we use the language of Euclidean coordinates and vectors to describe lines and planes in space. This chapter also introduces several examples of the types of problems from the sciences which will be used in the text. The emphasis is on using mathematical language to describe these different phenomena.

The first five discussionss in this chapter are in-class discussions and do not require access to the computer. Students should be able to complete most of the work on each of these discussions in one class period, although they may need time outside of class to complete a write-up of their results. Discussion 6 is an extended assignment which explores an application of the language of vectors to optics. This discussion can be assigned as an out-of-class collaborative exercise or it can be begun in class and then be completed outside of class.

The following is a daily schedule for Chapter 1 incorporating discussions, lectures and reading assignments.

Day 1

Discussion 1: Learning Mathematics or Discussion 2: Functions of Several Variables in class. Read Section 1.1: Euclidean Coordinates for homework.

Day 2

A brief lecture on Section 1.1 emphasizing set notation, followed by Discussion 3: The Coordinate System in ${\bf R}^{3}$ in class.

Day 3

A lecture on Section 1.2: Vectors emphasizing the correpondence between points and vectors. Read Section 1.2 for homework.

Day 4

Discussion 4: Using the Language of Vectors in class, followed by a summary of students' results. Read Section 1.3: Vector Products for homework.

Day 5

A lecture on Section 1.3.

Day 6

Finish Section 1.3, lecture on lines in the plane and in space.

Day 7

Discussion 5: Planes in class. Read Section 1.4: Lines and Planes for homework.

Day 8

Finish chapter with lecture on planes in space. Assign Discussion 6: Introduction to Optics as an extended exercise.

A brief description of each discussion including a statement of the goals of the discussion, a synopsis of the discussion, suggested reading assignments and an indication of which questions or models will be used again in subsequent discussions is given below.