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MATH 100 Spring 2006-07

Introduction to Multivariable Calculus

Lecture Notes
Dr. Tony Yee
Department of Mathematics The Hong Kong University of Science and Technology
January 27, 2007

Contents
Table of Contents iii
1 Vectors and Geometry of Space 1
1.1 Three-DimensionalCoordinateSystems ......................... 1

1.2 Vectors ........................................... 5

1.3 TheDotProduct ...................................... 10

1.4 TheCrossProduct ..................................... 13

1.5 EquationsofLines ..................................... 18

1.6 EquationsofPlanes .................................... 22

1.7 QuadricSurfaces ...................................... 29

Chapter 1
Vectors and Geometry of Space

To apply calculus in many real-world situations and in higher mathematics, we need a mathematical de-scription of three-dimensional space. In this beginning chapter we introduce three-dimensional coordinate systems and vectors. Building on what we already know about coordinates in the xy-plane, we establish coordinates in space by adding a third axis that measures distance above and below the xy-plane. Vectors are used to study the analytic geometry of space, where they give simple ways to describe lines, planes, surfaces, and curves in space. We use these geometric ideas later to study motion in space and the calculus of several variables, with their many important applications in science, engineering, and higher mathematics.
1.1 Three-Dimensional Coordinate Systems
In this section we will have a fairly short discussion introducing the three-dimensional coordinate system and conventions that we will be using. We will also take a brief look at how the di.erent coordinate systems can change the graph of an equation. Remark that the 3-dimensional coordinate system is often denoted by R3 . Likewise the 2-dimensional coordinate system is often denoted by R2 and the 1-dimensional by R.
R3
To locate a point in , we use three mutually perpendicular coordinate axes, arranged as in the following .gure. The axes shown there make a right-handed coordinate frame. When you hold your right hand so that the .ngers curl from the positive x-axis toward the positive y-axis, your thumb points along the positive z-axis.
z
. R = (0, y, z) S =(x, 0,z)
.
. P =(x, y, z)
y
Q =(x, y, 0) .
x
Let us look at the basic coordinate system. It is assumed that only the positive directions are shown by the axes. If we need the negative axis for any reason we will put them in as needed.
1. Vectors and Geometry of Space
Also note the various points in the .gure. The point P of the Cartesian coordinates (x, y, z) is the general point sitting out in 3-dimensional space R3 . If we start at P and drop straight down until we reach a z-coordinate of zero we arrive that the point Q. We say that Q sits in the xy-plane. The xy-plane corresponds to all the points which have a zero z-coordinate. We can also start